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Theorem mulsuniflem 28149

Description: Lemma for mulsunif 28150. 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	cutscld 27783	. . . 4 ⊢ (𝜑 → (𝐿 \|s 𝑅) ∈ No )
4	1, 3	eqeltrd 2837	. . 3 ⊢ (𝜑 → 𝐴 ∈ No )
5		mulsuniflem.4	. . . 4 ⊢ (𝜑 → 𝐵 = (𝑀 \|s 𝑆))
6		mulsuniflem.2	. . . . 5 ⊢ (𝜑 → 𝑀 <<s 𝑆)
7	6	cutscld 27783	. . . 4 ⊢ (𝜑 → (𝑀 \|s 𝑆) ∈ No )
8	5, 7	eqeltrd 2837	. . 3 ⊢ (𝜑 → 𝐵 ∈ No )
9		mulsval 28109	. . 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 585	. 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	mulcut2 28133	. . 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 27925	. . . . . 6 ⊢ (𝜑 → ∀𝑓 ∈ ( L ‘𝐴)∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝)
13	6, 5	cofcutr1d 27925	. . . . . . . . . 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 3209	. . . . . . . . . . . . . . 15 ⊢ (∃𝑝 ∈ 𝐿 ∃𝑞 ∈ 𝑀 (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞) ↔ (∃𝑝 ∈ 𝐿 𝑓 ≤s 𝑝 ∧ ∃𝑞 ∈ 𝑀 𝑔 ≤s 𝑞))
16		simprl 771	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑓 ∈ ( L ‘𝐴))
17	16	leftnod 27880	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑓 ∈ No )
18	17	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → 𝑓 ∈ No )
19	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → 𝐵 ∈ No )
20	18, 19	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (𝑓 ·_s 𝐵) ∈ No )
21	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → 𝐴 ∈ No )
22		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑔 ∈ ( L ‘𝐵))
23	22	leftnod 27880	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑔 ∈ No )
24	23	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → 𝑔 ∈ No )
25	21, 24	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (𝐴 ·_s 𝑔) ∈ No )
26	20, 25	addscld 27980	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → ((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) ∈ No )
27	18, 24	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (𝑓 ·_s 𝑔) ∈ No )
28	26, 27	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ∈ No )
29	28	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . 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 )
30		sltsss1 27765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐿 <<s 𝑅 → 𝐿 ⊆ No )
31	2, 30	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝐿 ⊆ No )
32	31	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝐿 ⊆ No )
33		simprl 771	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑝 ∈ 𝐿)
34	32, 33	sseldd 3935	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑝 ∈ No )
35	34	adantrl 717	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → 𝑝 ∈ No )
36	35, 19	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (𝑝 ·_s 𝐵) ∈ No )
37	36, 25	addscld 27980	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → ((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) ∈ No )
38	35, 24	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (𝑝 ·_s 𝑔) ∈ No )
39	37, 38	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑝 ·_s 𝑔)) ∈ No )
40	39	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . 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 )
41		sltsss1 27765	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑀 <<s 𝑆 → 𝑀 ⊆ No )
42	6, 41	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑀 ⊆ No )
43	42	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑀 ⊆ No )
44		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑞 ∈ 𝑀)
45	43, 44	sseldd 3935	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑞 ∈ No )
46	45	adantrl 717	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → 𝑞 ∈ No )
47	21, 46	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (𝐴 ·_s 𝑞) ∈ No )
48	36, 47	addscld 27980	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → ((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) ∈ No )
49	35, 46	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (𝑝 ·_s 𝑞) ∈ No )
50	48, 49	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) ∈ No )
51	50	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . 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 )
52	17	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑓 ∈ No )
53	35	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑝 ∈ No )
54	23	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑔 ∈ No )
55	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝐵 ∈ No )
56		simprrl 781	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞))) → 𝑓 ≤s 𝑝)
57	56	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑓 ≤s 𝑝)
58	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝐵 ∈ No )
59		sltsleft 27860	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐵 ∈ No → ( L ‘𝐵) <<s {𝐵})
60	8, 59	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → ( L ‘𝐵) <<s {𝐵})
61	60	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → ( L ‘𝐵) <<s {𝐵})
62		snidg 4618	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐵 ∈ No → 𝐵 ∈ {𝐵})
63	8, 62	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝐵 ∈ {𝐵})
64	63	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝐵 ∈ {𝐵})
65	61, 22, 64	sltssepcd 27772	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑔 <s 𝐵)
66	23, 58, 65	ltlesd 27745	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵))) → 𝑔 ≤s 𝐵)
67	66	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑔 ≤s 𝐵)
68	52, 53, 54, 55, 57, 67	lemulsd 28138	. . . . . . . . . . . . . . . . . . . . . . . 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)))
69	20, 27	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → ((𝑓 ·_s 𝐵) -_s (𝑓 ·_s 𝑔)) ∈ No )
70	36, 38	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → ((𝑝 ·_s 𝐵) -_s (𝑝 ·_s 𝑔)) ∈ No )
71	69, 70, 25	leadds1d 27995	. . . . . . . . . . . . . . . . . . . . . . . . 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))))
72	71	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . 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))))
73	68, 72	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)))
74	20, 25, 27	addsubsd 28082	. . . . . . . . . . . . . . . . . . . . . . . 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)))
75	74	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . 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)))
76	36, 25, 38	addsubsd 28082	. . . . . . . . . . . . . . . . . . . . . . . 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)))
77	76	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . 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)))
78	73, 75, 77	3brtr4d 5131	. . . . . . . . . . . . . . . . . . . . . 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)))
79	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝐴 ∈ No )
80	46	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑞 ∈ No )
81	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝐴 ∈ No )
82		cutcuts 27781	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐿 <<s 𝑅 → ((𝐿 \|s 𝑅) ∈ No ∧ 𝐿 <<s {(𝐿 \|s 𝑅)} ∧ {(𝐿 \|s 𝑅)} <<s 𝑅))
83	2, 82	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → ((𝐿 \|s 𝑅) ∈ No ∧ 𝐿 <<s {(𝐿 \|s 𝑅)} ∧ {(𝐿 \|s 𝑅)} <<s 𝑅))
84	83	simp2d 1144	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝐿 <<s {(𝐿 \|s 𝑅)})
85	84	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝐿 <<s {(𝐿 \|s 𝑅)})
86		ovex 7393	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐿 \|s 𝑅) ∈ V
87	86	snid 4620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝐿 \|s 𝑅) ∈ {(𝐿 \|s 𝑅)}
88	1, 87	eqeltrdi 2845	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝐴 ∈ {(𝐿 \|s 𝑅)})
89	88	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝐴 ∈ {(𝐿 \|s 𝑅)})
90	85, 33, 89	sltssepcd 27772	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑝 <s 𝐴)
91	34, 81, 90	ltlesd 27745	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀)) → 𝑝 ≤s 𝐴)
92	91	adantrl 717	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → 𝑝 ≤s 𝐴)
93	92	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑝 ≤s 𝐴)
94		simprrr 782	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞))) → 𝑔 ≤s 𝑞)
95	94	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ ((𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀) ∧ (𝑓 ≤s 𝑝 ∧ 𝑔 ≤s 𝑞)))) → 𝑔 ≤s 𝑞)
96	53, 79, 54, 80, 93, 95	lemulsd 28138	. . . . . . . . . . . . . . . . . . . . . . . 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)))
97	49, 47, 38, 25	lesubsubs3bd 28088	. . . . . . . . . . . . . . . . . . . . . . . . . 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))))
98	25, 38	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → ((𝐴 ·_s 𝑔) -_s (𝑝 ·_s 𝑔)) ∈ No )
99	47, 49	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑓 ∈ ( L ‘𝐴) ∧ 𝑔 ∈ ( L ‘𝐵)) ∧ (𝑝 ∈ 𝐿 ∧ 𝑞 ∈ 𝑀))) → ((𝐴 ·_s 𝑞) -_s (𝑝 ·_s 𝑞)) ∈ No )
100	98, 99, 36	leadds2d 27996	. . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
101	97, 100	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)))))
102	101	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . 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)))))
103	96, 102	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))))
104	36, 25, 38	addsubsassd 28081	. . . . . . . . . . . . . . . . . . . . . . . 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))))
105	104	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . 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))))
106	36, 47, 49	addsubsassd 28081	. . . . . . . . . . . . . . . . . . . . . . . 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))))
107	106	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . 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))))
108	103, 105, 107	3brtr4d 5131	. . . . . . . . . . . . . . . . . . . . . 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)))
109	29, 40, 51, 78, 108	lestrd 27738	. . . . . . . . . . . . . . . . . . . . 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)))
110	109	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)))
111	110	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))))
112	111	reximdvva 3185	. . . . . . . . . . . . . . . . . 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))))
113	112	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)))))
114	113	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)))))
115	114	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))))
116	15, 115	sylan2br 596	. . . . . . . . . . . . . 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))))
117	116	an4s 661	. . . . . . . . . . . . 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))))
118	117	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)))
119	118	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)))
120	119	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))))
121	120	ralimdva 3149	. . . . . . . . 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))))
122	14, 121	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)))
123	122	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))))
124	123	ralimdva 3149	. . . . . 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))))
125	12, 124	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)))
126		eqeq1 2741	. . . . . . . . . 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))))
127	126	2rexbidv 3202	. . . . . . . . 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))))
128	127	rexab 3654	. . . . . . . 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))
129		r19.41vv 3207	. . . . . . . . 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))
130	129	exbii 1850	. . . . . . . 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))
131		rexcom4 3264	. . . . . . . . 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))
132		rexcom4 3264	. . . . . . . . . . 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))
133		ovex 7393	. . . . . . . . . . . . 13 ⊢ (((𝑝 ·_s 𝐵) +_s (𝐴 ·_s 𝑞)) -_s (𝑝 ·_s 𝑞)) ∈ V
134		breq2 5103	. . . . . . . . . . . . 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))))
135	133, 134	ceqsexv 3491	. . . . . . . . . . . 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)))
136	135	rexbii 3084	. . . . . . . . . . 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)))
137	132, 136	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)))
138	137	rexbii 3084	. . . . . . . . 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)))
139	131, 138	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)))
140	128, 130, 139	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)))
141		ssun1 4131	. . . . . . . 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))})
142		ssrexv 4004	. . . . . . . 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))
143	141, 142	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)
144	140, 143	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)
145	144	2ralimi 3107	. . . . 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)
146	125, 145	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)
147	2, 1	cofcutr2d 27926	. . . . . 6 ⊢ (𝜑 → ∀𝑖 ∈ ( R ‘𝐴)∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖)
148	6, 5	cofcutr2d 27926	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑗 ∈ ( R ‘𝐵)∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑗)
149	148	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ ∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖)) → ∀𝑗 ∈ ( R ‘𝐵)∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑗)
150		reeanv 3209	. . . . . . . . . . . . . . 15 ⊢ (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗) ↔ (∃𝑟 ∈ 𝑅 𝑟 ≤s 𝑖 ∧ ∃𝑠 ∈ 𝑆 𝑠 ≤s 𝑗))
151		simprl 771	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝑖 ∈ ( R ‘𝐴))
152	151	rightnod 27882	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝑖 ∈ No )
153	152	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝑖 ∈ No )
154	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝐵 ∈ No )
155	153, 154	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (𝑖 ·_s 𝐵) ∈ No )
156	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝐴 ∈ No )
157		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝑗 ∈ ( R ‘𝐵))
158	157	rightnod 27882	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝑗 ∈ No )
159	158	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝑗 ∈ No )
160	156, 159	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (𝐴 ·_s 𝑗) ∈ No )
161	155, 160	addscld 27980	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) ∈ No )
162	153, 159	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (𝑖 ·_s 𝑗) ∈ No )
163	161, 162	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ∈ No )
164	163	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . 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 )
165		sltsss2 27766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐿 <<s 𝑅 → 𝑅 ⊆ No )
166	2, 165	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑅 ⊆ No )
167	166	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑅 ⊆ No )
168		simprl 771	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑟 ∈ 𝑅)
169	167, 168	sseldd 3935	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑟 ∈ No )
170	169	adantrl 717	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝑟 ∈ No )
171	170, 154	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (𝑟 ·_s 𝐵) ∈ No )
172	171, 160	addscld 27980	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) ∈ No )
173	170, 159	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (𝑟 ·_s 𝑗) ∈ No )
174	172, 173	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑟 ·_s 𝑗)) ∈ No )
175	174	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . 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 )
176		sltsss2 27766	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑀 <<s 𝑆 → 𝑆 ⊆ No )
177	6, 176	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝑆 ⊆ No )
178	177	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑆 ⊆ No )
179		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑠 ∈ 𝑆)
180	178, 179	sseldd 3935	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝑠 ∈ No )
181	180	adantrl 717	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝑠 ∈ No )
182	156, 181	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (𝐴 ·_s 𝑠) ∈ No )
183	171, 182	addscld 27980	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) ∈ No )
184	169, 180	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → (𝑟 ·_s 𝑠) ∈ No )
185	184	adantrl 717	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (𝑟 ·_s 𝑠) ∈ No )
186	183, 185	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) ∈ No )
187	186	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . 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 )
188	170	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝑟 ∈ No )
189	152	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝑖 ∈ No )
190	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝐵 ∈ No )
191	158	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝑗 ∈ No )
192		simprrl 781	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗))) → 𝑟 ≤s 𝑖)
193	192	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝑟 ≤s 𝑖)
194	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝐵 ∈ No )
195		sltsright 27861	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐵 ∈ No → {𝐵} <<s ( R ‘𝐵))
196	8, 195	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → {𝐵} <<s ( R ‘𝐵))
197	196	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → {𝐵} <<s ( R ‘𝐵))
198	63	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝐵 ∈ {𝐵})
199	197, 198, 157	sltssepcd 27772	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝐵 <s 𝑗)
200	194, 158, 199	ltlesd 27745	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵))) → 𝐵 ≤s 𝑗)
201	200	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝐵 ≤s 𝑗)
202	188, 189, 190, 191, 193, 201	lemulsd 28138	. . . . . . . . . . . . . . . . . . . . . . . 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)))
203	173, 171, 162, 155	lesubsubs2bd 28087	. . . . . . . . . . . . . . . . . . . . . . . . . 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))))
204	155, 162	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((𝑖 ·_s 𝐵) -_s (𝑖 ·_s 𝑗)) ∈ No )
205	171, 173	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((𝑟 ·_s 𝐵) -_s (𝑟 ·_s 𝑗)) ∈ No )
206	204, 205, 160	leadds1d 27995	. . . . . . . . . . . . . . . . . . . . . . . . . 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))))
207	203, 206	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))))
208	207	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . 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))))
209	202, 208	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)))
210	155, 160, 162	addsubsd 28082	. . . . . . . . . . . . . . . . . . . . . . . 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)))
211	210	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . 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)))
212	171, 160, 173	addsubsd 28082	. . . . . . . . . . . . . . . . . . . . . . . 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)))
213	212	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . 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)))
214	209, 211, 213	3brtr4d 5131	. . . . . . . . . . . . . . . . . . . . . 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)))
215	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝐴 ∈ No )
216	181	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝑠 ∈ No )
217	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝐴 ∈ No )
218	83	simp3d 1145	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → {(𝐿 \|s 𝑅)} <<s 𝑅)
219	218	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → {(𝐿 \|s 𝑅)} <<s 𝑅)
220	88	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝐴 ∈ {(𝐿 \|s 𝑅)})
221	219, 220, 168	sltssepcd 27772	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝐴 <s 𝑟)
222	217, 169, 221	ltlesd 27745	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆)) → 𝐴 ≤s 𝑟)
223	222	adantrl 717	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → 𝐴 ≤s 𝑟)
224	223	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝐴 ≤s 𝑟)
225		simprrr 782	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗))) → 𝑠 ≤s 𝑗)
226	225	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) ∧ (𝑟 ≤s 𝑖 ∧ 𝑠 ≤s 𝑗)))) → 𝑠 ≤s 𝑗)
227	215, 188, 216, 191, 224, 226	lemulsd 28138	. . . . . . . . . . . . . . . . . . . . . . . 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)))
228	160, 173, 182, 185	lesubsubsbd 28086	. . . . . . . . . . . . . . . . . . . . . . . . . 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))))
229	160, 173	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((𝐴 ·_s 𝑗) -_s (𝑟 ·_s 𝑗)) ∈ No )
230	182, 185	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑖 ∈ ( R ‘𝐴) ∧ 𝑗 ∈ ( R ‘𝐵)) ∧ (𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆))) → ((𝐴 ·_s 𝑠) -_s (𝑟 ·_s 𝑠)) ∈ No )
231	229, 230, 171	leadds2d 27996	. . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
232	228, 231	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)))))
233	232	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . 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)))))
234	227, 233	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))))
235	171, 160, 173	addsubsassd 28081	. . . . . . . . . . . . . . . . . . . . . . . 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))))
236	235	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . 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))))
237	171, 182, 185	addsubsassd 28081	. . . . . . . . . . . . . . . . . . . . . . . 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))))
238	237	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . 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))))
239	234, 236, 238	3brtr4d 5131	. . . . . . . . . . . . . . . . . . . . . 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)))
240	164, 175, 187, 214, 239	lestrd 27738	. . . . . . . . . . . . . . . . . . . . 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)))
241	240	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)))
242	241	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))))
243	242	reximdvva 3185	. . . . . . . . . . . . . . . . . 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))))
244	243	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)))))
245	244	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)))))
246	245	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))))
247	150, 246	sylan2br 596	. . . . . . . . . . . . . 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))))
248	247	an4s 661	. . . . . . . . . . . . 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))))
249	248	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)))
250	249	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)))
251	250	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))))
252	251	ralimdva 3149	. . . . . . . . 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))))
253	149, 252	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)))
254	253	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))))
255	254	ralimdva 3149	. . . . . 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))))
256	147, 255	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)))
257		eqeq1 2741	. . . . . . . . . 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))))
258	257	2rexbidv 3202	. . . . . . . . 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))))
259	258	rexab 3654	. . . . . . . 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))
260		r19.41vv 3207	. . . . . . . . 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))
261	260	exbii 1850	. . . . . . . 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))
262		rexcom4 3264	. . . . . . . . 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))
263		rexcom4 3264	. . . . . . . . . . 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))
264		ovex 7393	. . . . . . . . . . . . 13 ⊢ (((𝑟 ·_s 𝐵) +_s (𝐴 ·_s 𝑠)) -_s (𝑟 ·_s 𝑠)) ∈ V
265		breq2 5103	. . . . . . . . . . . . 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))))
266	264, 265	ceqsexv 3491	. . . . . . . . . . . 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)))
267	266	rexbii 3084	. . . . . . . . . . 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)))
268	263, 267	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)))
269	268	rexbii 3084	. . . . . . . . 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)))
270	262, 269	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)))
271	259, 261, 270	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)))
272		ssun2 4132	. . . . . . . 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))})
273		ssrexv 4004	. . . . . . . 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))
274	272, 273	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)
275	271, 274	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)
276	275	2ralimi 3107	. . . . 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)
277	256, 276	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)
278		ralunb 4150	. . . . 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))
279		eqeq1 2741	. . . . . . . . 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))))
280	279	2rexbidv 3202	. . . . . . . 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))))
281	280	ralab 3652	. . . . . . 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))
282		r19.23v 3164	. . . . . . . . . 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))
283	282	ralbii 3083	. . . . . . . . 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))
284		r19.23v 3164	. . . . . . . . 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))
285	283, 284	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))
286	285	albii 1821	. . . . . . 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))
287		ralcom4 3263	. . . . . . . 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))
288		ralcom4 3263	. . . . . . . . . 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))
289		ovex 7393	. . . . . . . . . . . 12 ⊢ (((𝑓 ·_s 𝐵) +_s (𝐴 ·_s 𝑔)) -_s (𝑓 ·_s 𝑔)) ∈ V
290		breq1 5102	. . . . . . . . . . . . 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))
291	290	rexbidv 3161	. . . . . . . . . . . 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))
292	289, 291	ceqsalv 3481	. . . . . . . . . . 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)
293	292	ralbii 3083	. . . . . . . . . 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)
294	288, 293	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)
295	294	ralbii 3083	. . . . . . . 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)
296	287, 295	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)
297	281, 286, 296	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)
298		eqeq1 2741	. . . . . . . . 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))))
299	298	2rexbidv 3202	. . . . . . . 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))))
300	299	ralab 3652	. . . . . . 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))
301		r19.23v 3164	. . . . . . . . . 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))
302	301	ralbii 3083	. . . . . . . . 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))
303		r19.23v 3164	. . . . . . . . 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))
304	302, 303	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))
305	304	albii 1821	. . . . . . 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))
306		ralcom4 3263	. . . . . . . 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))
307		ralcom4 3263	. . . . . . . . . 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))
308		ovex 7393	. . . . . . . . . . . 12 ⊢ (((𝑖 ·_s 𝐵) +_s (𝐴 ·_s 𝑗)) -_s (𝑖 ·_s 𝑗)) ∈ V
309		breq1 5102	. . . . . . . . . . . . 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))
310	309	rexbidv 3161	. . . . . . . . . . . 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))
311	308, 310	ceqsalv 3481	. . . . . . . . . . 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)
312	311	ralbii 3083	. . . . . . . . . 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)
313	307, 312	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)
314	313	ralbii 3083	. . . . . . . 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)
315	306, 314	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)
316	300, 305, 315	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)
317	297, 316	anbi12i 629	. . . . 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))
318	278, 317	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))
319	146, 277, 318	sylanbrc 584	. . 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)
320	2, 1	cofcutr1d 27925	. . . . . 6 ⊢ (𝜑 → ∀𝑙 ∈ ( L ‘𝐴)∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡)
321	6, 5	cofcutr2d 27926	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ ( R ‘𝐵)∃𝑢 ∈ 𝑆 𝑢 ≤s 𝑚)
322	321	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ ∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡)) → ∀𝑚 ∈ ( R ‘𝐵)∃𝑢 ∈ 𝑆 𝑢 ≤s 𝑚)
323		reeanv 3209	. . . . . . . . . . . . . . 15 ⊢ (∃𝑡 ∈ 𝐿 ∃𝑢 ∈ 𝑆 (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚) ↔ (∃𝑡 ∈ 𝐿 𝑙 ≤s 𝑡 ∧ ∃𝑢 ∈ 𝑆 𝑢 ≤s 𝑚))
324	31	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐿 ⊆ No )
325		simprl 771	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑡 ∈ 𝐿)
326	324, 325	sseldd 3935	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑡 ∈ No )
327	326	adantrl 717	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → 𝑡 ∈ No )
328	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → 𝐵 ∈ No )
329	327, 328	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (𝑡 ·_s 𝐵) ∈ No )
330	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → 𝐴 ∈ No )
331	177	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑆 ⊆ No )
332		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑢 ∈ 𝑆)
333	331, 332	sseldd 3935	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝑢 ∈ No )
334	333	adantrl 717	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → 𝑢 ∈ No )
335	330, 334	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (𝐴 ·_s 𝑢) ∈ No )
336	329, 335	addscld 27980	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → ((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) ∈ No )
337	327, 334	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (𝑡 ·_s 𝑢) ∈ No )
338	336, 337	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ∈ No )
339	338	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . 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 )
340		simprl 771	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑙 ∈ ( L ‘𝐴))
341	340	leftnod 27880	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑙 ∈ No )
342	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝐵 ∈ No )
343	341, 342	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → (𝑙 ·_s 𝐵) ∈ No )
344	343	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (𝑙 ·_s 𝐵) ∈ No )
345	344, 335	addscld 27980	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → ((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) ∈ No )
346	341	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → 𝑙 ∈ No )
347	346, 334	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (𝑙 ·_s 𝑢) ∈ No )
348	345, 347	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑙 ·_s 𝑢)) ∈ No )
349	348	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . 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 )
350	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝐴 ∈ No )
351		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑚 ∈ ( R ‘𝐵))
352	351	rightnod 27882	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑚 ∈ No )
353	350, 352	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → (𝐴 ·_s 𝑚) ∈ No )
354	353	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (𝐴 ·_s 𝑚) ∈ No )
355	344, 354	addscld 27980	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → ((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) ∈ No )
356	341, 352	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → (𝑙 ·_s 𝑚) ∈ No )
357	356	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (𝑙 ·_s 𝑚) ∈ No )
358	355, 357	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) ∈ No )
359	358	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . 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 )
360	341	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝑙 ∈ No )
361	327	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝑡 ∈ No )
362	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝐵 ∈ No )
363	334	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝑢 ∈ No )
364		simprrl 781	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚))) → 𝑙 ≤s 𝑡)
365	364	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝑙 ≤s 𝑡)
366	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 ∈ No )
367		cutcuts 27781	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑀 <<s 𝑆 → ((𝑀 \|s 𝑆) ∈ No ∧ 𝑀 <<s {(𝑀 \|s 𝑆)} ∧ {(𝑀 \|s 𝑆)} <<s 𝑆))
368	6, 367	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝜑 → ((𝑀 \|s 𝑆) ∈ No ∧ 𝑀 <<s {(𝑀 \|s 𝑆)} ∧ {(𝑀 \|s 𝑆)} <<s 𝑆))
369	368	simp3d 1145	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → {(𝑀 \|s 𝑆)} <<s 𝑆)
370	369	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → {(𝑀 \|s 𝑆)} <<s 𝑆)
371		ovex 7393	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑀 \|s 𝑆) ∈ V
372	371	snid 4620	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑀 \|s 𝑆) ∈ {(𝑀 \|s 𝑆)}
373	5, 372	eqeltrdi 2845	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → 𝐵 ∈ {(𝑀 \|s 𝑆)})
374	373	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 ∈ {(𝑀 \|s 𝑆)})
375	370, 374, 332	sltssepcd 27772	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 <s 𝑢)
376	366, 333, 375	ltlesd 27745	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆)) → 𝐵 ≤s 𝑢)
377	376	adantrl 717	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → 𝐵 ≤s 𝑢)
378	377	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝐵 ≤s 𝑢)
379	360, 361, 362, 363, 365, 378	lemulsd 28138	. . . . . . . . . . . . . . . . . . . . . . . 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)))
380	347, 344, 337, 329	lesubsubs2bd 28087	. . . . . . . . . . . . . . . . . . . . . . . . . 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))))
381	329, 337	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → ((𝑡 ·_s 𝐵) -_s (𝑡 ·_s 𝑢)) ∈ No )
382	344, 347	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → ((𝑙 ·_s 𝐵) -_s (𝑙 ·_s 𝑢)) ∈ No )
383	381, 382, 335	leadds1d 27995	. . . . . . . . . . . . . . . . . . . . . . . . . 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))))
384	380, 383	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))))
385	384	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . 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))))
386	379, 385	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)))
387	329, 335, 337	addsubsd 28082	. . . . . . . . . . . . . . . . . . . . . . . 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)))
388	387	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . 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)))
389	344, 335, 347	addsubsd 28082	. . . . . . . . . . . . . . . . . . . . . . . 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)))
390	389	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . 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)))
391	386, 388, 390	3brtr4d 5131	. . . . . . . . . . . . . . . . . . . . . 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)))
392	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝐴 ∈ No )
393	352	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝑚 ∈ No )
394		sltsleft 27860	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐴 ∈ No → ( L ‘𝐴) <<s {𝐴})
395	4, 394	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → ( L ‘𝐴) <<s {𝐴})
396	395	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → ( L ‘𝐴) <<s {𝐴})
397		snidg 4618	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝐴 ∈ No → 𝐴 ∈ {𝐴})
398	4, 397	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
399	398	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝐴 ∈ {𝐴})
400	396, 340, 399	sltssepcd 27772	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑙 <s 𝐴)
401	341, 350, 400	ltlesd 27745	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵))) → 𝑙 ≤s 𝐴)
402	401	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝑙 ≤s 𝐴)
403		simprrr 782	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚))) → 𝑢 ≤s 𝑚)
404	403	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ ((𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆) ∧ (𝑙 ≤s 𝑡 ∧ 𝑢 ≤s 𝑚)))) → 𝑢 ≤s 𝑚)
405	360, 392, 363, 393, 402, 404	lemulsd 28138	. . . . . . . . . . . . . . . . . . . . . . . 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)))
406	357, 354, 347, 335	lesubsubs3bd 28088	. . . . . . . . . . . . . . . . . . . . . . . . . 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))))
407	335, 347	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → ((𝐴 ·_s 𝑢) -_s (𝑙 ·_s 𝑢)) ∈ No )
408	354, 357	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑙 ∈ ( L ‘𝐴) ∧ 𝑚 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ 𝐿 ∧ 𝑢 ∈ 𝑆))) → ((𝐴 ·_s 𝑚) -_s (𝑙 ·_s 𝑚)) ∈ No )
409	407, 408, 344	leadds2d 27996	. . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
410	406, 409	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)))))
411	410	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . 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)))))
412	405, 411	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))))
413	344, 335, 347	addsubsassd 28081	. . . . . . . . . . . . . . . . . . . . . . . 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))))
414	413	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . 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))))
415	344, 354, 357	addsubsassd 28081	. . . . . . . . . . . . . . . . . . . . . . . 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))))
416	415	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . 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))))
417	412, 414, 416	3brtr4d 5131	. . . . . . . . . . . . . . . . . . . . . 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)))
418	339, 349, 359, 391, 417	lestrd 27738	. . . . . . . . . . . . . . . . . . . . 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)))
419	418	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)))
420	419	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))))
421	420	reximdvva 3185	. . . . . . . . . . . . . . . . . 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))))
422	421	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)))))
423	422	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)))))
424	423	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))))
425	323, 424	sylan2br 596	. . . . . . . . . . . . . 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))))
426	425	an4s 661	. . . . . . . . . . . . 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))))
427	426	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)))
428	427	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)))
429	428	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))))
430	429	ralimdva 3149	. . . . . . . . 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))))
431	322, 430	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)))
432	431	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))))
433	432	ralimdva 3149	. . . . . 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))))
434	320, 433	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)))
435		eqeq1 2741	. . . . . . . . . 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))))
436	435	2rexbidv 3202	. . . . . . . . 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))))
437	436	rexab 3654	. . . . . . . 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))))
438		r19.41vv 3207	. . . . . . . . 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))))
439	438	exbii 1850	. . . . . . . 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))))
440		rexcom4 3264	. . . . . . . . 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))))
441		rexcom4 3264	. . . . . . . . . . 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))))
442		ovex 7393	. . . . . . . . . . . . 13 ⊢ (((𝑡 ·_s 𝐵) +_s (𝐴 ·_s 𝑢)) -_s (𝑡 ·_s 𝑢)) ∈ V
443		breq1 5102	. . . . . . . . . . . . 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))))
444	442, 443	ceqsexv 3491	. . . . . . . . . . . 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)))
445	444	rexbii 3084	. . . . . . . . . . 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)))
446	441, 445	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)))
447	446	rexbii 3084	. . . . . . . . 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)))
448	440, 447	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)))
449	437, 439, 448	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)))
450		ssun1 4131	. . . . . . . 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))})
451		ssrexv 4004	. . . . . . . 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))))
452	450, 451	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)))
453	449, 452	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)))
454	453	2ralimi 3107	. . . . 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)))
455	434, 454	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)))
456	2, 1	cofcutr2d 27926	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ ( R ‘𝐴)∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥)
457	6, 5	cofcutr1d 27925	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑦 ∈ ( L ‘𝐵)∃𝑤 ∈ 𝑀 𝑦 ≤s 𝑤)
458	457	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ ∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥)) → ∀𝑦 ∈ ( L ‘𝐵)∃𝑤 ∈ 𝑀 𝑦 ≤s 𝑤)
459		reeanv 3209	. . . . . . . . . . . . . . 15 ⊢ (∃𝑣 ∈ 𝑅 ∃𝑤 ∈ 𝑀 (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤) ↔ (∃𝑣 ∈ 𝑅 𝑣 ≤s 𝑥 ∧ ∃𝑤 ∈ 𝑀 𝑦 ≤s 𝑤))
460	166	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑅 ⊆ No )
461		simprl 771	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑣 ∈ 𝑅)
462	460, 461	sseldd 3935	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑣 ∈ No )
463	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐵 ∈ No )
464	462, 463	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑣 ·_s 𝐵) ∈ No )
465	464	adantrl 717	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (𝑣 ·_s 𝐵) ∈ No )
466	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝐴 ∈ No )
467	42	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑀 ⊆ No )
468		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑤 ∈ 𝑀)
469	467, 468	sseldd 3935	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → 𝑤 ∈ No )
470	466, 469	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝐴 ·_s 𝑤) ∈ No )
471	470	adantrl 717	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (𝐴 ·_s 𝑤) ∈ No )
472	465, 471	addscld 27980	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → ((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) ∈ No )
473	462, 469	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀)) → (𝑣 ·_s 𝑤) ∈ No )
474	473	adantrl 717	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (𝑣 ·_s 𝑤) ∈ No )
475	472, 474	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ∈ No )
476	475	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . 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 )
477	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝐴 ∈ No )
478		simprr 773	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑦 ∈ ( L ‘𝐵))
479	478	leftnod 27880	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑦 ∈ No )
480	479	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝑦 ∈ No )
481	477, 480	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (𝐴 ·_s 𝑦) ∈ No )
482	465, 481	addscld 27980	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → ((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) ∈ No )
483	462	adantrl 717	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝑣 ∈ No )
484	483, 480	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (𝑣 ·_s 𝑦) ∈ No )
485	482, 484	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑣 ·_s 𝑦)) ∈ No )
486	485	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . 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 )
487		simprl 771	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑥 ∈ ( R ‘𝐴))
488	487	rightnod 27882	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑥 ∈ No )
489	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝐵 ∈ No )
490	488, 489	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → (𝑥 ·_s 𝐵) ∈ No )
491	490	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (𝑥 ·_s 𝐵) ∈ No )
492	491, 481	addscld 27980	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → ((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) ∈ No )
493	488, 479	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → (𝑥 ·_s 𝑦) ∈ No )
494	493	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (𝑥 ·_s 𝑦) ∈ No )
495	492, 494	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) ∈ No )
496	495	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . 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 )
497	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝐴 ∈ No )
498	483	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝑣 ∈ No )
499	479	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝑦 ∈ No )
500	469	adantrl 717	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝑤 ∈ No )
501	500	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝑤 ∈ No )
502	1	sneqd 4593	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝜑 → {𝐴} = {(𝐿 \|s 𝑅)})
503	502, 218	eqbrtrd 5121	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝜑 → {𝐴} <<s 𝑅)
504	503	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → {𝐴} <<s 𝑅)
505	477, 397	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝐴 ∈ {𝐴})
506		simprrl 781	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝑣 ∈ 𝑅)
507	504, 505, 506	sltssepcd 27772	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝐴 <s 𝑣)
508	477, 483, 507	ltlesd 27745	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → 𝐴 ≤s 𝑣)
509	508	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝐴 ≤s 𝑣)
510		simprrr 782	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤))) → 𝑦 ≤s 𝑤)
511	510	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝑦 ≤s 𝑤)
512	497, 498, 499, 501, 509, 511	lemulsd 28138	. . . . . . . . . . . . . . . . . . . . . . . 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)))
513	471, 474, 481, 484	lesubsubsbd 28086	. . . . . . . . . . . . . . . . . . . . . . . . . 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))))
514	471, 474	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → ((𝐴 ·_s 𝑤) -_s (𝑣 ·_s 𝑤)) ∈ No )
515	481, 484	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → ((𝐴 ·_s 𝑦) -_s (𝑣 ·_s 𝑦)) ∈ No )
516	514, 515, 465	leadds2d 27996	. . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
517	513, 516	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)))))
518	517	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . 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)))))
519	512, 518	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))))
520	465, 471, 474	addsubsassd 28081	. . . . . . . . . . . . . . . . . . . . . . . 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))))
521	520	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . 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))))
522	465, 481, 484	addsubsassd 28081	. . . . . . . . . . . . . . . . . . . . . . . 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))))
523	522	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . 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))))
524	519, 521, 523	3brtr4d 5131	. . . . . . . . . . . . . . . . . . . . . 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)))
525	488	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝑥 ∈ No )
526	8	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝐵 ∈ No )
527		simprrl 781	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤))) → 𝑣 ≤s 𝑥)
528	527	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝑣 ≤s 𝑥)
529	489, 59	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → ( L ‘𝐵) <<s {𝐵})
530	63	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝐵 ∈ {𝐵})
531	529, 478, 530	sltssepcd 27772	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑦 <s 𝐵)
532	479, 489, 531	ltlesd 27745	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝑦 ≤s 𝐵)
533	532	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → 𝑦 ≤s 𝐵)
534	498, 525, 499, 526, 528, 533	lemulsd 28138	. . . . . . . . . . . . . . . . . . . . . . . 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)))
535	465, 484	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → ((𝑣 ·_s 𝐵) -_s (𝑣 ·_s 𝑦)) ∈ No )
536	535	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 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 )
537	491, 494	subscld 28063	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀))) → ((𝑥 ·_s 𝐵) -_s (𝑥 ·_s 𝑦)) ∈ No )
538	537	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 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 )
539	481	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ ((𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵)) ∧ ((𝑣 ∈ 𝑅 ∧ 𝑤 ∈ 𝑀) ∧ (𝑣 ≤s 𝑥 ∧ 𝑦 ≤s 𝑤)))) → (𝐴 ·_s 𝑦) ∈ No )
540	536, 538, 539	leadds1d 27995	. . . . . . . . . . . . . . . . . . . . . . . 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))))
541	534, 540	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)))
542	465, 481, 484	addsubsd 28082	. . . . . . . . . . . . . . . . . . . . . . . 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)))
543	542	adantrrr 726	. . . . . . . . . . . . . . . . . . . . . . 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)))
544	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → 𝐴 ∈ No )
545	544, 479	mulscld 28135	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑥 ∈ ( R ‘𝐴) ∧ 𝑦 ∈ ( L ‘𝐵))) → (𝐴 ·_s 𝑦) ∈ No )
546	490, 545, 493	addsubsd 28082	. . . . . . . . . . . . . . . . . . . . . . . 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)))
547	546	adantrr 718	. . . . . . . . . . . . . . . . . . . . . . 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)))
548	541, 543, 547	3brtr4d 5131	. . . . . . . . . . . . . . . . . . . . . 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)))
549	476, 486, 496, 524, 548	lestrd 27738	. . . . . . . . . . . . . . . . . . . . 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)))
550	549	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)))
551	550	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))))
552	551	reximdvva 3185	. . . . . . . . . . . . . . . . . 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))))
553	552	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)))))
554	553	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)))))
555	554	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))))
556	459, 555	sylan2br 596	. . . . . . . . . . . . . 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))))
557	556	an4s 661	. . . . . . . . . . . . 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))))
558	557	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)))
559	558	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)))
560	559	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))))
561	560	ralimdva 3149	. . . . . . . . 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))))
562	458, 561	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)))
563	562	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))))
564	563	ralimdva 3149	. . . . . 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))))
565	456, 564	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)))
566		eqeq1 2741	. . . . . . . . . 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))))
567	566	2rexbidv 3202	. . . . . . . . 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))))
568	567	rexab 3654	. . . . . . . 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))))
569		r19.41vv 3207	. . . . . . . . 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))))
570	569	exbii 1850	. . . . . . . 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))))
571		rexcom4 3264	. . . . . . . . 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))))
572		rexcom4 3264	. . . . . . . . . . 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))))
573		ovex 7393	. . . . . . . . . . . . 13 ⊢ (((𝑣 ·_s 𝐵) +_s (𝐴 ·_s 𝑤)) -_s (𝑣 ·_s 𝑤)) ∈ V
574		breq1 5102	. . . . . . . . . . . . 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))))
575	573, 574	ceqsexv 3491	. . . . . . . . . . . 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)))
576	575	rexbii 3084	. . . . . . . . . . 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)))
577	572, 576	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)))
578	577	rexbii 3084	. . . . . . . . 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)))
579	571, 578	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)))
580	568, 570, 579	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)))
581		ssun2 4132	. . . . . . . 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))})
582		ssrexv 4004	. . . . . . . 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))))
583	581, 582	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)))
584	580, 583	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)))
585	584	2ralimi 3107	. . . . 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)))
586	565, 585	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)))
587		ralunb 4150	. . . . 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))
588		eqeq1 2741	. . . . . . . . 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))))
589	588	2rexbidv 3202	. . . . . . . 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))))
590	589	ralab 3652	. . . . . . 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))
591		r19.23v 3164	. . . . . . . . . 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))
592	591	ralbii 3083	. . . . . . . . 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))
593		r19.23v 3164	. . . . . . . . 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))
594	592, 593	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))
595	594	albii 1821	. . . . . . 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))
596		ralcom4 3263	. . . . . . . 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))
597		ralcom4 3263	. . . . . . . . . 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))
598		ovex 7393	. . . . . . . . . . . 12 ⊢ (((𝑙 ·_s 𝐵) +_s (𝐴 ·_s 𝑚)) -_s (𝑙 ·_s 𝑚)) ∈ V
599		breq2 5103	. . . . . . . . . . . . 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))))
600	599	rexbidv 3161	. . . . . . . . . . . 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))))
601	598, 600	ceqsalv 3481	. . . . . . . . . . 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)))
602	601	ralbii 3083	. . . . . . . . . 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)))
603	597, 602	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)))
604	603	ralbii 3083	. . . . . . . 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)))
605	596, 604	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)))
606	590, 595, 605	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)))
607		eqeq1 2741	. . . . . . . . 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))))
608	607	2rexbidv 3202	. . . . . . . 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))))
609	608	ralab 3652	. . . . . . 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))
610		r19.23v 3164	. . . . . . . . . 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))
611	610	ralbii 3083	. . . . . . . . 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))
612		r19.23v 3164	. . . . . . . . 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))
613	611, 612	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))
614	613	albii 1821	. . . . . . 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))
615		ralcom4 3263	. . . . . . . 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))
616		ralcom4 3263	. . . . . . . . . 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))
617		ovex 7393	. . . . . . . . . . . 12 ⊢ (((𝑥 ·_s 𝐵) +_s (𝐴 ·_s 𝑦)) -_s (𝑥 ·_s 𝑦)) ∈ V
618		breq2 5103	. . . . . . . . . . . . 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))))
619	618	rexbidv 3161	. . . . . . . . . . . 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))))
620	617, 619	ceqsalv 3481	. . . . . . . . . . 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)))
621	620	ralbii 3083	. . . . . . . . . 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)))
622	616, 621	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)))
623	622	ralbii 3083	. . . . . . . 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)))
624	615, 623	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)))
625	609, 614, 624	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)))
626	606, 625	anbi12i 629	. . . . 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))))
627	587, 626	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))))
628	455, 586, 627	sylanbrc 584	. . 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)
629	2, 6, 1, 5	sltmuls1 28147	. . . 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)})
630	10	sneqd 4593	. . . 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))}))})
631	629, 630	breqtrd 5125	. . 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))}))})
632	2, 6, 1, 5	sltmuls2 28148	. . . 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))}))
633	630, 632	eqbrtrrd 5123	. . 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))}))
634	11, 319, 628, 631, 633	cofcut1d 27921	. 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))})))
635	10, 634	eqtrd 2772	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 1087 ∀wal 1540 = wceq 1542 ∃wex 1781 ∈ wcel 2114 {cab 2715 ∀wral 3052 ∃wrex 3061 ∪ cun 3900 ⊆ wss 3902 {csn 4581 class class class wbr 5099 ‘cfv 6493 (class class class)co 7360 No csur 27611 ≤s cles 27716 <<s cslts 27757 |s ccuts 27759 L cleft 27825 R cright 27826 +_s cadds 27959 -_s csubs 28020 ·_s cmuls 28106

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-ot 4590 df-uni 4865 df-int 4904 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-1o 8399 df-2o 8400 df-nadd 8596 df-no 27614 df-lts 27615 df-bday 27616 df-les 27717 df-slts 27758 df-cuts 27760 df-0s 27807 df-made 27827 df-old 27828 df-left 27830 df-right 27831 df-norec 27938 df-norec2 27949 df-adds 27960 df-negs 28021 df-subs 28022 df-muls 28107

This theorem is referenced by: mulsunif 28150

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