Theorem mulsproplem9 28124

Description: Lemma for surreal multiplication. Show that the cut involved in surreal multiplication makes sense. (Contributed by Scott Fenton, 5-Mar-2025.)

Hypotheses

Ref	Expression
mulsproplem.1	⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))
mulsproplem9.1	⊢ (𝜑 → 𝐴 ∈ No )
mulsproplem9.2	⊢ (𝜑 → 𝐵 ∈ No )

Assertion

Ref	Expression
mulsproplem9	⊢ (𝜑 → ({𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) <<s ({𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}))

Distinct variable groups:   𝐴,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐵,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐶,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐷,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐸,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐹,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐴,𝑔,ℎ,𝑖,𝑗,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑣,𝑤,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝐵,𝑔,ℎ,𝑖,𝑗,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑣,𝑤   𝜑,𝑔,ℎ,𝑖,𝑗,𝑝,𝑞,𝑟,𝑠,𝑡,𝑢,𝑣,𝑤

Allowed substitution hints:   𝜑(𝑒,𝑓,𝑎,𝑏,𝑐,𝑑)   𝐶(𝑤,𝑣,𝑢,𝑡,𝑔,ℎ,𝑖,𝑗,𝑠,𝑟,𝑞,𝑝)   𝐷(𝑤,𝑣,𝑢,𝑡,𝑔,ℎ,𝑖,𝑗,𝑠,𝑟,𝑞,𝑝)   𝐸(𝑤,𝑣,𝑢,𝑡,𝑔,ℎ,𝑖,𝑗,𝑠,𝑟,𝑞,𝑝)   𝐹(𝑤,𝑣,𝑢,𝑡,𝑔,ℎ,𝑖,𝑗,𝑠,𝑟,𝑞,𝑝)

Proof of Theorem mulsproplem9

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . . . . 6 ⊢ (𝑝 ∈ ( L ‘𝐴), 𝑞 ∈ ( L ‘𝐵) ↦ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) = (𝑝 ∈ ( L ‘𝐴), 𝑞 ∈ ( L ‘𝐵) ↦ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))
2	1	rnmpo 7493	. . . . 5 ⊢ ran (𝑝 ∈ ( L ‘𝐴), 𝑞 ∈ ( L ‘𝐵) ↦ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) = {𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))}
3		fvex 6848	. . . . . . 7 ⊢ ( L ‘𝐴) ∈ V
4		fvex 6848	. . . . . . 7 ⊢ ( L ‘𝐵) ∈ V
5	3, 4	mpoex 8025	. . . . . 6 ⊢ (𝑝 ∈ ( L ‘𝐴), 𝑞 ∈ ( L ‘𝐵) ↦ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) ∈ V
6	5	rnex 7854	. . . . 5 ⊢ ran (𝑝 ∈ ( L ‘𝐴), 𝑞 ∈ ( L ‘𝐵) ↦ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) ∈ V
7	2, 6	eqeltrri 2834	. . . 4 ⊢ {𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∈ V
8		eqid 2737	. . . . . 6 ⊢ (𝑟 ∈ ( R ‘𝐴), 𝑠 ∈ ( R ‘𝐵) ↦ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) = (𝑟 ∈ ( R ‘𝐴), 𝑠 ∈ ( R ‘𝐵) ↦ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))
9	8	rnmpo 7493	. . . . 5 ⊢ ran (𝑟 ∈ ( R ‘𝐴), 𝑠 ∈ ( R ‘𝐵) ↦ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) = {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}
10		fvex 6848	. . . . . . 7 ⊢ ( R ‘𝐴) ∈ V
11		fvex 6848	. . . . . . 7 ⊢ ( R ‘𝐵) ∈ V
12	10, 11	mpoex 8025	. . . . . 6 ⊢ (𝑟 ∈ ( R ‘𝐴), 𝑠 ∈ ( R ‘𝐵) ↦ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) ∈ V
13	12	rnex 7854	. . . . 5 ⊢ ran (𝑟 ∈ ( R ‘𝐴), 𝑠 ∈ ( R ‘𝐵) ↦ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) ∈ V
14	9, 13	eqeltrri 2834	. . . 4 ⊢ {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))} ∈ V
15	7, 14	unex 7691	. . 3 ⊢ ({𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) ∈ V
16	15	a1i 11	. 2 ⊢ (𝜑 → ({𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) ∈ V)
17		eqid 2737	. . . . . 6 ⊢ (𝑡 ∈ ( L ‘𝐴), 𝑢 ∈ ( R ‘𝐵) ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) = (𝑡 ∈ ( L ‘𝐴), 𝑢 ∈ ( R ‘𝐵) ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)))
18	17	rnmpo 7493	. . . . 5 ⊢ ran (𝑡 ∈ ( L ‘𝐴), 𝑢 ∈ ( R ‘𝐵) ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) = {𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))}
19	3, 11	mpoex 8025	. . . . . 6 ⊢ (𝑡 ∈ ( L ‘𝐴), 𝑢 ∈ ( R ‘𝐵) ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) ∈ V
20	19	rnex 7854	. . . . 5 ⊢ ran (𝑡 ∈ ( L ‘𝐴), 𝑢 ∈ ( R ‘𝐵) ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) ∈ V
21	18, 20	eqeltrri 2834	. . . 4 ⊢ {𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∈ V
22		eqid 2737	. . . . . 6 ⊢ (𝑣 ∈ ( R ‘𝐴), 𝑤 ∈ ( L ‘𝐵) ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))) = (𝑣 ∈ ( R ‘𝐴), 𝑤 ∈ ( L ‘𝐵) ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))
23	22	rnmpo 7493	. . . . 5 ⊢ ran (𝑣 ∈ ( R ‘𝐴), 𝑤 ∈ ( L ‘𝐵) ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))) = {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}
24	10, 4	mpoex 8025	. . . . . 6 ⊢ (𝑣 ∈ ( R ‘𝐴), 𝑤 ∈ ( L ‘𝐵) ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))) ∈ V
25	24	rnex 7854	. . . . 5 ⊢ ran (𝑣 ∈ ( R ‘𝐴), 𝑤 ∈ ( L ‘𝐵) ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))) ∈ V
26	23, 25	eqeltrri 2834	. . . 4 ⊢ {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))} ∈ V
27	21, 26	unex 7691	. . 3 ⊢ ({𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) ∈ V
28	27	a1i 11	. 2 ⊢ (𝜑 → ({𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) ∈ V)
29		mulsproplem.1	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))
30	29	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))
31		simprl 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝑝 ∈ ( L ‘𝐴))
32	31	leftoldd 27879	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝑝 ∈ ( O ‘( bday ‘𝐴)))
33		mulsproplem9.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ No )
34	33	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝐵 ∈ No )
35	30, 32, 34	mulsproplem2 28117	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (𝑝 ·s 𝐵) ∈ No )
36		mulsproplem9.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ No )
37	36	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝐴 ∈ No )
38		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝑞 ∈ ( L ‘𝐵))
39	38	leftoldd 27879	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝑞 ∈ ( O ‘( bday ‘𝐵)))
40	30, 37, 39	mulsproplem3 28118	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (𝐴 ·s 𝑞) ∈ No )
41	35, 40	addscld 27980	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → ((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) ∈ No )
42	30, 32, 39	mulsproplem4 28119	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (𝑝 ·s 𝑞) ∈ No )
43	41, 42	subscld 28063	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∈ No )
44		eleq1 2825	. . . . . 6 ⊢ (𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) → (𝑔 ∈ No ↔ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∈ No ))
45	43, 44	syl5ibrcom 247	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) → 𝑔 ∈ No ))
46	45	rexlimdvva 3194	. . . 4 ⊢ (𝜑 → (∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) → 𝑔 ∈ No ))
47	46	abssdv 4020	. . 3 ⊢ (𝜑 → {𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ⊆ No )
48	29	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))
49		simprl 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝑟 ∈ ( R ‘𝐴))
50	49	rightoldd 27881	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝑟 ∈ ( O ‘( bday ‘𝐴)))
51	33	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝐵 ∈ No )
52	48, 50, 51	mulsproplem2 28117	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (𝑟 ·s 𝐵) ∈ No )
53	36	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝐴 ∈ No )
54		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝑠 ∈ ( R ‘𝐵))
55	54	rightoldd 27881	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝑠 ∈ ( O ‘( bday ‘𝐵)))
56	48, 53, 55	mulsproplem3 28118	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (𝐴 ·s 𝑠) ∈ No )
57	52, 56	addscld 27980	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → ((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) ∈ No )
58	48, 50, 55	mulsproplem4 28119	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (𝑟 ·s 𝑠) ∈ No )
59	57, 58	subscld 28063	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∈ No )
60		eleq1 2825	. . . . . 6 ⊢ (ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) → (ℎ ∈ No ↔ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∈ No ))
61	59, 60	syl5ibrcom 247	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) → ℎ ∈ No ))
62	61	rexlimdvva 3194	. . . 4 ⊢ (𝜑 → (∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) → ℎ ∈ No ))
63	62	abssdv 4020	. . 3 ⊢ (𝜑 → {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))} ⊆ No )
64	47, 63	unssd 4145	. 2 ⊢ (𝜑 → ({𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) ⊆ No )
65	29	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))
66		simprl 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝑡 ∈ ( L ‘𝐴))
67	66	leftoldd 27879	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝑡 ∈ ( O ‘( bday ‘𝐴)))
68	33	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝐵 ∈ No )
69	65, 67, 68	mulsproplem2 28117	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (𝑡 ·s 𝐵) ∈ No )
70	36	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝐴 ∈ No )
71		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝑢 ∈ ( R ‘𝐵))
72	71	rightoldd 27881	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝑢 ∈ ( O ‘( bday ‘𝐵)))
73	65, 70, 72	mulsproplem3 28118	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (𝐴 ·s 𝑢) ∈ No )
74	69, 73	addscld 27980	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → ((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) ∈ No )
75	65, 67, 72	mulsproplem4 28119	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (𝑡 ·s 𝑢) ∈ No )
76	74, 75	subscld 28063	. . . . . 6 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∈ No )
77		eleq1 2825	. . . . . 6 ⊢ (𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → (𝑖 ∈ No ↔ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∈ No ))
78	76, 77	syl5ibrcom 247	. . . . 5 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → 𝑖 ∈ No ))
79	78	rexlimdvva 3194	. . . 4 ⊢ (𝜑 → (∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → 𝑖 ∈ No ))
80	79	abssdv 4020	. . 3 ⊢ (𝜑 → {𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ⊆ No )
81	29	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))
82		simprl 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝑣 ∈ ( R ‘𝐴))
83	82	rightoldd 27881	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝑣 ∈ ( O ‘( bday ‘𝐴)))
84	33	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝐵 ∈ No )
85	81, 83, 84	mulsproplem2 28117	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (𝑣 ·s 𝐵) ∈ No )
86	36	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝐴 ∈ No )
87		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝑤 ∈ ( L ‘𝐵))
88	87	leftoldd 27879	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝑤 ∈ ( O ‘( bday ‘𝐵)))
89	81, 86, 88	mulsproplem3 28118	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (𝐴 ·s 𝑤) ∈ No )
90	85, 89	addscld 27980	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → ((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) ∈ No )
91	81, 83, 88	mulsproplem4 28119	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (𝑣 ·s 𝑤) ∈ No )
92	90, 91	subscld 28063	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∈ No )
93		eleq1 2825	. . . . . 6 ⊢ (𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → (𝑗 ∈ No ↔ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∈ No ))
94	92, 93	syl5ibrcom 247	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → 𝑗 ∈ No ))
95	94	rexlimdvva 3194	. . . 4 ⊢ (𝜑 → (∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → 𝑗 ∈ No ))
96	95	abssdv 4020	. . 3 ⊢ (𝜑 → {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))} ⊆ No )
97	80, 96	unssd 4145	. 2 ⊢ (𝜑 → ({𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) ⊆ No )
98		elun 4106	. . . . . . 7 ⊢ (𝑥 ∈ ({𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) ↔ (𝑥 ∈ {𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∨ 𝑥 ∈ {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}))
99		vex 3445	. . . . . . . . 9 ⊢ 𝑥 ∈ V
100		eqeq1 2741	. . . . . . . . . 10 ⊢ (𝑔 = 𝑥 → (𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ 𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))))
101	100	2rexbidv 3202	. . . . . . . . 9 ⊢ (𝑔 = 𝑥 → (∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ↔ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))))
102	99, 101	elab 3635	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ↔ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)))
103		eqeq1 2741	. . . . . . . . . 10 ⊢ (ℎ = 𝑥 → (ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ↔ 𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))))
104	103	2rexbidv 3202	. . . . . . . . 9 ⊢ (ℎ = 𝑥 → (∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ↔ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))))
105	99, 104	elab 3635	. . . . . . . 8 ⊢ (𝑥 ∈ {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))} ↔ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)))
106	102, 105	orbi12i 915	. . . . . . 7 ⊢ ((𝑥 ∈ {𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∨ 𝑥 ∈ {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) ↔ (∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∨ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))))
107	98, 106	bitri 275	. . . . . 6 ⊢ (𝑥 ∈ ({𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) ↔ (∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∨ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))))
108		elun 4106	. . . . . . 7 ⊢ (𝑦 ∈ ({𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) ↔ (𝑦 ∈ {𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∨ 𝑦 ∈ {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}))
109		vex 3445	. . . . . . . . 9 ⊢ 𝑦 ∈ V
110		eqeq1 2741	. . . . . . . . . 10 ⊢ (𝑖 = 𝑦 → (𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ↔ 𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))))
111	110	2rexbidv 3202	. . . . . . . . 9 ⊢ (𝑖 = 𝑦 → (∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ↔ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))))
112	109, 111	elab 3635	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ↔ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)))
113		eqeq1 2741	. . . . . . . . . 10 ⊢ (𝑗 = 𝑦 → (𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ↔ 𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))))
114	113	2rexbidv 3202	. . . . . . . . 9 ⊢ (𝑗 = 𝑦 → (∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ↔ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))))
115	109, 114	elab 3635	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))} ↔ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))
116	112, 115	orbi12i 915	. . . . . . 7 ⊢ ((𝑦 ∈ {𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∨ 𝑦 ∈ {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) ↔ (∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∨ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))))
117	108, 116	bitri 275	. . . . . 6 ⊢ (𝑦 ∈ ({𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}) ↔ (∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∨ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))))
118	107, 117	anbi12i 629	. . . . 5 ⊢ ((𝑥 ∈ ({𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) ∧ 𝑦 ∈ ({𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})) ↔ ((∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∨ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) ∧ (∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∨ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))))
119		anddi 1013	. . . . 5 ⊢ (((∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∨ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) ∧ (∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∨ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))) ↔ (((∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) ∨ (∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))) ∨ ((∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) ∨ (∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))))))
120	118, 119	bitri 275	. . . 4 ⊢ ((𝑥 ∈ ({𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) ∧ 𝑦 ∈ ({𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})) ↔ (((∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) ∨ (∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))) ∨ ((∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) ∨ (∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))))))
121	29	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))
122	36	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝐴 ∈ No )
123	33	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝐵 ∈ No )
124		simprll 779	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑝 ∈ ( L ‘𝐴))
125		simprlr 780	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑞 ∈ ( L ‘𝐵))
126		simprrl 781	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑡 ∈ ( L ‘𝐴))
127		simprrr 782	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑢 ∈ ( R ‘𝐵))
128	121, 122, 123, 124, 125, 126, 127	mulsproplem5 28120	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) <s (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)))
129		breq2 5103	. . . . . . . . . . . 12 ⊢ (𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → ((((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) <s 𝑦 ↔ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) <s (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))))
130	128, 129	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → (𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) <s 𝑦))
131	130	anassrs 467	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) <s 𝑦))
132	131	rexlimdvva 3194	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) <s 𝑦))
133		breq1 5102	. . . . . . . . . 10 ⊢ (𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) → (𝑥 <s 𝑦 ↔ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) <s 𝑦))
134	133	imbi2d 340	. . . . . . . . 9 ⊢ (𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) → ((∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → 𝑥 <s 𝑦) ↔ (∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) <s 𝑦)))
135	132, 134	syl5ibrcom 247	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) → (∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → 𝑥 <s 𝑦)))
136	135	rexlimdvva 3194	. . . . . . 7 ⊢ (𝜑 → (∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) → (∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → 𝑥 <s 𝑦)))
137	136	impd 410	. . . . . 6 ⊢ (𝜑 → ((∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) → 𝑥 <s 𝑦))
138	29	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))
139	36	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝐴 ∈ No )
140	33	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝐵 ∈ No )
141		simprll 779	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑝 ∈ ( L ‘𝐴))
142		simprlr 780	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑞 ∈ ( L ‘𝐵))
143		simprrl 781	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑣 ∈ ( R ‘𝐴))
144		simprrr 782	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑤 ∈ ( L ‘𝐵))
145	138, 139, 140, 141, 142, 143, 144	mulsproplem6 28121	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) <s (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))
146		breq2 5103	. . . . . . . . . . . 12 ⊢ (𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → ((((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) <s 𝑦 ↔ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) <s (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))))
147	145, 146	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → (𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) <s 𝑦))
148	147	anassrs 467	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) <s 𝑦))
149	148	rexlimdvva 3194	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) <s 𝑦))
150	133	imbi2d 340	. . . . . . . . 9 ⊢ (𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) → ((∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → 𝑥 <s 𝑦) ↔ (∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) <s 𝑦)))
151	149, 150	syl5ibrcom 247	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) → (∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → 𝑥 <s 𝑦)))
152	151	rexlimdvva 3194	. . . . . . 7 ⊢ (𝜑 → (∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) → (∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → 𝑥 <s 𝑦)))
153	152	impd 410	. . . . . 6 ⊢ (𝜑 → ((∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))) → 𝑥 <s 𝑦))
154	137, 153	jaod 860	. . . . 5 ⊢ (𝜑 → (((∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) ∨ (∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))) → 𝑥 <s 𝑦))
155	29	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))
156	36	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝐴 ∈ No )
157	33	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝐵 ∈ No )
158		simprll 779	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑟 ∈ ( R ‘𝐴))
159		simprlr 780	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑠 ∈ ( R ‘𝐵))
160		simprrl 781	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑡 ∈ ( L ‘𝐴))
161		simprrr 782	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑢 ∈ ( R ‘𝐵))
162	155, 156, 157, 158, 159, 160, 161	mulsproplem7 28122	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) <s (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)))
163		breq2 5103	. . . . . . . . . . . 12 ⊢ (𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → ((((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) <s 𝑦 ↔ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) <s (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))))
164	162, 163	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → (𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) <s 𝑦))
165	164	anassrs 467	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) <s 𝑦))
166	165	rexlimdvva 3194	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) <s 𝑦))
167		breq1 5102	. . . . . . . . . 10 ⊢ (𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) → (𝑥 <s 𝑦 ↔ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) <s 𝑦))
168	167	imbi2d 340	. . . . . . . . 9 ⊢ (𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) → ((∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → 𝑥 <s 𝑦) ↔ (∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) <s 𝑦)))
169	166, 168	syl5ibrcom 247	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) → (∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → 𝑥 <s 𝑦)))
170	169	rexlimdvva 3194	. . . . . . 7 ⊢ (𝜑 → (∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) → (∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → 𝑥 <s 𝑦)))
171	170	impd 410	. . . . . 6 ⊢ (𝜑 → ((∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) → 𝑥 <s 𝑦))
172	29	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (((( bday ‘𝑎) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝑎 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))
173	36	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝐴 ∈ No )
174	33	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝐵 ∈ No )
175		simprll 779	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑟 ∈ ( R ‘𝐴))
176		simprlr 780	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑠 ∈ ( R ‘𝐵))
177		simprrl 781	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑣 ∈ ( R ‘𝐴))
178		simprrr 782	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑤 ∈ ( L ‘𝐵))
179	172, 173, 174, 175, 176, 177, 178	mulsproplem8 28123	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) <s (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))
180		breq2 5103	. . . . . . . . . . . 12 ⊢ (𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → ((((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) <s 𝑦 ↔ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) <s (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))))
181	179, 180	syl5ibrcom 247	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → (𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) <s 𝑦))
182	181	anassrs 467	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) <s 𝑦))
183	182	rexlimdvva 3194	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) <s 𝑦))
184	167	imbi2d 340	. . . . . . . . 9 ⊢ (𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) → ((∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → 𝑥 <s 𝑦) ↔ (∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) <s 𝑦)))
185	183, 184	syl5ibrcom 247	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) → (∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → 𝑥 <s 𝑦)))
186	185	rexlimdvva 3194	. . . . . . 7 ⊢ (𝜑 → (∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) → (∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → 𝑥 <s 𝑦)))
187	186	impd 410	. . . . . 6 ⊢ (𝜑 → ((∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))) → 𝑥 <s 𝑦))
188	171, 187	jaod 860	. . . . 5 ⊢ (𝜑 → (((∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) ∨ (∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))) → 𝑥 <s 𝑦))
189	154, 188	jaod 860	. . . 4 ⊢ (𝜑 → ((((∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) ∨ (∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑥 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∧ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)))) ∨ ((∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑦 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) ∨ (∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)𝑥 = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∧ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑦 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))))) → 𝑥 <s 𝑦))
190	120, 189	biimtrid 242	. . 3 ⊢ (𝜑 → ((𝑥 ∈ ({𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) ∧ 𝑦 ∈ ({𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})) → 𝑥 <s 𝑦))
191	190	3impib 1117	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) ∧ 𝑦 ∈ ({𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))})) → 𝑥 <s 𝑦)
192	16, 28, 64, 97, 191	sltsd 27768	1 ⊢ (𝜑 → ({𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∪ {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))}) <<s ({𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∪ {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))}))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3061  Vcvv 3441   ∪ cun 3900   class class class wbr 5099  ran crn 5626  ‘cfv 6493  (class class class)co 7360   ∈ cmpo 7362   +no cnadd 8595   No csur 27611   <s clts 27612   bday cbday 27613   <<s cslts 27757   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

This theorem is referenced by:  mulsproplem10  28125