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Theorem mulsuniflem 28176

Description: Lemma for mulsunif 28177. State the theorem with some extra distinct variable conditions. (Contributed by Scott Fenton, 8-Mar-2025.)

**Hypotheses**
Ref	Expression
mulsuniflem.1	⊢ (𝜑 → 𝐿 <<s 𝑅)
mulsuniflem.2	⊢ (𝜑 → 𝑀 <<s 𝑆)
mulsuniflem.3	⊢ (𝜑 → 𝐴 = (𝐿 \|s 𝑅))
mulsuniflem.4	⊢ (𝜑 → 𝐵 = (𝑀 \|s 𝑆))

**Assertion**
Ref	Expression
mulsuniflem	⊢ (𝜑 → (𝐴 ·_s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))}) \|s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})))

Distinct variable groups: 𝐴,𝑎,𝑝,𝑞 𝐴,𝑏,𝑟,𝑠 𝐴,𝑐,𝑡,𝑢 𝐴,𝑑,𝑣,𝑤 𝐵,𝑎,𝑝,𝑞 𝐵,𝑏,𝑟,𝑠 𝐵,𝑐,𝑡,𝑢 𝐵,𝑑,𝑣,𝑤 𝐿,𝑎,𝑝,𝑞 𝐿,𝑐,𝑡,𝑢 𝑣,𝐿 𝐿,𝑟 𝑀,𝑎,𝑝,𝑞 𝑢,𝑀 𝑀,𝑑,𝑣,𝑤 𝑀,𝑠 𝑅,𝑝 𝑅,𝑏,𝑟,𝑠 𝑡,𝑅 𝑅,𝑑,𝑣,𝑤 𝑆,𝑏,𝑟,𝑠 𝑆,𝑐,𝑡,𝑢 𝑤,𝑆 𝑆,𝑞 𝜑,𝑎,𝑝,𝑞 𝜑,𝑏,𝑟,𝑠 𝜑,𝑐,𝑡,𝑢 𝜑,𝑑,𝑣,𝑤 Allowed substitution hints: 𝑅(𝑢,𝑞,𝑎,𝑐) 𝑆(𝑣,𝑝,𝑎,𝑑) 𝐿(𝑤,𝑠,𝑏,𝑑) 𝑀(𝑡,𝑟,𝑏,𝑐)

Proof of Theorem mulsuniflem Dummy variables 𝑒 𝑓 𝑔 ℎ 𝑖 𝑗 𝑘 𝑙 𝑚 𝑛 𝑥 𝑦 𝑧 𝑥_𝑂 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulsuniflem.3	. . . 4 ⊢ (𝜑 → 𝐴 = (𝐿 \|s 𝑅))
2		mulsuniflem.1	. . . . 5 ⊢ (𝜑 → 𝐿 <<s 𝑅)
3	2	scutcld 27849	. . . 4 ⊢ (𝜑 → (𝐿 \|s 𝑅) ∈ No )
4	1, 3	eqeltrd 2840	. . 3 ⊢ (𝜑 → 𝐴 ∈ No )
5		mulsuniflem.4	. . . 4 ⊢ (𝜑 → 𝐵 = (𝑀 \|s 𝑆))
6		mulsuniflem.2	. . . . 5 ⊢ (𝜑 → 𝑀 <<s 𝑆)
7	6	scutcld 27849	. . . 4 ⊢ (𝜑 → (𝑀 \|s 𝑆) ∈ No )
8	5, 7	eqeltrd 2840	. . 3 ⊢ (𝜑 → 𝐵 ∈ No )
9		mulsval 28136	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 ·_s 𝐵) = (({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗))}) \|s ({𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))} ∪ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))})))
10	4, 8, 9	syl2anc 584	. 2 ⊢ (𝜑 → (𝐴 ·_s 𝐵) = (({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗))}) \|s ({𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))} ∪ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))})))
11	4, 8	mulscut2 28160	. . 3 ⊢ (𝜑 → ({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗))}) <<s ({𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))} ∪ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))}))
12	2, 1	cofcutr1d 27960	. . . . . 6 ⊢ (𝜑 → ∀𝑓 ∈ ( L ‘𝐴)∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝)
13	6, 5	cofcutr1d 27960	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑔 ∈ ( L ‘𝐵)∃𝑞 ∈ 𝑀 𝑔 ≤s 𝑞)
14	13	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ ∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝)) → ∀𝑔 ∈ ( L ‘𝐵)∃𝑞 ∈ 𝑀 𝑔 ≤s 𝑞)
15		reeanv 3228	. . . . . . . . . . . . . . 15 ⊢ (∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞) ↔ (∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝 ∧ ∃𝑞 ∈ 𝑀 𝑔 ≤s 𝑞))
16		leftssno 27920	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ( L ‘𝐴) ⊆ No
17		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑓 ∈ ( L ‘𝐴))
18	16, 17	sselid 3980	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑓 ∈ No )
19	18	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → 𝑓 ∈ No )
20	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → 𝐵 ∈ No )
21	19, 20	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (𝑓 ·_s 𝐵) ∈ No )
22	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → 𝐴 ∈ No )
23		leftssno 27920	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ( L ‘𝐵) ⊆ No
24		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑔 ∈ ( L ‘𝐵))
25	23, 24	sselid 3980	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑔 ∈ No )
26	25	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → 𝑔 ∈ No )
27	22, 26	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (𝐴 ·_s 𝑔) ∈ No )
28	21, 27	addscld 28014	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → ((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) ∈ No )
29	19, 26	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (𝑓 ·_s 𝑔) ∈ No )
30	28, 29	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ∈ No )
31	30	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ∈ No )
32		ssltss1 27834	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐿 <<s 𝑅 → 𝐿 ⊆ No )
33	2, 32	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝐿 ⊆ No )
34	33	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝐿 ⊆ No )
35		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑝 ∈ 𝐿)
36	34, 35	sseldd 3983	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑝 ∈ No )
37	36	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → 𝑝 ∈ No )
38	37, 20	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (𝑝 ·_s 𝐵) ∈ No )
39	38, 27	addscld 28014	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → ((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) ∈ No )
40	37, 26	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (𝑝 ·_s 𝑔) ∈ No )
41	39, 40	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑝 ·_s 𝑔)) ∈ No )
42	41	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑝 ·_s 𝑔)) ∈ No )
43		ssltss1 27834	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑀 <<s 𝑆 → 𝑀 ⊆ No )
44	6, 43	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑀 ⊆ No )
45	44	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑀 ⊆ No )
46		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑞 ∈ 𝑀)
47	45, 46	sseldd 3983	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑞 ∈ No )
48	47	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → 𝑞 ∈ No )
49	22, 48	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (𝐴 ·_s 𝑞) ∈ No )
50	38, 49	addscld 28014	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → ((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) ∈ No )
51	37, 48	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (𝑝 ·_s 𝑞) ∈ No )
52	50, 51	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) ∈ No )
53	52	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) ∈ No )
54	18	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑓 ∈ No )
55	37	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑝 ∈ No )
56	25	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑔 ∈ No )
57	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝐵 ∈ No )
58		simprrl 780	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞))) → 𝑓 ≤s 𝑝)
59	58	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑓 ≤s 𝑝)
60	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝐵 ∈ No )
61		ssltleft 27910	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐵 ∈ No → ( L ‘𝐵) <<s {𝐵})
62	8, 61	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → ( L ‘𝐵) <<s {𝐵})
63	62	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → ( L ‘𝐵) <<s {𝐵})
64		snidg 4659	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐵 ∈ No → 𝐵 ∈ {𝐵})
65	8, 64	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝐵 ∈ {𝐵})
66	65	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝐵 ∈ {𝐵})
67	63, 24, 66	ssltsepcd 27840	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑔 <s 𝐵)
68	25, 60, 67	sltled 27815	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑔 ≤s 𝐵)
69	68	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑔 ≤s 𝐵)
70	54, 55, 56, 57, 59, 69	slemuld 28165	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → ((𝑓 ·_s 𝐵) -_s (𝑓 ·_s 𝑔)) ≤s ((𝑝 ·_s 𝐵) -_s (𝑝 ·_s 𝑔)))
71	21, 29	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → ((𝑓 ·_s 𝐵) -_s (𝑓 ·_s 𝑔)) ∈ No )
72	38, 40	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → ((𝑝 ·_s 𝐵) -_s (𝑝 ·_s 𝑔)) ∈ No )
73	71, 72, 27	sleadd1d 28029	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑓 ·_s 𝐵) -_s (𝑓 ·_s 𝑔)) ≤s ((𝑝 ·_s 𝐵) -_s (𝑝 ·_s 𝑔)) ↔ (((𝑓 ·_s 𝐵) -_s (𝑓 ·_s 𝑔)) +_s (𝐴 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) -_s (𝑝 ·_s 𝑔)) +_s (𝐴 ·_s 𝑔))))
74	73	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑓 ·_s 𝐵) -_s (𝑓 ·_s 𝑔)) ≤s ((𝑝 ·_s 𝐵) -_s (𝑝 ·_s 𝑔)) ↔ (((𝑓 ·_s 𝐵) -_s (𝑓 ·_s 𝑔)) +_s (𝐴 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) -_s (𝑝 ·_s 𝑔)) +_s (𝐴 ·_s 𝑔))))
75	70, 74	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑓 ·_s 𝐵) -_s (𝑓 ·_s 𝑔)) +_s (𝐴 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) -_s (𝑝 ·_s 𝑔)) +_s (𝐴 ·_s 𝑔)))
76	21, 27, 29	addsubsd 28113	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) = (((𝑓 ·_s 𝐵) -_s (𝑓 ·_s 𝑔)) +_s (𝐴 ·_s 𝑔)))
77	76	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) = (((𝑓 ·_s 𝐵) -_s (𝑓 ·_s 𝑔)) +_s (𝐴 ·_s 𝑔)))
78	38, 27, 40	addsubsd 28113	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑝 ·_s 𝑔)) = (((𝑝 ·_s 𝐵) -_s (𝑝 ·_s 𝑔)) +_s (𝐴 ·_s 𝑔)))
79	78	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑝 ·_s 𝑔)) = (((𝑝 ·_s 𝐵) -_s (𝑝 ·_s 𝑔)) +_s (𝐴 ·_s 𝑔)))
80	75, 77, 79	3brtr4d 5174	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑝 ·_s 𝑔)))
81	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝐴 ∈ No )
82	48	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑞 ∈ No )
83	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝐴 ∈ No )
84		scutcut 27847	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐿 <<s 𝑅 → ((𝐿 \|s 𝑅) ∈ No ∧ 𝐿 <<s {(𝐿 \|s 𝑅)} ∧ {(𝐿 \|s 𝑅)} <<s 𝑅))
85	2, 84	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → ((𝐿 \|s 𝑅) ∈ No ∧ 𝐿 <<s {(𝐿 \|s 𝑅)} ∧ {(𝐿 \|s 𝑅)} <<s 𝑅))
86	85	simp2d 1143	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝐿 <<s {(𝐿 \|s 𝑅)})
87	86	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝐿 <<s {(𝐿 \|s 𝑅)})
88		ovex 7465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐿 \|s 𝑅) ∈ V
89	88	snid 4661	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐿 \|s 𝑅) ∈ {(𝐿 \|s 𝑅)}
90	1, 89	eqeltrdi 2848	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝐴 ∈ {(𝐿 \|s 𝑅)})
91	90	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝐴 ∈ {(𝐿 \|s 𝑅)})
92	87, 35, 91	ssltsepcd 27840	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑝 <s 𝐴)
93	36, 83, 92	sltled 27815	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑝 ≤s 𝐴)
94	93	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → 𝑝 ≤s 𝐴)
95	94	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑝 ≤s 𝐴)
96		simprrr 781	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞))) → 𝑔 ≤s 𝑞)
97	96	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑔 ≤s 𝑞)
98	55, 81, 56, 82, 95, 97	slemuld 28165	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → ((𝑝 ·_s 𝑞) -_s (𝑝 ·_s 𝑔)) ≤s ((𝐴 ·_s 𝑞) -_s (𝐴 ·_s 𝑔)))
99	51, 49, 40, 27	slesubsub3bd 28119	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑝 ·_s 𝑞) -_s (𝑝 ·_s 𝑔)) ≤s ((𝐴 ·_s 𝑞) -_s (𝐴 ·_s 𝑔)) ↔ ((𝐴 ·_s 𝑔) -_s (𝑝 ·_s 𝑔)) ≤s ((𝐴 ·_s 𝑞) -_s (𝑝 ·_s 𝑞))))
100	27, 40	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → ((𝐴 ·_s 𝑔) -_s (𝑝 ·_s 𝑔)) ∈ No )
101	49, 51	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → ((𝐴 ·_s 𝑞) -_s (𝑝 ·_s 𝑞)) ∈ No )
102	100, 101, 38	sleadd2d 28030	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝐴 ·_s 𝑔) -_s (𝑝 ·_s 𝑔)) ≤s ((𝐴 ·_s 𝑞) -_s (𝑝 ·_s 𝑞)) ↔ ((𝑝 ·_s 𝐵) +_s ((𝐴 ·_s 𝑔) -_s (𝑝 ·_s 𝑔))) ≤s ((𝑝 ·_s 𝐵) +_s ((𝐴 ·_s 𝑞) -_s (𝑝 ·_s 𝑞)))))
103	99, 102	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑝 ·_s 𝑞) -_s (𝑝 ·_s 𝑔)) ≤s ((𝐴 ·_s 𝑞) -_s (𝐴 ·_s 𝑔)) ↔ ((𝑝 ·_s 𝐵) +_s ((𝐴 ·_s 𝑔) -_s (𝑝 ·_s 𝑔))) ≤s ((𝑝 ·_s 𝐵) +_s ((𝐴 ·_s 𝑞) -_s (𝑝 ·_s 𝑞)))))
104	103	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑝 ·_s 𝑞) -_s (𝑝 ·_s 𝑔)) ≤s ((𝐴 ·_s 𝑞) -_s (𝐴 ·_s 𝑔)) ↔ ((𝑝 ·_s 𝐵) +_s ((𝐴 ·_s 𝑔) -_s (𝑝 ·_s 𝑔))) ≤s ((𝑝 ·_s 𝐵) +_s ((𝐴 ·_s 𝑞) -_s (𝑝 ·_s 𝑞)))))
105	98, 104	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → ((𝑝 ·_s 𝐵) +_s ((𝐴 ·_s 𝑔) -_s (𝑝 ·_s 𝑔))) ≤s ((𝑝 ·_s 𝐵) +_s ((𝐴 ·_s 𝑞) -_s (𝑝 ·_s 𝑞))))
106	38, 27, 40	addsubsassd 28112	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑝 ·_s 𝑔)) = ((𝑝 ·_s 𝐵) +_s ((𝐴 ·_s 𝑔) -_s (𝑝 ·_s 𝑔))))
107	106	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑝 ·_s 𝑔)) = ((𝑝 ·_s 𝐵) +_s ((𝐴 ·_s 𝑔) -_s (𝑝 ·_s 𝑔))))
108	38, 49, 51	addsubsassd 28112	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) = ((𝑝 ·_s 𝐵) +_s ((𝐴 ·_s 𝑞) -_s (𝑝 ·_s 𝑞))))
109	108	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) = ((𝑝 ·_s 𝐵) +_s ((𝐴 ·_s 𝑞) -_s (𝑝 ·_s 𝑞))))
110	105, 107, 109	3brtr4d 5174	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑝 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)))
111	31, 42, 53, 80, 110	sletrd 27808	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)))
112	111	anassrs 467	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞))) → (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)))
113	112	expr 456	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → ((𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞) → (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))))
114	113	reximdvva 3206	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → (∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞) → ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))))
115	114	expcom 413	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) → (𝜑 → (∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞) → ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)))))
116	115	com23 86	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) → (∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞) → (𝜑 → ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)))))
117	116	imp 406	. . . . . . . . . . . . . . 15 ⊢ (((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)) → (𝜑 → ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))))
118	15, 117	sylan2br 595	. . . . . . . . . . . . . 14 ⊢ (((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝 ∧ ∃𝑞 ∈ 𝑀 𝑔 ≤s 𝑞)) → (𝜑 → ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))))
119	118	an4s 660	. . . . . . . . . . . . 13 ⊢ (((𝑓 ∈ ( L ‘𝐴) ∧ ∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝) ∧ (𝑔 ∈ ( L ‘𝐵) ∧ ∃𝑞 ∈ 𝑀 𝑔 ≤s 𝑞)) → (𝜑 → ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))))
120	119	impcom 407	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ ∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝) ∧ (𝑔 ∈ ( L ‘𝐵) ∧ ∃𝑞 ∈ 𝑀 𝑔 ≤s 𝑞))) → ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)))
121	120	anassrs 467	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ ∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝)) ∧ (𝑔 ∈ ( L ‘𝐵) ∧ ∃𝑞 ∈ 𝑀 𝑔 ≤s 𝑞)) → ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)))
122	121	expr 456	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ ∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝)) ∧ 𝑔 ∈ ( L ‘𝐵)) → (∃𝑞 ∈ 𝑀 𝑔 ≤s 𝑞 → ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))))
123	122	ralimdva 3166	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ ∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝)) → (∀𝑔 ∈ ( L ‘𝐵)∃𝑞 ∈ 𝑀 𝑔 ≤s 𝑞 → ∀𝑔 ∈ ( L ‘𝐵)∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))))
124	14, 123	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ ∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝)) → ∀𝑔 ∈ ( L ‘𝐵)∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)))
125	124	expr 456	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ ( L ‘𝐴)) → (∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝 → ∀𝑔 ∈ ( L ‘𝐵)∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))))
126	125	ralimdva 3166	. . . . . 6 ⊢ (𝜑 → (∀𝑓 ∈ ( L ‘𝐴)∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝 → ∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))))
127	12, 126	mpd 15	. . . . 5 ⊢ (𝜑 → ∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)))
128		eqeq1 2740	. . . . . . . . . 10 ⊢ (𝑎 = 𝑧 → (𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) ↔ 𝑧 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))))
129	128	2rexbidv 3221	. . . . . . . . 9 ⊢ (𝑎 = 𝑧 → (∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) ↔ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑧 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))))
130	129	rexab 3699	. . . . . . . 8 ⊢ (∃𝑧 ∈ {𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧 ↔ ∃𝑧(∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑧 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) ∧ (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧))
131		r19.41vv 3226	. . . . . . . . 9 ⊢ (∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (𝑧 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) ∧ (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧) ↔ (∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑧 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) ∧ (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧))
132	131	exbii 1847	. . . . . . . 8 ⊢ (∃𝑧∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (𝑧 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) ∧ (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧) ↔ ∃𝑧(∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑧 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) ∧ (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧))
133		rexcom4 3287	. . . . . . . . 9 ⊢ (∃𝑝 ∈ 𝐿 ∃𝑧∃𝑞 ∈ 𝑀 (𝑧 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) ∧ (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧) ↔ ∃𝑧∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (𝑧 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) ∧ (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧))
134		rexcom4 3287	. . . . . . . . . . 11 ⊢ (∃𝑞 ∈ 𝑀 ∃𝑧(𝑧 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) ∧ (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧) ↔ ∃𝑧∃𝑞 ∈ 𝑀 (𝑧 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) ∧ (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧))
135		ovex 7465	. . . . . . . . . . . . 13 ⊢ (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) ∈ V
136		breq2 5146	. . . . . . . . . . . . 13 ⊢ (𝑧 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) → ((((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧 ↔ (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))))
137	135, 136	ceqsexv 3531	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑧 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) ∧ (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧) ↔ (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)))
138	137	rexbii 3093	. . . . . . . . . . 11 ⊢ (∃𝑞 ∈ 𝑀 ∃𝑧(𝑧 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) ∧ (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧) ↔ ∃𝑞 ∈ 𝑀 (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)))
139	134, 138	bitr3i 277	. . . . . . . . . 10 ⊢ (∃𝑧∃𝑞 ∈ 𝑀 (𝑧 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) ∧ (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧) ↔ ∃𝑞 ∈ 𝑀 (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)))
140	139	rexbii 3093	. . . . . . . . 9 ⊢ (∃𝑝 ∈ 𝐿 ∃𝑧∃𝑞 ∈ 𝑀 (𝑧 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) ∧ (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧) ↔ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)))
141	133, 140	bitr3i 277	. . . . . . . 8 ⊢ (∃𝑧∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (𝑧 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) ∧ (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧) ↔ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)))
142	130, 132, 141	3bitr2i 299	. . . . . . 7 ⊢ (∃𝑧 ∈ {𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧 ↔ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)))
143		ssun1 4177	. . . . . . . 8 ⊢ {𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ⊆ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})
144		ssrexv 4052	. . . . . . . 8 ⊢ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ⊆ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))}) → (∃𝑧 ∈ {𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧 → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧))
145	143, 144	ax-mp 5	. . . . . . 7 ⊢ (∃𝑧 ∈ {𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧 → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧)
146	142, 145	sylbir 235	. . . . . 6 ⊢ (∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧)
147	146	2ralimi 3122	. . . . 5 ⊢ (∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) → ∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧)
148	127, 147	syl 17	. . . 4 ⊢ (𝜑 → ∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧)
149	2, 1	cofcutr2d 27961	. . . . . 6 ⊢ (𝜑 → ∀𝑖 ∈ ( R ‘𝐴)∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖)
150	6, 5	cofcutr2d 27961	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑗 ∈ ( R ‘𝐵)∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑗)
151	150	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖)) → ∀𝑗 ∈ ( R ‘𝐵)∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑗)
152		reeanv 3228	. . . . . . . . . . . . . . 15 ⊢ (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗) ↔ (∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖 ∧ ∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑗))
153		rightssno 27921	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ( R ‘𝐴) ⊆ No
154		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝑖 ∈ ( R ‘𝐴))
155	153, 154	sselid 3980	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝑖 ∈ No )
156	155	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝑖 ∈ No )
157	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝐵 ∈ No )
158	156, 157	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (𝑖 ·_s 𝐵) ∈ No )
159	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝐴 ∈ No )
160		rightssno 27921	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ( R ‘𝐵) ⊆ No
161		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝑗 ∈ ( R ‘𝐵))
162	160, 161	sselid 3980	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝑗 ∈ No )
163	162	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝑗 ∈ No )
164	159, 163	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (𝐴 ·_s 𝑗) ∈ No )
165	158, 164	addscld 28014	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) ∈ No )
166	156, 163	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (𝑖 ·_s 𝑗) ∈ No )
167	165, 166	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ∈ No )
168	167	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ∈ No )
169		ssltss2 27835	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐿 <<s 𝑅 → 𝑅 ⊆ No )
170	2, 169	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑅 ⊆ No )
171	170	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑅 ⊆ No )
172		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑟 ∈ 𝑅)
173	171, 172	sseldd 3983	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑟 ∈ No )
174	173	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝑟 ∈ No )
175	174, 157	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (𝑟 ·_s 𝐵) ∈ No )
176	175, 164	addscld 28014	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) ∈ No )
177	174, 163	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (𝑟 ·_s 𝑗) ∈ No )
178	176, 177	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑟 ·_s 𝑗)) ∈ No )
179	178	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑟 ·_s 𝑗)) ∈ No )
180		ssltss2 27835	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑀 <<s 𝑆 → 𝑆 ⊆ No )
181	6, 180	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑆 ⊆ No )
182	181	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑆 ⊆ No )
183		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑠 ∈ 𝑆)
184	182, 183	sseldd 3983	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑠 ∈ No )
185	184	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝑠 ∈ No )
186	159, 185	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (𝐴 ·_s 𝑠) ∈ No )
187	175, 186	addscld 28014	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) ∈ No )
188	173, 184	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝑟 ·_s 𝑠) ∈ No )
189	188	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (𝑟 ·_s 𝑠) ∈ No )
190	187, 189	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) ∈ No )
191	190	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) ∈ No )
192	174	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝑟 ∈ No )
193	155	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝑖 ∈ No )
194	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝐵 ∈ No )
195	162	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝑗 ∈ No )
196		simprrl 780	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗))) → 𝑟 ≤s 𝑖)
197	196	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝑟 ≤s 𝑖)
198	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝐵 ∈ No )
199		ssltright 27911	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐵 ∈ No → {𝐵} <<s ( R ‘𝐵))
200	8, 199	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → {𝐵} <<s ( R ‘𝐵))
201	200	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → {𝐵} <<s ( R ‘𝐵))
202	65	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝐵 ∈ {𝐵})
203	201, 202, 161	ssltsepcd 27840	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝐵 <s 𝑗)
204	198, 162, 203	sltled 27815	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝐵 ≤s 𝑗)
205	204	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝐵 ≤s 𝑗)
206	192, 193, 194, 195, 197, 205	slemuld 28165	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → ((𝑟 ·_s 𝑗) -_s (𝑟 ·_s 𝐵)) ≤s ((𝑖 ·_s 𝑗) -_s (𝑖 ·_s 𝐵)))
207	177, 175, 166, 158	slesubsub2bd 28118	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑟 ·_s 𝑗) -_s (𝑟 ·_s 𝐵)) ≤s ((𝑖 ·_s 𝑗) -_s (𝑖 ·_s 𝐵)) ↔ ((𝑖 ·_s 𝐵) -_s (𝑖 ·_s 𝑗)) ≤s ((𝑟 ·_s 𝐵) -_s (𝑟 ·_s 𝑗))))
208	158, 166	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((𝑖 ·_s 𝐵) -_s (𝑖 ·_s 𝑗)) ∈ No )
209	175, 177	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((𝑟 ·_s 𝐵) -_s (𝑟 ·_s 𝑗)) ∈ No )
210	208, 209, 164	sleadd1d 28029	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑖 ·_s 𝐵) -_s (𝑖 ·_s 𝑗)) ≤s ((𝑟 ·_s 𝐵) -_s (𝑟 ·_s 𝑗)) ↔ (((𝑖 ·_s 𝐵) -_s (𝑖 ·_s 𝑗)) +_s (𝐴 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) -_s (𝑟 ·_s 𝑗)) +_s (𝐴 ·_s 𝑗))))
211	207, 210	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑟 ·_s 𝑗) -_s (𝑟 ·_s 𝐵)) ≤s ((𝑖 ·_s 𝑗) -_s (𝑖 ·_s 𝐵)) ↔ (((𝑖 ·_s 𝐵) -_s (𝑖 ·_s 𝑗)) +_s (𝐴 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) -_s (𝑟 ·_s 𝑗)) +_s (𝐴 ·_s 𝑗))))
212	211	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑟 ·_s 𝑗) -_s (𝑟 ·_s 𝐵)) ≤s ((𝑖 ·_s 𝑗) -_s (𝑖 ·_s 𝐵)) ↔ (((𝑖 ·_s 𝐵) -_s (𝑖 ·_s 𝑗)) +_s (𝐴 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) -_s (𝑟 ·_s 𝑗)) +_s (𝐴 ·_s 𝑗))))
213	206, 212	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑖 ·_s 𝐵) -_s (𝑖 ·_s 𝑗)) +_s (𝐴 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) -_s (𝑟 ·_s 𝑗)) +_s (𝐴 ·_s 𝑗)))
214	158, 164, 166	addsubsd 28113	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) = (((𝑖 ·_s 𝐵) -_s (𝑖 ·_s 𝑗)) +_s (𝐴 ·_s 𝑗)))
215	214	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) = (((𝑖 ·_s 𝐵) -_s (𝑖 ·_s 𝑗)) +_s (𝐴 ·_s 𝑗)))
216	175, 164, 177	addsubsd 28113	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑟 ·_s 𝑗)) = (((𝑟 ·_s 𝐵) -_s (𝑟 ·_s 𝑗)) +_s (𝐴 ·_s 𝑗)))
217	216	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑟 ·_s 𝑗)) = (((𝑟 ·_s 𝐵) -_s (𝑟 ·_s 𝑗)) +_s (𝐴 ·_s 𝑗)))
218	213, 215, 217	3brtr4d 5174	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑟 ·_s 𝑗)))
219	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝐴 ∈ No )
220	185	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝑠 ∈ No )
221	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝐴 ∈ No )
222	85	simp3d 1144	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → {(𝐿 \|s 𝑅)} <<s 𝑅)
223	222	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → {(𝐿 \|s 𝑅)} <<s 𝑅)
224	90	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝐴 ∈ {(𝐿 \|s 𝑅)})
225	223, 224, 172	ssltsepcd 27840	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝐴 <s 𝑟)
226	221, 173, 225	sltled 27815	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝐴 ≤s 𝑟)
227	226	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝐴 ≤s 𝑟)
228	227	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝐴 ≤s 𝑟)
229		simprrr 781	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗))) → 𝑠 ≤s 𝑗)
230	229	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝑠 ≤s 𝑗)
231	219, 192, 220, 195, 228, 230	slemuld 28165	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → ((𝐴 ·_s 𝑗) -_s (𝐴 ·_s 𝑠)) ≤s ((𝑟 ·_s 𝑗) -_s (𝑟 ·_s 𝑠)))
232	164, 177, 186, 189	slesubsubbd 28117	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝐴 ·_s 𝑗) -_s (𝐴 ·_s 𝑠)) ≤s ((𝑟 ·_s 𝑗) -_s (𝑟 ·_s 𝑠)) ↔ ((𝐴 ·_s 𝑗) -_s (𝑟 ·_s 𝑗)) ≤s ((𝐴 ·_s 𝑠) -_s (𝑟 ·_s 𝑠))))
233	164, 177	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((𝐴 ·_s 𝑗) -_s (𝑟 ·_s 𝑗)) ∈ No )
234	186, 189	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((𝐴 ·_s 𝑠) -_s (𝑟 ·_s 𝑠)) ∈ No )
235	233, 234, 175	sleadd2d 28030	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝐴 ·_s 𝑗) -_s (𝑟 ·_s 𝑗)) ≤s ((𝐴 ·_s 𝑠) -_s (𝑟 ·_s 𝑠)) ↔ ((𝑟 ·_s 𝐵) +_s ((𝐴 ·_s 𝑗) -_s (𝑟 ·_s 𝑗))) ≤s ((𝑟 ·_s 𝐵) +_s ((𝐴 ·_s 𝑠) -_s (𝑟 ·_s 𝑠)))))
236	232, 235	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝐴 ·_s 𝑗) -_s (𝐴 ·_s 𝑠)) ≤s ((𝑟 ·_s 𝑗) -_s (𝑟 ·_s 𝑠)) ↔ ((𝑟 ·_s 𝐵) +_s ((𝐴 ·_s 𝑗) -_s (𝑟 ·_s 𝑗))) ≤s ((𝑟 ·_s 𝐵) +_s ((𝐴 ·_s 𝑠) -_s (𝑟 ·_s 𝑠)))))
237	236	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝐴 ·_s 𝑗) -_s (𝐴 ·_s 𝑠)) ≤s ((𝑟 ·_s 𝑗) -_s (𝑟 ·_s 𝑠)) ↔ ((𝑟 ·_s 𝐵) +_s ((𝐴 ·_s 𝑗) -_s (𝑟 ·_s 𝑗))) ≤s ((𝑟 ·_s 𝐵) +_s ((𝐴 ·_s 𝑠) -_s (𝑟 ·_s 𝑠)))))
238	231, 237	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → ((𝑟 ·_s 𝐵) +_s ((𝐴 ·_s 𝑗) -_s (𝑟 ·_s 𝑗))) ≤s ((𝑟 ·_s 𝐵) +_s ((𝐴 ·_s 𝑠) -_s (𝑟 ·_s 𝑠))))
239	175, 164, 177	addsubsassd 28112	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑟 ·_s 𝑗)) = ((𝑟 ·_s 𝐵) +_s ((𝐴 ·_s 𝑗) -_s (𝑟 ·_s 𝑗))))
240	239	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑟 ·_s 𝑗)) = ((𝑟 ·_s 𝐵) +_s ((𝐴 ·_s 𝑗) -_s (𝑟 ·_s 𝑗))))
241	175, 186, 189	addsubsassd 28112	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) = ((𝑟 ·_s 𝐵) +_s ((𝐴 ·_s 𝑠) -_s (𝑟 ·_s 𝑠))))
242	241	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) = ((𝑟 ·_s 𝐵) +_s ((𝐴 ·_s 𝑠) -_s (𝑟 ·_s 𝑠))))
243	238, 240, 242	3brtr4d 5174	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑟 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)))
244	168, 179, 191, 218, 243	sletrd 27808	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)))
245	244	anassrs 467	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗))) → (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)))
246	245	expr 456	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → ((𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗) → (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))))
247	246	reximdvva 3206	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗) → ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))))
248	247	expcom 413	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) → (𝜑 → (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗) → ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)))))
249	248	com23 86	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) → (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗) → (𝜑 → ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)))))
250	249	imp 406	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)) → (𝜑 → ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))))
251	152, 250	sylan2br 595	. . . . . . . . . . . . . 14 ⊢ (((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖 ∧ ∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑗)) → (𝜑 → ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))))
252	251	an4s 660	. . . . . . . . . . . . 13 ⊢ (((𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖) ∧ (𝑗 ∈ ( R ‘𝐵) ∧ ∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑗)) → (𝜑 → ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))))
253	252	impcom 407	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖) ∧ (𝑗 ∈ ( R ‘𝐵) ∧ ∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑗))) → ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)))
254	253	anassrs 467	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖)) ∧ (𝑗 ∈ ( R ‘𝐵) ∧ ∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑗)) → ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)))
255	254	expr 456	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖)) ∧ 𝑗 ∈ ( R ‘𝐵)) → (∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑗 → ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))))
256	255	ralimdva 3166	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖)) → (∀𝑗 ∈ ( R ‘𝐵)∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑗 → ∀𝑗 ∈ ( R ‘𝐵)∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))))
257	151, 256	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖)) → ∀𝑗 ∈ ( R ‘𝐵)∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)))
258	257	expr 456	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ ( R ‘𝐴)) → (∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖 → ∀𝑗 ∈ ( R ‘𝐵)∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))))
259	258	ralimdva 3166	. . . . . 6 ⊢ (𝜑 → (∀𝑖 ∈ ( R ‘𝐴)∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖 → ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))))
260	149, 259	mpd 15	. . . . 5 ⊢ (𝜑 → ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)))
261		eqeq1 2740	. . . . . . . . . 10 ⊢ (𝑏 = 𝑧 → (𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) ↔ 𝑧 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))))
262	261	2rexbidv 3221	. . . . . . . . 9 ⊢ (𝑏 = 𝑧 → (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑧 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))))
263	262	rexab 3699	. . . . . . . 8 ⊢ (∃𝑧 ∈ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))} (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧 ↔ ∃𝑧(∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑧 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) ∧ (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧))
264		r19.41vv 3226	. . . . . . . . 9 ⊢ (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) ∧ (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧) ↔ (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑧 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) ∧ (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧))
265	264	exbii 1847	. . . . . . . 8 ⊢ (∃𝑧∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) ∧ (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧) ↔ ∃𝑧(∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑧 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) ∧ (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧))
266		rexcom4 3287	. . . . . . . . 9 ⊢ (∃𝑟 ∈ 𝑅 ∃𝑧∃𝑠 ∈ 𝑆 (𝑧 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) ∧ (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧) ↔ ∃𝑧∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) ∧ (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧))
267		rexcom4 3287	. . . . . . . . . . 11 ⊢ (∃𝑠 ∈ 𝑆 ∃𝑧(𝑧 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) ∧ (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧) ↔ ∃𝑧∃𝑠 ∈ 𝑆 (𝑧 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) ∧ (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧))
268		ovex 7465	. . . . . . . . . . . . 13 ⊢ (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) ∈ V
269		breq2 5146	. . . . . . . . . . . . 13 ⊢ (𝑧 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) → ((((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧 ↔ (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))))
270	268, 269	ceqsexv 3531	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑧 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) ∧ (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧) ↔ (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)))
271	270	rexbii 3093	. . . . . . . . . . 11 ⊢ (∃𝑠 ∈ 𝑆 ∃𝑧(𝑧 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) ∧ (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧) ↔ ∃𝑠 ∈ 𝑆 (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)))
272	267, 271	bitr3i 277	. . . . . . . . . 10 ⊢ (∃𝑧∃𝑠 ∈ 𝑆 (𝑧 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) ∧ (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧) ↔ ∃𝑠 ∈ 𝑆 (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)))
273	272	rexbii 3093	. . . . . . . . 9 ⊢ (∃𝑟 ∈ 𝑅 ∃𝑧∃𝑠 ∈ 𝑆 (𝑧 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) ∧ (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)))
274	266, 273	bitr3i 277	. . . . . . . 8 ⊢ (∃𝑧∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) ∧ (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)))
275	263, 265, 274	3bitr2i 299	. . . . . . 7 ⊢ (∃𝑧 ∈ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))} (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧 ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)))
276		ssun2 4178	. . . . . . . 8 ⊢ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))} ⊆ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})
277		ssrexv 4052	. . . . . . . 8 ⊢ ({𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))} ⊆ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))}) → (∃𝑧 ∈ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))} (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧 → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧))
278	276, 277	ax-mp 5	. . . . . . 7 ⊢ (∃𝑧 ∈ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))} (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧 → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧)
279	275, 278	sylbir 235	. . . . . 6 ⊢ (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧)
280	279	2ralimi 3122	. . . . 5 ⊢ (∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) → ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧)
281	260, 280	syl 17	. . . 4 ⊢ (𝜑 → ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧)
282		ralunb 4196	. . . . 5 ⊢ (∀𝑥_𝑂 ∈ ({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗))})∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧 ↔ (∀𝑥_𝑂 ∈ {𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧 ∧ ∀𝑥_𝑂 ∈ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧))
283		eqeq1 2740	. . . . . . . . 9 ⊢ (𝑒 = 𝑥_𝑂 → (𝑒 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ↔ 𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔))))
284	283	2rexbidv 3221	. . . . . . . 8 ⊢ (𝑒 = 𝑥_𝑂 → (∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ↔ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔))))
285	284	ralab 3696	. . . . . . 7 ⊢ (∀𝑥_𝑂 ∈ {𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧 ↔ ∀𝑥_𝑂(∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧))
286		r19.23v 3182	. . . . . . . . . 10 ⊢ (∀𝑔 ∈ ( L ‘𝐵)(𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ (∃𝑔 ∈ ( L ‘𝐵)𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧))
287	286	ralbii 3092	. . . . . . . . 9 ⊢ (∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)(𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ ∀𝑓 ∈ ( L ‘𝐴)(∃𝑔 ∈ ( L ‘𝐵)𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧))
288		r19.23v 3182	. . . . . . . . 9 ⊢ (∀𝑓 ∈ ( L ‘𝐴)(∃𝑔 ∈ ( L ‘𝐵)𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ (∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧))
289	287, 288	bitri 275	. . . . . . . 8 ⊢ (∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)(𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ (∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧))
290	289	albii 1818	. . . . . . 7 ⊢ (∀𝑥_𝑂∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)(𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ ∀𝑥_𝑂(∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧))
291		ralcom4 3285	. . . . . . . 8 ⊢ (∀𝑓 ∈ ( L ‘𝐴)∀𝑥_𝑂∀𝑔 ∈ ( L ‘𝐵)(𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ ∀𝑥_𝑂∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)(𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧))
292		ralcom4 3285	. . . . . . . . . 10 ⊢ (∀𝑔 ∈ ( L ‘𝐵)∀𝑥_𝑂(𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ ∀𝑥_𝑂∀𝑔 ∈ ( L ‘𝐵)(𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧))
293		ovex 7465	. . . . . . . . . . . 12 ⊢ (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ∈ V
294		breq1 5145	. . . . . . . . . . . . 13 ⊢ (𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → (𝑥_𝑂 ≤s 𝑧 ↔ (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧))
295	294	rexbidv 3178	. . . . . . . . . . . 12 ⊢ (𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → (∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧 ↔ ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧))
296	293, 295	ceqsalv 3520	. . . . . . . . . . 11 ⊢ (∀𝑥_𝑂(𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧)
297	296	ralbii 3092	. . . . . . . . . 10 ⊢ (∀𝑔 ∈ ( L ‘𝐵)∀𝑥_𝑂(𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ ∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧)
298	292, 297	bitr3i 277	. . . . . . . . 9 ⊢ (∀𝑥_𝑂∀𝑔 ∈ ( L ‘𝐵)(𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ ∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧)
299	298	ralbii 3092	. . . . . . . 8 ⊢ (∀𝑓 ∈ ( L ‘𝐴)∀𝑥_𝑂∀𝑔 ∈ ( L ‘𝐵)(𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ ∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧)
300	291, 299	bitr3i 277	. . . . . . 7 ⊢ (∀𝑥_𝑂∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)(𝑥_𝑂 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ ∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧)
301	285, 290, 300	3bitr2i 299	. . . . . 6 ⊢ (∀𝑥_𝑂 ∈ {𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧 ↔ ∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧)
302		eqeq1 2740	. . . . . . . . 9 ⊢ (ℎ = 𝑥_𝑂 → (ℎ = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ↔ 𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗))))
303	302	2rexbidv 3221	. . . . . . . 8 ⊢ (ℎ = 𝑥_𝑂 → (∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ↔ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗))))
304	303	ralab 3696	. . . . . . 7 ⊢ (∀𝑥_𝑂 ∈ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧 ↔ ∀𝑥_𝑂(∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧))
305		r19.23v 3182	. . . . . . . . . 10 ⊢ (∀𝑗 ∈ ( R ‘𝐵)(𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ (∃𝑗 ∈ ( R ‘𝐵)𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧))
306	305	ralbii 3092	. . . . . . . . 9 ⊢ (∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)(𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ ∀𝑖 ∈ ( R ‘𝐴)(∃𝑗 ∈ ( R ‘𝐵)𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧))
307		r19.23v 3182	. . . . . . . . 9 ⊢ (∀𝑖 ∈ ( R ‘𝐴)(∃𝑗 ∈ ( R ‘𝐵)𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ (∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧))
308	306, 307	bitri 275	. . . . . . . 8 ⊢ (∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)(𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ (∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧))
309	308	albii 1818	. . . . . . 7 ⊢ (∀𝑥_𝑂∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)(𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ ∀𝑥_𝑂(∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧))
310		ralcom4 3285	. . . . . . . 8 ⊢ (∀𝑖 ∈ ( R ‘𝐴)∀𝑥_𝑂∀𝑗 ∈ ( R ‘𝐵)(𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ ∀𝑥_𝑂∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)(𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧))
311		ralcom4 3285	. . . . . . . . . 10 ⊢ (∀𝑗 ∈ ( R ‘𝐵)∀𝑥_𝑂(𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ ∀𝑥_𝑂∀𝑗 ∈ ( R ‘𝐵)(𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧))
312		ovex 7465	. . . . . . . . . . . 12 ⊢ (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ∈ V
313		breq1 5145	. . . . . . . . . . . . 13 ⊢ (𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → (𝑥_𝑂 ≤s 𝑧 ↔ (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧))
314	313	rexbidv 3178	. . . . . . . . . . . 12 ⊢ (𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → (∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧 ↔ ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧))
315	312, 314	ceqsalv 3520	. . . . . . . . . . 11 ⊢ (∀𝑥_𝑂(𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧)
316	315	ralbii 3092	. . . . . . . . . 10 ⊢ (∀𝑗 ∈ ( R ‘𝐵)∀𝑥_𝑂(𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ ∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧)
317	311, 316	bitr3i 277	. . . . . . . . 9 ⊢ (∀𝑥_𝑂∀𝑗 ∈ ( R ‘𝐵)(𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ ∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧)
318	317	ralbii 3092	. . . . . . . 8 ⊢ (∀𝑖 ∈ ( R ‘𝐴)∀𝑥_𝑂∀𝑗 ∈ ( R ‘𝐵)(𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧)
319	310, 318	bitr3i 277	. . . . . . 7 ⊢ (∀𝑥_𝑂∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)(𝑥_𝑂 = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) → ∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧)
320	304, 309, 319	3bitr2i 299	. . . . . 6 ⊢ (∀𝑥_𝑂 ∈ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧 ↔ ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧)
321	301, 320	anbi12i 628	. . . . 5 ⊢ ((∀𝑥_𝑂 ∈ {𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧 ∧ ∀𝑥_𝑂 ∈ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗))}∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧) ↔ (∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧 ∧ ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧))
322	282, 321	bitri 275	. . . 4 ⊢ (∀𝑥_𝑂 ∈ ({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗))})∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧 ↔ (∀𝑓 ∈ ( L ‘𝐴)∀𝑔 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ≤s 𝑧 ∧ ∀𝑖 ∈ ( R ‘𝐴)∀𝑗 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})(((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ≤s 𝑧))
323	148, 281, 322	sylanbrc 583	. . 3 ⊢ (𝜑 → ∀𝑥_𝑂 ∈ ({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗))})∃𝑧 ∈ ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))})𝑥_𝑂 ≤s 𝑧)
324	2, 1	cofcutr1d 27960	. . . . . 6 ⊢ (𝜑 → ∀𝑙 ∈ ( L ‘𝐴)∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡)
325	6, 5	cofcutr2d 27961	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ( R ‘𝐵)∃𝑢 ∈ 𝑆 𝑢 ≤s 𝑚)
326	325	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡)) → ∀𝑚 ∈ ( R ‘𝐵)∃𝑢 ∈ 𝑆 𝑢 ≤s 𝑚)
327		reeanv 3228	. . . . . . . . . . . . . . 15 ⊢ (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚) ↔ (∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡 ∧ ∃𝑢 ∈ 𝑆 𝑢 ≤s 𝑚))
328	33	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐿 ⊆ No )
329		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑡 ∈ 𝐿)
330	328, 329	sseldd 3983	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑡 ∈ No )
331	330	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → 𝑡 ∈ No )
332	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → 𝐵 ∈ No )
333	331, 332	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (𝑡 ·_s 𝐵) ∈ No )
334	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → 𝐴 ∈ No )
335	181	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑆 ⊆ No )
336		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑢 ∈ 𝑆)
337	335, 336	sseldd 3983	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑢 ∈ No )
338	337	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → 𝑢 ∈ No )
339	334, 338	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (𝐴 ·_s 𝑢) ∈ No )
340	333, 339	addscld 28014	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → ((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) ∈ No )
341	331, 338	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (𝑡 ·_s 𝑢) ∈ No )
342	340, 341	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ∈ No )
343	342	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ∈ No )
344		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑙 ∈ ( L ‘𝐴))
345	16, 344	sselid 3980	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑙 ∈ No )
346	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝐵 ∈ No )
347	345, 346	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → (𝑙 ·_s 𝐵) ∈ No )
348	347	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (𝑙 ·_s 𝐵) ∈ No )
349	348, 339	addscld 28014	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → ((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) ∈ No )
350	345	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → 𝑙 ∈ No )
351	350, 338	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (𝑙 ·_s 𝑢) ∈ No )
352	349, 351	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑙 ·_s 𝑢)) ∈ No )
353	352	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑙 ·_s 𝑢)) ∈ No )
354	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝐴 ∈ No )
355		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑚 ∈ ( R ‘𝐵))
356	160, 355	sselid 3980	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑚 ∈ No )
357	354, 356	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → (𝐴 ·_s 𝑚) ∈ No )
358	357	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (𝐴 ·_s 𝑚) ∈ No )
359	348, 358	addscld 28014	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → ((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) ∈ No )
360	345, 356	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → (𝑙 ·_s 𝑚) ∈ No )
361	360	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (𝑙 ·_s 𝑚) ∈ No )
362	359, 361	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) ∈ No )
363	362	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) ∈ No )
364	345	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝑙 ∈ No )
365	331	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝑡 ∈ No )
366	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝐵 ∈ No )
367	338	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝑢 ∈ No )
368		simprrl 780	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚))) → 𝑙 ≤s 𝑡)
369	368	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝑙 ≤s 𝑡)
370	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 ∈ No )
371		scutcut 27847	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑀 <<s 𝑆 → ((𝑀 \|s 𝑆) ∈ No ∧ 𝑀 <<s {(𝑀 \|s 𝑆)} ∧ {(𝑀 \|s 𝑆)} <<s 𝑆))
372	6, 371	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → ((𝑀 \|s 𝑆) ∈ No ∧ 𝑀 <<s {(𝑀 \|s 𝑆)} ∧ {(𝑀 \|s 𝑆)} <<s 𝑆))
373	372	simp3d 1144	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → {(𝑀 \|s 𝑆)} <<s 𝑆)
374	373	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → {(𝑀 \|s 𝑆)} <<s 𝑆)
375		ovex 7465	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑀 \|s 𝑆) ∈ V
376	375	snid 4661	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑀 \|s 𝑆) ∈ {(𝑀 \|s 𝑆)}
377	5, 376	eqeltrdi 2848	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝐵 ∈ {(𝑀 \|s 𝑆)})
378	377	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 ∈ {(𝑀 \|s 𝑆)})
379	374, 378, 336	ssltsepcd 27840	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 <s 𝑢)
380	370, 337, 379	sltled 27815	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 ≤s 𝑢)
381	380	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → 𝐵 ≤s 𝑢)
382	381	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝐵 ≤s 𝑢)
383	364, 365, 366, 367, 369, 382	slemuld 28165	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → ((𝑙 ·_s 𝑢) -_s (𝑙 ·_s 𝐵)) ≤s ((𝑡 ·_s 𝑢) -_s (𝑡 ·_s 𝐵)))
384	351, 348, 341, 333	slesubsub2bd 28118	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑙 ·_s 𝑢) -_s (𝑙 ·_s 𝐵)) ≤s ((𝑡 ·_s 𝑢) -_s (𝑡 ·_s 𝐵)) ↔ ((𝑡 ·_s 𝐵) -_s (𝑡 ·_s 𝑢)) ≤s ((𝑙 ·_s 𝐵) -_s (𝑙 ·_s 𝑢))))
385	333, 341	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → ((𝑡 ·_s 𝐵) -_s (𝑡 ·_s 𝑢)) ∈ No )
386	348, 351	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → ((𝑙 ·_s 𝐵) -_s (𝑙 ·_s 𝑢)) ∈ No )
387	385, 386, 339	sleadd1d 28029	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑡 ·_s 𝐵) -_s (𝑡 ·_s 𝑢)) ≤s ((𝑙 ·_s 𝐵) -_s (𝑙 ·_s 𝑢)) ↔ (((𝑡 ·_s 𝐵) -_s (𝑡 ·_s 𝑢)) +_s (𝐴 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) -_s (𝑙 ·_s 𝑢)) +_s (𝐴 ·_s 𝑢))))
388	384, 387	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑙 ·_s 𝑢) -_s (𝑙 ·_s 𝐵)) ≤s ((𝑡 ·_s 𝑢) -_s (𝑡 ·_s 𝐵)) ↔ (((𝑡 ·_s 𝐵) -_s (𝑡 ·_s 𝑢)) +_s (𝐴 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) -_s (𝑙 ·_s 𝑢)) +_s (𝐴 ·_s 𝑢))))
389	388	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑙 ·_s 𝑢) -_s (𝑙 ·_s 𝐵)) ≤s ((𝑡 ·_s 𝑢) -_s (𝑡 ·_s 𝐵)) ↔ (((𝑡 ·_s 𝐵) -_s (𝑡 ·_s 𝑢)) +_s (𝐴 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) -_s (𝑙 ·_s 𝑢)) +_s (𝐴 ·_s 𝑢))))
390	383, 389	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑡 ·_s 𝐵) -_s (𝑡 ·_s 𝑢)) +_s (𝐴 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) -_s (𝑙 ·_s 𝑢)) +_s (𝐴 ·_s 𝑢)))
391	333, 339, 341	addsubsd 28113	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) = (((𝑡 ·_s 𝐵) -_s (𝑡 ·_s 𝑢)) +_s (𝐴 ·_s 𝑢)))
392	391	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) = (((𝑡 ·_s 𝐵) -_s (𝑡 ·_s 𝑢)) +_s (𝐴 ·_s 𝑢)))
393	348, 339, 351	addsubsd 28113	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑙 ·_s 𝑢)) = (((𝑙 ·_s 𝐵) -_s (𝑙 ·_s 𝑢)) +_s (𝐴 ·_s 𝑢)))
394	393	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑙 ·_s 𝑢)) = (((𝑙 ·_s 𝐵) -_s (𝑙 ·_s 𝑢)) +_s (𝐴 ·_s 𝑢)))
395	390, 392, 394	3brtr4d 5174	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑙 ·_s 𝑢)))
396	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝐴 ∈ No )
397	356	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝑚 ∈ No )
398		ssltleft 27910	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐴 ∈ No → ( L ‘𝐴) <<s {𝐴})
399	4, 398	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → ( L ‘𝐴) <<s {𝐴})
400	399	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → ( L ‘𝐴) <<s {𝐴})
401		snidg 4659	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐴 ∈ No → 𝐴 ∈ {𝐴})
402	4, 401	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
403	402	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝐴 ∈ {𝐴})
404	400, 344, 403	ssltsepcd 27840	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑙 <s 𝐴)
405	345, 354, 404	sltled 27815	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑙 ≤s 𝐴)
406	405	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝑙 ≤s 𝐴)
407		simprrr 781	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚))) → 𝑢 ≤s 𝑚)
408	407	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝑢 ≤s 𝑚)
409	364, 396, 367, 397, 406, 408	slemuld 28165	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → ((𝑙 ·_s 𝑚) -_s (𝑙 ·_s 𝑢)) ≤s ((𝐴 ·_s 𝑚) -_s (𝐴 ·_s 𝑢)))
410	361, 358, 351, 339	slesubsub3bd 28119	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑙 ·_s 𝑚) -_s (𝑙 ·_s 𝑢)) ≤s ((𝐴 ·_s 𝑚) -_s (𝐴 ·_s 𝑢)) ↔ ((𝐴 ·_s 𝑢) -_s (𝑙 ·_s 𝑢)) ≤s ((𝐴 ·_s 𝑚) -_s (𝑙 ·_s 𝑚))))
411	339, 351	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → ((𝐴 ·_s 𝑢) -_s (𝑙 ·_s 𝑢)) ∈ No )
412	358, 361	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → ((𝐴 ·_s 𝑚) -_s (𝑙 ·_s 𝑚)) ∈ No )
413	411, 412, 348	sleadd2d 28030	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝐴 ·_s 𝑢) -_s (𝑙 ·_s 𝑢)) ≤s ((𝐴 ·_s 𝑚) -_s (𝑙 ·_s 𝑚)) ↔ ((𝑙 ·_s 𝐵) +_s ((𝐴 ·_s 𝑢) -_s (𝑙 ·_s 𝑢))) ≤s ((𝑙 ·_s 𝐵) +_s ((𝐴 ·_s 𝑚) -_s (𝑙 ·_s 𝑚)))))
414	410, 413	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑙 ·_s 𝑚) -_s (𝑙 ·_s 𝑢)) ≤s ((𝐴 ·_s 𝑚) -_s (𝐴 ·_s 𝑢)) ↔ ((𝑙 ·_s 𝐵) +_s ((𝐴 ·_s 𝑢) -_s (𝑙 ·_s 𝑢))) ≤s ((𝑙 ·_s 𝐵) +_s ((𝐴 ·_s 𝑚) -_s (𝑙 ·_s 𝑚)))))
415	414	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑙 ·_s 𝑚) -_s (𝑙 ·_s 𝑢)) ≤s ((𝐴 ·_s 𝑚) -_s (𝐴 ·_s 𝑢)) ↔ ((𝑙 ·_s 𝐵) +_s ((𝐴 ·_s 𝑢) -_s (𝑙 ·_s 𝑢))) ≤s ((𝑙 ·_s 𝐵) +_s ((𝐴 ·_s 𝑚) -_s (𝑙 ·_s 𝑚)))))
416	409, 415	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → ((𝑙 ·_s 𝐵) +_s ((𝐴 ·_s 𝑢) -_s (𝑙 ·_s 𝑢))) ≤s ((𝑙 ·_s 𝐵) +_s ((𝐴 ·_s 𝑚) -_s (𝑙 ·_s 𝑚))))
417	348, 339, 351	addsubsassd 28112	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑙 ·_s 𝑢)) = ((𝑙 ·_s 𝐵) +_s ((𝐴 ·_s 𝑢) -_s (𝑙 ·_s 𝑢))))
418	417	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑙 ·_s 𝑢)) = ((𝑙 ·_s 𝐵) +_s ((𝐴 ·_s 𝑢) -_s (𝑙 ·_s 𝑢))))
419	348, 358, 361	addsubsassd 28112	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) = ((𝑙 ·_s 𝐵) +_s ((𝐴 ·_s 𝑚) -_s (𝑙 ·_s 𝑚))))
420	419	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) = ((𝑙 ·_s 𝐵) +_s ((𝐴 ·_s 𝑚) -_s (𝑙 ·_s 𝑚))))
421	416, 418, 420	3brtr4d 5174	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑙 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
422	343, 353, 363, 395, 421	sletrd 27808	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
423	422	anassrs 467	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚))) → (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
424	423	expr 456	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → ((𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚) → (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))))
425	424	reximdvva 3206	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚) → ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))))
426	425	expcom 413	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) → (𝜑 → (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚) → ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))))
427	426	com23 86	. . . . . . . . . . . . . . . 16 ⊢ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) → (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚) → (𝜑 → ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))))
428	427	imp 406	. . . . . . . . . . . . . . 15 ⊢ (((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)) → (𝜑 → ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))))
429	327, 428	sylan2br 595	. . . . . . . . . . . . . 14 ⊢ (((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡 ∧ ∃𝑢 ∈ 𝑆 𝑢 ≤s 𝑚)) → (𝜑 → ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))))
430	429	an4s 660	. . . . . . . . . . . . 13 ⊢ (((𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡) ∧ (𝑚 ∈ ( R ‘𝐵) ∧ ∃𝑢 ∈ 𝑆 𝑢 ≤s 𝑚)) → (𝜑 → ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))))
431	430	impcom 407	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡) ∧ (𝑚 ∈ ( R ‘𝐵) ∧ ∃𝑢 ∈ 𝑆 𝑢 ≤s 𝑚))) → ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
432	431	anassrs 467	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡)) ∧ (𝑚 ∈ ( R ‘𝐵) ∧ ∃𝑢 ∈ 𝑆 𝑢 ≤s 𝑚)) → ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
433	432	expr 456	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡)) ∧ 𝑚 ∈ ( R ‘𝐵)) → (∃𝑢 ∈ 𝑆 𝑢 ≤s 𝑚 → ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))))
434	433	ralimdva 3166	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡)) → (∀𝑚 ∈ ( R ‘𝐵)∃𝑢 ∈ 𝑆 𝑢 ≤s 𝑚 → ∀𝑚 ∈ ( R ‘𝐵)∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))))
435	326, 434	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡)) → ∀𝑚 ∈ ( R ‘𝐵)∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
436	435	expr 456	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑙 ∈ ( L ‘𝐴)) → (∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡 → ∀𝑚 ∈ ( R ‘𝐵)∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))))
437	436	ralimdva 3166	. . . . . 6 ⊢ (𝜑 → (∀𝑙 ∈ ( L ‘𝐴)∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡 → ∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))))
438	324, 437	mpd 15	. . . . 5 ⊢ (𝜑 → ∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
439		eqeq1 2740	. . . . . . . . . 10 ⊢ (𝑐 = 𝑧 → (𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ↔ 𝑧 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))))
440	439	2rexbidv 3221	. . . . . . . . 9 ⊢ (𝑐 = 𝑧 → (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ↔ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑧 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))))
441	440	rexab 3699	. . . . . . . 8 ⊢ (∃𝑧 ∈ {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))}𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) ↔ ∃𝑧(∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑧 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))))
442		r19.41vv 3226	. . . . . . . . 9 ⊢ (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (𝑧 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))) ↔ (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑧 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))))
443	442	exbii 1847	. . . . . . . 8 ⊢ (∃𝑧∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (𝑧 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))) ↔ ∃𝑧(∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑧 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))))
444		rexcom4 3287	. . . . . . . . 9 ⊢ (∃𝑡 ∈ 𝐿 ∃𝑧∃𝑢 ∈ 𝑆 (𝑧 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))) ↔ ∃𝑧∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (𝑧 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))))
445		rexcom4 3287	. . . . . . . . . . 11 ⊢ (∃𝑢 ∈ 𝑆 ∃𝑧(𝑧 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))) ↔ ∃𝑧∃𝑢 ∈ 𝑆 (𝑧 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))))
446		ovex 7465	. . . . . . . . . . . . 13 ⊢ (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ∈ V
447		breq1 5145	. . . . . . . . . . . . 13 ⊢ (𝑧 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) → (𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) ↔ (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))))
448	446, 447	ceqsexv 3531	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑧 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))) ↔ (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
449	448	rexbii 3093	. . . . . . . . . . 11 ⊢ (∃𝑢 ∈ 𝑆 ∃𝑧(𝑧 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))) ↔ ∃𝑢 ∈ 𝑆 (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
450	445, 449	bitr3i 277	. . . . . . . . . 10 ⊢ (∃𝑧∃𝑢 ∈ 𝑆 (𝑧 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))) ↔ ∃𝑢 ∈ 𝑆 (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
451	450	rexbii 3093	. . . . . . . . 9 ⊢ (∃𝑡 ∈ 𝐿 ∃𝑧∃𝑢 ∈ 𝑆 (𝑧 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))) ↔ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
452	444, 451	bitr3i 277	. . . . . . . 8 ⊢ (∃𝑧∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (𝑧 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ∧ 𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))) ↔ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
453	441, 443, 452	3bitr2i 299	. . . . . . 7 ⊢ (∃𝑧 ∈ {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))}𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) ↔ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
454		ssun1 4177	. . . . . . . 8 ⊢ {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ⊆ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})
455		ssrexv 4052	. . . . . . . 8 ⊢ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ⊆ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))}) → (∃𝑧 ∈ {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))}𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))))
456	454, 455	ax-mp 5	. . . . . . 7 ⊢ (∃𝑧 ∈ {𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))}𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
457	453, 456	sylbir 235	. . . . . 6 ⊢ (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
458	457	2ralimi 3122	. . . . 5 ⊢ (∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
459	438, 458	syl 17	. . . 4 ⊢ (𝜑 → ∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
460	2, 1	cofcutr2d 27961	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ ( R ‘𝐴)∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥)
461	6, 5	cofcutr1d 27960	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑦 ∈ ( L ‘𝐵)∃𝑤 ∈ 𝑀 𝑦 ≤s 𝑤)
462	461	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥)) → ∀𝑦 ∈ ( L ‘𝐵)∃𝑤 ∈ 𝑀 𝑦 ≤s 𝑤)
463		reeanv 3228	. . . . . . . . . . . . . . 15 ⊢ (∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤) ↔ (∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥 ∧ ∃𝑤 ∈ 𝑀 𝑦 ≤s 𝑤))
464	170	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑅 ⊆ No )
465		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑣 ∈ 𝑅)
466	464, 465	sseldd 3983	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑣 ∈ No )
467	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐵 ∈ No )
468	466, 467	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑣 ·_s 𝐵) ∈ No )
469	468	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (𝑣 ·_s 𝐵) ∈ No )
470	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐴 ∈ No )
471	44	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑀 ⊆ No )
472		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑤 ∈ 𝑀)
473	471, 472	sseldd 3983	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑤 ∈ No )
474	470, 473	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝐴 ·_s 𝑤) ∈ No )
475	474	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (𝐴 ·_s 𝑤) ∈ No )
476	469, 475	addscld 28014	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → ((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) ∈ No )
477	466, 473	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑣 ·_s 𝑤) ∈ No )
478	477	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (𝑣 ·_s 𝑤) ∈ No )
479	476, 478	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ∈ No )
480	479	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ∈ No )
481	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝐴 ∈ No )
482		simprr 772	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑦 ∈ ( L ‘𝐵))
483	23, 482	sselid 3980	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑦 ∈ No )
484	483	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝑦 ∈ No )
485	481, 484	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (𝐴 ·_s 𝑦) ∈ No )
486	469, 485	addscld 28014	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → ((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) ∈ No )
487	466	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝑣 ∈ No )
488	487, 484	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (𝑣 ·_s 𝑦) ∈ No )
489	486, 488	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑣 ·_s 𝑦)) ∈ No )
490	489	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑣 ·_s 𝑦)) ∈ No )
491		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑥 ∈ ( R ‘𝐴))
492	153, 491	sselid 3980	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑥 ∈ No )
493	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝐵 ∈ No )
494	492, 493	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → (𝑥 ·_s 𝐵) ∈ No )
495	494	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (𝑥 ·_s 𝐵) ∈ No )
496	495, 485	addscld 28014	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → ((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) ∈ No )
497	492, 483	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → (𝑥 ·_s 𝑦) ∈ No )
498	497	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (𝑥 ·_s 𝑦) ∈ No )
499	496, 498	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) ∈ No )
500	499	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) ∈ No )
501	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝐴 ∈ No )
502	487	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝑣 ∈ No )
503	483	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝑦 ∈ No )
504	473	adantrl 716	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝑤 ∈ No )
505	504	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝑤 ∈ No )
506	1	sneqd 4637	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → {𝐴} = {(𝐿 \|s 𝑅)})
507	506, 222	eqbrtrd 5164	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → {𝐴} <<s 𝑅)
508	507	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → {𝐴} <<s 𝑅)
509	481, 401	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝐴 ∈ {𝐴})
510		simprrl 780	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝑣 ∈ 𝑅)
511	508, 509, 510	ssltsepcd 27840	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝐴 <s 𝑣)
512	481, 487, 511	sltled 27815	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝐴 ≤s 𝑣)
513	512	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝐴 ≤s 𝑣)
514		simprrr 781	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤))) → 𝑦 ≤s 𝑤)
515	514	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝑦 ≤s 𝑤)
516	501, 502, 503, 505, 513, 515	slemuld 28165	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → ((𝐴 ·_s 𝑤) -_s (𝐴 ·_s 𝑦)) ≤s ((𝑣 ·_s 𝑤) -_s (𝑣 ·_s 𝑦)))
517	475, 478, 485, 488	slesubsubbd 28117	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (((𝐴 ·_s 𝑤) -_s (𝐴 ·_s 𝑦)) ≤s ((𝑣 ·_s 𝑤) -_s (𝑣 ·_s 𝑦)) ↔ ((𝐴 ·_s 𝑤) -_s (𝑣 ·_s 𝑤)) ≤s ((𝐴 ·_s 𝑦) -_s (𝑣 ·_s 𝑦))))
518	475, 478	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → ((𝐴 ·_s 𝑤) -_s (𝑣 ·_s 𝑤)) ∈ No )
519	485, 488	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → ((𝐴 ·_s 𝑦) -_s (𝑣 ·_s 𝑦)) ∈ No )
520	518, 519, 469	sleadd2d 28030	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (((𝐴 ·_s 𝑤) -_s (𝑣 ·_s 𝑤)) ≤s ((𝐴 ·_s 𝑦) -_s (𝑣 ·_s 𝑦)) ↔ ((𝑣 ·_s 𝐵) +_s ((𝐴 ·_s 𝑤) -_s (𝑣 ·_s 𝑤))) ≤s ((𝑣 ·_s 𝐵) +_s ((𝐴 ·_s 𝑦) -_s (𝑣 ·_s 𝑦)))))
521	517, 520	bitrd 279	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (((𝐴 ·_s 𝑤) -_s (𝐴 ·_s 𝑦)) ≤s ((𝑣 ·_s 𝑤) -_s (𝑣 ·_s 𝑦)) ↔ ((𝑣 ·_s 𝐵) +_s ((𝐴 ·_s 𝑤) -_s (𝑣 ·_s 𝑤))) ≤s ((𝑣 ·_s 𝐵) +_s ((𝐴 ·_s 𝑦) -_s (𝑣 ·_s 𝑦)))))
522	521	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝐴 ·_s 𝑤) -_s (𝐴 ·_s 𝑦)) ≤s ((𝑣 ·_s 𝑤) -_s (𝑣 ·_s 𝑦)) ↔ ((𝑣 ·_s 𝐵) +_s ((𝐴 ·_s 𝑤) -_s (𝑣 ·_s 𝑤))) ≤s ((𝑣 ·_s 𝐵) +_s ((𝐴 ·_s 𝑦) -_s (𝑣 ·_s 𝑦)))))
523	516, 522	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → ((𝑣 ·_s 𝐵) +_s ((𝐴 ·_s 𝑤) -_s (𝑣 ·_s 𝑤))) ≤s ((𝑣 ·_s 𝐵) +_s ((𝐴 ·_s 𝑦) -_s (𝑣 ·_s 𝑦))))
524	469, 475, 478	addsubsassd 28112	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) = ((𝑣 ·_s 𝐵) +_s ((𝐴 ·_s 𝑤) -_s (𝑣 ·_s 𝑤))))
525	524	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) = ((𝑣 ·_s 𝐵) +_s ((𝐴 ·_s 𝑤) -_s (𝑣 ·_s 𝑤))))
526	469, 485, 488	addsubsassd 28112	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑣 ·_s 𝑦)) = ((𝑣 ·_s 𝐵) +_s ((𝐴 ·_s 𝑦) -_s (𝑣 ·_s 𝑦))))
527	526	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑣 ·_s 𝑦)) = ((𝑣 ·_s 𝐵) +_s ((𝐴 ·_s 𝑦) -_s (𝑣 ·_s 𝑦))))
528	523, 525, 527	3brtr4d 5174	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑣 ·_s 𝑦)))
529	492	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝑥 ∈ No )
530	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝐵 ∈ No )
531		simprrl 780	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤))) → 𝑣 ≤s 𝑥)
532	531	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝑣 ≤s 𝑥)
533	493, 61	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → ( L ‘𝐵) <<s {𝐵})
534	65	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝐵 ∈ {𝐵})
535	533, 482, 534	ssltsepcd 27840	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑦 <s 𝐵)
536	483, 493, 535	sltled 27815	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑦 ≤s 𝐵)
537	536	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝑦 ≤s 𝐵)
538	502, 529, 503, 530, 532, 537	slemuld 28165	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → ((𝑣 ·_s 𝐵) -_s (𝑣 ·_s 𝑦)) ≤s ((𝑥 ·_s 𝐵) -_s (𝑥 ·_s 𝑦)))
539	469, 488	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → ((𝑣 ·_s 𝐵) -_s (𝑣 ·_s 𝑦)) ∈ No )
540	539	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → ((𝑣 ·_s 𝐵) -_s (𝑣 ·_s 𝑦)) ∈ No )
541	495, 498	subscld 28094	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → ((𝑥 ·_s 𝐵) -_s (𝑥 ·_s 𝑦)) ∈ No )
542	541	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → ((𝑥 ·_s 𝐵) -_s (𝑥 ·_s 𝑦)) ∈ No )
543	485	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (𝐴 ·_s 𝑦) ∈ No )
544	540, 542, 543	sleadd1d 28029	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑣 ·_s 𝐵) -_s (𝑣 ·_s 𝑦)) ≤s ((𝑥 ·_s 𝐵) -_s (𝑥 ·_s 𝑦)) ↔ (((𝑣 ·_s 𝐵) -_s (𝑣 ·_s 𝑦)) +_s (𝐴 ·_s 𝑦)) ≤s (((𝑥 ·_s 𝐵) -_s (𝑥 ·_s 𝑦)) +_s (𝐴 ·_s 𝑦))))
545	538, 544	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑣 ·_s 𝐵) -_s (𝑣 ·_s 𝑦)) +_s (𝐴 ·_s 𝑦)) ≤s (((𝑥 ·_s 𝐵) -_s (𝑥 ·_s 𝑦)) +_s (𝐴 ·_s 𝑦)))
546	469, 485, 488	addsubsd 28113	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑣 ·_s 𝑦)) = (((𝑣 ·_s 𝐵) -_s (𝑣 ·_s 𝑦)) +_s (𝐴 ·_s 𝑦)))
547	546	adantrrr 725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑣 ·_s 𝑦)) = (((𝑣 ·_s 𝐵) -_s (𝑣 ·_s 𝑦)) +_s (𝐴 ·_s 𝑦)))
548	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝐴 ∈ No )
549	548, 483	mulscld 28162	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → (𝐴 ·_s 𝑦) ∈ No )
550	494, 549, 497	addsubsd 28113	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) = (((𝑥 ·_s 𝐵) -_s (𝑥 ·_s 𝑦)) +_s (𝐴 ·_s 𝑦)))
551	550	adantrr 717	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) = (((𝑥 ·_s 𝐵) -_s (𝑥 ·_s 𝑦)) +_s (𝐴 ·_s 𝑦)))
552	545, 547, 551	3brtr4d 5174	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑣 ·_s 𝑦)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
553	480, 490, 500, 528, 552	sletrd 27808	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
554	553	anassrs 467	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤))) → (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
555	554	expr 456	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → ((𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤) → (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
556	555	reximdvva 3206	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → (∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤) → ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
557	556	expcom 413	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) → (𝜑 → (∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤) → ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))))
558	557	com23 86	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) → (∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤) → (𝜑 → ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))))
559	558	imp 406	. . . . . . . . . . . . . . 15 ⊢ (((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)) → (𝜑 → ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
560	463, 559	sylan2br 595	. . . . . . . . . . . . . 14 ⊢ (((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥 ∧ ∃𝑤 ∈ 𝑀 𝑦 ≤s 𝑤)) → (𝜑 → ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
561	560	an4s 660	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥) ∧ (𝑦 ∈ ( L ‘𝐵) ∧ ∃𝑤 ∈ 𝑀 𝑦 ≤s 𝑤)) → (𝜑 → ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
562	561	impcom 407	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥) ∧ (𝑦 ∈ ( L ‘𝐵) ∧ ∃𝑤 ∈ 𝑀 𝑦 ≤s 𝑤))) → ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
563	562	anassrs 467	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥)) ∧ (𝑦 ∈ ( L ‘𝐵) ∧ ∃𝑤 ∈ 𝑀 𝑦 ≤s 𝑤)) → ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
564	563	expr 456	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥)) ∧ 𝑦 ∈ ( L ‘𝐵)) → (∃𝑤 ∈ 𝑀 𝑦 ≤s 𝑤 → ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
565	564	ralimdva 3166	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥)) → (∀𝑦 ∈ ( L ‘𝐵)∃𝑤 ∈ 𝑀 𝑦 ≤s 𝑤 → ∀𝑦 ∈ ( L ‘𝐵)∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
566	462, 565	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥)) → ∀𝑦 ∈ ( L ‘𝐵)∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
567	566	expr 456	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ( R ‘𝐴)) → (∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥 → ∀𝑦 ∈ ( L ‘𝐵)∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
568	567	ralimdva 3166	. . . . . 6 ⊢ (𝜑 → (∀𝑥 ∈ ( R ‘𝐴)∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥 → ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
569	460, 568	mpd 15	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
570		eqeq1 2740	. . . . . . . . . 10 ⊢ (𝑑 = 𝑧 → (𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ↔ 𝑧 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))))
571	570	2rexbidv 3221	. . . . . . . . 9 ⊢ (𝑑 = 𝑧 → (∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ↔ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑧 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))))
572	571	rexab 3699	. . . . . . . 8 ⊢ (∃𝑧 ∈ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))}𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) ↔ ∃𝑧(∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑧 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
573		r19.41vv 3226	. . . . . . . . 9 ⊢ (∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (𝑧 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))) ↔ (∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑧 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
574	573	exbii 1847	. . . . . . . 8 ⊢ (∃𝑧∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (𝑧 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))) ↔ ∃𝑧(∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑧 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
575		rexcom4 3287	. . . . . . . . 9 ⊢ (∃𝑣 ∈ 𝑅 ∃𝑧∃𝑤 ∈ 𝑀 (𝑧 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))) ↔ ∃𝑧∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (𝑧 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
576		rexcom4 3287	. . . . . . . . . . 11 ⊢ (∃𝑤 ∈ 𝑀 ∃𝑧(𝑧 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))) ↔ ∃𝑧∃𝑤 ∈ 𝑀 (𝑧 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
577		ovex 7465	. . . . . . . . . . . . 13 ⊢ (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ∈ V
578		breq1 5145	. . . . . . . . . . . . 13 ⊢ (𝑧 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) → (𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) ↔ (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
579	577, 578	ceqsexv 3531	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑧 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))) ↔ (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
580	579	rexbii 3093	. . . . . . . . . . 11 ⊢ (∃𝑤 ∈ 𝑀 ∃𝑧(𝑧 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))) ↔ ∃𝑤 ∈ 𝑀 (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
581	576, 580	bitr3i 277	. . . . . . . . . 10 ⊢ (∃𝑧∃𝑤 ∈ 𝑀 (𝑧 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))) ↔ ∃𝑤 ∈ 𝑀 (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
582	581	rexbii 3093	. . . . . . . . 9 ⊢ (∃𝑣 ∈ 𝑅 ∃𝑧∃𝑤 ∈ 𝑀 (𝑧 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))) ↔ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
583	575, 582	bitr3i 277	. . . . . . . 8 ⊢ (∃𝑧∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (𝑧 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ∧ 𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))) ↔ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
584	572, 574, 583	3bitr2i 299	. . . . . . 7 ⊢ (∃𝑧 ∈ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))}𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) ↔ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
585		ssun2 4178	. . . . . . . 8 ⊢ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))} ⊆ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})
586		ssrexv 4052	. . . . . . . 8 ⊢ ({𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))} ⊆ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))}) → (∃𝑧 ∈ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))}𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
587	585, 586	ax-mp 5	. . . . . . 7 ⊢ (∃𝑧 ∈ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))}𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
588	584, 587	sylbir 235	. . . . . 6 ⊢ (∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
589	588	2ralimi 3122	. . . . 5 ⊢ (∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
590	569, 589	syl 17	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
591		ralunb 4196	. . . . 5 ⊢ (∀𝑥_𝑂 ∈ ({𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))} ∪ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))})∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂 ↔ (∀𝑥_𝑂 ∈ {𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))}∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂 ∧ ∀𝑥_𝑂 ∈ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))}∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂))
592		eqeq1 2740	. . . . . . . . 9 ⊢ (𝑘 = 𝑥_𝑂 → (𝑘 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) ↔ 𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))))
593	592	2rexbidv 3221	. . . . . . . 8 ⊢ (𝑘 = 𝑥_𝑂 → (∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) ↔ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))))
594	593	ralab 3696	. . . . . . 7 ⊢ (∀𝑥_𝑂 ∈ {𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))}∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂 ↔ ∀𝑥_𝑂(∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂))
595		r19.23v 3182	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ ( R ‘𝐵)(𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ (∃𝑚 ∈ ( R ‘𝐵)𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂))
596	595	ralbii 3092	. . . . . . . . 9 ⊢ (∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)(𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ ∀𝑙 ∈ ( L ‘𝐴)(∃𝑚 ∈ ( R ‘𝐵)𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂))
597		r19.23v 3182	. . . . . . . . 9 ⊢ (∀𝑙 ∈ ( L ‘𝐴)(∃𝑚 ∈ ( R ‘𝐵)𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ (∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂))
598	596, 597	bitri 275	. . . . . . . 8 ⊢ (∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)(𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ (∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂))
599	598	albii 1818	. . . . . . 7 ⊢ (∀𝑥_𝑂∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)(𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ ∀𝑥_𝑂(∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂))
600		ralcom4 3285	. . . . . . . 8 ⊢ (∀𝑙 ∈ ( L ‘𝐴)∀𝑥_𝑂∀𝑚 ∈ ( R ‘𝐵)(𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ ∀𝑥_𝑂∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)(𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂))
601		ralcom4 3285	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ ( R ‘𝐵)∀𝑥_𝑂(𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ ∀𝑥_𝑂∀𝑚 ∈ ( R ‘𝐵)(𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂))
602		ovex 7465	. . . . . . . . . . . 12 ⊢ (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) ∈ V
603		breq2 5146	. . . . . . . . . . . . 13 ⊢ (𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → (𝑧 ≤s 𝑥_𝑂 ↔ 𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))))
604	603	rexbidv 3178	. . . . . . . . . . . 12 ⊢ (𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → (∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂 ↔ ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))))
605	602, 604	ceqsalv 3520	. . . . . . . . . . 11 ⊢ (∀𝑥_𝑂(𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
606	605	ralbii 3092	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ ( R ‘𝐵)∀𝑥_𝑂(𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ ∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
607	601, 606	bitr3i 277	. . . . . . . . 9 ⊢ (∀𝑥_𝑂∀𝑚 ∈ ( R ‘𝐵)(𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ ∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
608	607	ralbii 3092	. . . . . . . 8 ⊢ (∀𝑙 ∈ ( L ‘𝐴)∀𝑥_𝑂∀𝑚 ∈ ( R ‘𝐵)(𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ ∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
609	600, 608	bitr3i 277	. . . . . . 7 ⊢ (∀𝑥_𝑂∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)(𝑥_𝑂 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ ∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
610	594, 599, 609	3bitr2i 299	. . . . . 6 ⊢ (∀𝑥_𝑂 ∈ {𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))}∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂 ↔ ∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)))
611		eqeq1 2740	. . . . . . . . 9 ⊢ (𝑛 = 𝑥_𝑂 → (𝑛 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) ↔ 𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
612	611	2rexbidv 3221	. . . . . . . 8 ⊢ (𝑛 = 𝑥_𝑂 → (∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) ↔ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
613	612	ralab 3696	. . . . . . 7 ⊢ (∀𝑥_𝑂 ∈ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))}∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂 ↔ ∀𝑥_𝑂(∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂))
614		r19.23v 3182	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ ( L ‘𝐵)(𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ (∃𝑦 ∈ ( L ‘𝐵)𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂))
615	614	ralbii 3092	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)(𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ ∀𝑥 ∈ ( R ‘𝐴)(∃𝑦 ∈ ( L ‘𝐵)𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂))
616		r19.23v 3182	. . . . . . . . 9 ⊢ (∀𝑥 ∈ ( R ‘𝐴)(∃𝑦 ∈ ( L ‘𝐵)𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ (∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂))
617	615, 616	bitri 275	. . . . . . . 8 ⊢ (∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)(𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ (∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂))
618	617	albii 1818	. . . . . . 7 ⊢ (∀𝑥_𝑂∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)(𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ ∀𝑥_𝑂(∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂))
619		ralcom4 3285	. . . . . . . 8 ⊢ (∀𝑥 ∈ ( R ‘𝐴)∀𝑥_𝑂∀𝑦 ∈ ( L ‘𝐵)(𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ ∀𝑥_𝑂∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)(𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂))
620		ralcom4 3285	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ ( L ‘𝐵)∀𝑥_𝑂(𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ ∀𝑥_𝑂∀𝑦 ∈ ( L ‘𝐵)(𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂))
621		ovex 7465	. . . . . . . . . . . 12 ⊢ (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) ∈ V
622		breq2 5146	. . . . . . . . . . . . 13 ⊢ (𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → (𝑧 ≤s 𝑥_𝑂 ↔ 𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
623	622	rexbidv 3178	. . . . . . . . . . . 12 ⊢ (𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → (∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂 ↔ ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
624	621, 623	ceqsalv 3520	. . . . . . . . . . 11 ⊢ (∀𝑥_𝑂(𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
625	624	ralbii 3092	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ ( L ‘𝐵)∀𝑥_𝑂(𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ ∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
626	620, 625	bitr3i 277	. . . . . . . . 9 ⊢ (∀𝑥_𝑂∀𝑦 ∈ ( L ‘𝐵)(𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ ∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
627	626	ralbii 3092	. . . . . . . 8 ⊢ (∀𝑥 ∈ ( R ‘𝐴)∀𝑥_𝑂∀𝑦 ∈ ( L ‘𝐵)(𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
628	619, 627	bitr3i 277	. . . . . . 7 ⊢ (∀𝑥_𝑂∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)(𝑥_𝑂 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) → ∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
629	613, 618, 628	3bitr2i 299	. . . . . 6 ⊢ (∀𝑥_𝑂 ∈ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))}∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂 ↔ ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)))
630	610, 629	anbi12i 628	. . . . 5 ⊢ ((∀𝑥_𝑂 ∈ {𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))}∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂 ∧ ∀𝑥_𝑂 ∈ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))}∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂) ↔ (∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) ∧ ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
631	591, 630	bitri 275	. . . 4 ⊢ (∀𝑥_𝑂 ∈ ({𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))} ∪ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))})∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂 ↔ (∀𝑙 ∈ ( L ‘𝐴)∀𝑚 ∈ ( R ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) ∧ ∀𝑥 ∈ ( R ‘𝐴)∀𝑦 ∈ ( L ‘𝐵)∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))))
632	459, 590, 631	sylanbrc 583	. . 3 ⊢ (𝜑 → ∀𝑥_𝑂 ∈ ({𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))} ∪ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))})∃𝑧 ∈ ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})𝑧 ≤s 𝑥_𝑂)
633	2, 6, 1, 5	ssltmul1 28174	. . . 4 ⊢ (𝜑 → ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))}) <<s {(𝐴 ·_s 𝐵)})
634	10	sneqd 4637	. . . 4 ⊢ (𝜑 → {(𝐴 ·_s 𝐵)} = {(({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗))}) \|s ({𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))} ∪ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))}))})
635	633, 634	breqtrd 5168	. . 3 ⊢ (𝜑 → ({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))}) <<s {(({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗))}) \|s ({𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))} ∪ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))}))})
636	2, 6, 1, 5	ssltmul2 28175	. . . 4 ⊢ (𝜑 → {(𝐴 ·_s 𝐵)} <<s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))}))
637	634, 636	eqbrtrrd 5166	. . 3 ⊢ (𝜑 → {(({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗))}) \|s ({𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))} ∪ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))}))} <<s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))}))
638	11, 323, 632, 635, 637	cofcut1d 27956	. 2 ⊢ (𝜑 → (({𝑒 ∣ ∃𝑓 ∈ ( L ‘𝐴)∃𝑔 ∈ ( L ‘𝐵)𝑒 = (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔))} ∪ {ℎ ∣ ∃𝑖 ∈ ( R ‘𝐴)∃𝑗 ∈ ( R ‘𝐵)ℎ = (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗))}) \|s ({𝑘 ∣ ∃𝑙 ∈ ( L ‘𝐴)∃𝑚 ∈ ( R ‘𝐵)𝑘 = (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚))} ∪ {𝑛 ∣ ∃𝑥 ∈ ( R ‘𝐴)∃𝑦 ∈ ( L ‘𝐵)𝑛 = (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦))})) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))}) \|s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})))
639	10, 638	eqtrd 2776	1 ⊢ (𝜑 → (𝐴 ·_s 𝐵) = (({𝑎 ∣ ∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 𝑎 = (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞))} ∪ {𝑏 ∣ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 𝑏 = (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠))}) \|s ({𝑐 ∣ ∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 𝑐 = (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢))} ∪ {𝑑 ∣ ∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 𝑑 = (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤))})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∀wal 1537 = wceq 1539 ∃wex 1778 ∈ wcel 2107 {cab 2713 ∀wral 3060 ∃wrex 3069 ∪ cun 3948 ⊆ wss 3950 {csn 4625 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 No csur 27685 ≤s csle 27790 <<s csslt 27826 |s cscut 27828 L cleft 27885 R cright 27886 +_s cadds 27993 -_s csubs 28053 ·_s cmuls 28133

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-ot 4634 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-1o 8507 df-2o 8508 df-nadd 8705 df-no 27688 df-slt 27689 df-bday 27690 df-sle 27791 df-sslt 27827 df-scut 27829 df-0s 27870 df-made 27887 df-old 27888 df-left 27890 df-right 27891 df-norec 27972 df-norec2 27983 df-adds 27994 df-negs 28054 df-subs 28055 df-muls 28134

This theorem is referenced by: mulsunif 28177

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