Theorem mulsproplem9 28165

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 2735	. . . . . 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 7566	. . . . 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 6920	. . . . . . 7 ⊢ ( L ‘𝐴) ∈ V
4		fvex 6920	. . . . . . 7 ⊢ ( L ‘𝐵) ∈ V
5	3, 4	mpoex 8103	. . . . . 6 ⊢ (𝑝 ∈ ( L ‘𝐴), 𝑞 ∈ ( L ‘𝐵) ↦ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) ∈ V
6	5	rnex 7933	. . . . 5 ⊢ ran (𝑝 ∈ ( L ‘𝐴), 𝑞 ∈ ( L ‘𝐵) ↦ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) ∈ V
7	2, 6	eqeltrri 2836	. . . 4 ⊢ {𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∈ V
8		eqid 2735	. . . . . 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 7566	. . . . 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 6920	. . . . . . 7 ⊢ ( R ‘𝐴) ∈ V
11		fvex 6920	. . . . . . 7 ⊢ ( R ‘𝐵) ∈ V
12	10, 11	mpoex 8103	. . . . . 6 ⊢ (𝑟 ∈ ( R ‘𝐴), 𝑠 ∈ ( R ‘𝐵) ↦ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) ∈ V
13	12	rnex 7933	. . . . 5 ⊢ ran (𝑟 ∈ ( R ‘𝐴), 𝑠 ∈ ( R ‘𝐵) ↦ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) ∈ V
14	9, 13	eqeltrri 2836	. . . 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 7763	. . 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 2735	. . . . . 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 7566	. . . . 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 8103	. . . . . 6 ⊢ (𝑡 ∈ ( L ‘𝐴), 𝑢 ∈ ( R ‘𝐵) ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) ∈ V
20	19	rnex 7933	. . . . 5 ⊢ ran (𝑡 ∈ ( L ‘𝐴), 𝑢 ∈ ( R ‘𝐵) ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) ∈ V
21	18, 20	eqeltrri 2836	. . . 4 ⊢ {𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∈ V
22		eqid 2735	. . . . . 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 7566	. . . . 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 8103	. . . . . 6 ⊢ (𝑣 ∈ ( R ‘𝐴), 𝑤 ∈ ( L ‘𝐵) ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))) ∈ V
25	24	rnex 7933	. . . . 5 ⊢ ran (𝑣 ∈ ( R ‘𝐴), 𝑤 ∈ ( L ‘𝐵) ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))) ∈ V
26	23, 25	eqeltrri 2836	. . . 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 7763	. . 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		leftssold 27932	. . . . . . . . . 10 ⊢ ( L ‘𝐴) ⊆ ( O ‘( bday ‘𝐴))
32		simprl 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝑝 ∈ ( L ‘𝐴))
33	31, 32	sselid 3993	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝑝 ∈ ( O ‘( bday ‘𝐴)))
34		mulsproplem9.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ No )
35	34	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝐵 ∈ No )
36	30, 33, 35	mulsproplem2 28158	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (𝑝 ·s 𝐵) ∈ No )
37		mulsproplem9.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ No )
38	37	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝐴 ∈ No )
39		leftssold 27932	. . . . . . . . . 10 ⊢ ( L ‘𝐵) ⊆ ( O ‘( bday ‘𝐵))
40		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝑞 ∈ ( L ‘𝐵))
41	39, 40	sselid 3993	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝑞 ∈ ( O ‘( bday ‘𝐵)))
42	30, 38, 41	mulsproplem3 28159	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (𝐴 ·s 𝑞) ∈ No )
43	36, 42	addscld 28028	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → ((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) ∈ No )
44	30, 33, 41	mulsproplem4 28160	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (𝑝 ·s 𝑞) ∈ No )
45	43, 44	subscld 28108	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∈ No )
46		eleq1 2827	. . . . . 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 ))
47	45, 46	syl5ibrcom 247	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) → 𝑔 ∈ No ))
48	47	rexlimdvva 3211	. . . 4 ⊢ (𝜑 → (∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) → 𝑔 ∈ No ))
49	48	abssdv 4078	. . 3 ⊢ (𝜑 → {𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ⊆ No )
50	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))))))
51		rightssold 27933	. . . . . . . . . 10 ⊢ ( R ‘𝐴) ⊆ ( O ‘( bday ‘𝐴))
52		simprl 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝑟 ∈ ( R ‘𝐴))
53	51, 52	sselid 3993	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝑟 ∈ ( O ‘( bday ‘𝐴)))
54	34	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝐵 ∈ No )
55	50, 53, 54	mulsproplem2 28158	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (𝑟 ·s 𝐵) ∈ No )
56	37	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝐴 ∈ No )
57		rightssold 27933	. . . . . . . . . 10 ⊢ ( R ‘𝐵) ⊆ ( O ‘( bday ‘𝐵))
58		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝑠 ∈ ( R ‘𝐵))
59	57, 58	sselid 3993	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝑠 ∈ ( O ‘( bday ‘𝐵)))
60	50, 56, 59	mulsproplem3 28159	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (𝐴 ·s 𝑠) ∈ No )
61	55, 60	addscld 28028	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → ((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) ∈ No )
62	50, 53, 59	mulsproplem4 28160	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (𝑟 ·s 𝑠) ∈ No )
63	61, 62	subscld 28108	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∈ No )
64		eleq1 2827	. . . . . 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 ))
65	63, 64	syl5ibrcom 247	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) → ℎ ∈ No ))
66	65	rexlimdvva 3211	. . . 4 ⊢ (𝜑 → (∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) → ℎ ∈ No ))
67	66	abssdv 4078	. . 3 ⊢ (𝜑 → {ℎ ∣ ∃𝑟 ∈ ( R ‘𝐴)∃𝑠 ∈ ( R ‘𝐵)ℎ = (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))} ⊆ No )
68	49, 67	unssd 4202	. 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 )
69	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))))))
70		simprl 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝑡 ∈ ( L ‘𝐴))
71	31, 70	sselid 3993	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝑡 ∈ ( O ‘( bday ‘𝐴)))
72	34	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝐵 ∈ No )
73	69, 71, 72	mulsproplem2 28158	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (𝑡 ·s 𝐵) ∈ No )
74	37	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝐴 ∈ No )
75		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝑢 ∈ ( R ‘𝐵))
76	57, 75	sselid 3993	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝑢 ∈ ( O ‘( bday ‘𝐵)))
77	69, 74, 76	mulsproplem3 28159	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (𝐴 ·s 𝑢) ∈ No )
78	73, 77	addscld 28028	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → ((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) ∈ No )
79	69, 71, 76	mulsproplem4 28160	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (𝑡 ·s 𝑢) ∈ No )
80	78, 79	subscld 28108	. . . . . 6 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∈ No )
81		eleq1 2827	. . . . . 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 ))
82	80, 81	syl5ibrcom 247	. . . . 5 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → 𝑖 ∈ No ))
83	82	rexlimdvva 3211	. . . 4 ⊢ (𝜑 → (∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) → 𝑖 ∈ No ))
84	83	abssdv 4078	. . 3 ⊢ (𝜑 → {𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ⊆ No )
85	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))))))
86		simprl 771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝑣 ∈ ( R ‘𝐴))
87	51, 86	sselid 3993	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝑣 ∈ ( O ‘( bday ‘𝐴)))
88	34	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝐵 ∈ No )
89	85, 87, 88	mulsproplem2 28158	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (𝑣 ·s 𝐵) ∈ No )
90	37	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝐴 ∈ No )
91		simprr 773	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝑤 ∈ ( L ‘𝐵))
92	39, 91	sselid 3993	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝑤 ∈ ( O ‘( bday ‘𝐵)))
93	85, 90, 92	mulsproplem3 28159	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (𝐴 ·s 𝑤) ∈ No )
94	89, 93	addscld 28028	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → ((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) ∈ No )
95	85, 87, 92	mulsproplem4 28160	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (𝑣 ·s 𝑤) ∈ No )
96	94, 95	subscld 28108	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∈ No )
97		eleq1 2827	. . . . . 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 ))
98	96, 97	syl5ibrcom 247	. . . . 5 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → 𝑗 ∈ No ))
99	98	rexlimdvva 3211	. . . 4 ⊢ (𝜑 → (∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) → 𝑗 ∈ No ))
100	99	abssdv 4078	. . 3 ⊢ (𝜑 → {𝑗 ∣ ∃𝑣 ∈ ( R ‘𝐴)∃𝑤 ∈ ( L ‘𝐵)𝑗 = (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))} ⊆ No )
101	84, 100	unssd 4202	. 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 )
102		elun 4163	. . . . . . 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))}))
103		vex 3482	. . . . . . . . 9 ⊢ 𝑥 ∈ V
104		eqeq1 2739	. . . . . . . . . 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))))
105	104	2rexbidv 3220	. . . . . . . . 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))))
106	103, 105	elab 3681	. . . . . . . 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)))
107		eqeq1 2739	. . . . . . . . . 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))))
108	107	2rexbidv 3220	. . . . . . . . 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))))
109	103, 108	elab 3681	. . . . . . . 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)))
110	106, 109	orbi12i 914	. . . . . . 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))))
111	102, 110	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))))
112		elun 4163	. . . . . . 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))}))
113		vex 3482	. . . . . . . . 9 ⊢ 𝑦 ∈ V
114		eqeq1 2739	. . . . . . . . . 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))))
115	114	2rexbidv 3220	. . . . . . . . 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))))
116	113, 115	elab 3681	. . . . . . . 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)))
117		eqeq1 2739	. . . . . . . . . 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))))
118	117	2rexbidv 3220	. . . . . . . . 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))))
119	113, 118	elab 3681	. . . . . . . 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)))
120	116, 119	orbi12i 914	. . . . . . 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))))
121	112, 120	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))))
122	111, 121	anbi12i 628	. . . . 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)))))
123		anddi 1012	. . . . 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))))))
124	122, 123	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))))))
125	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))))))
126	37	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝐴 ∈ No )
127	34	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝐵 ∈ No )
128		simprll 779	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑝 ∈ ( L ‘𝐴))
129		simprlr 780	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑞 ∈ ( L ‘𝐵))
130		simprrl 781	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑡 ∈ ( L ‘𝐴))
131		simprrr 782	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑢 ∈ ( R ‘𝐵))
132	125, 126, 127, 128, 129, 130, 131	mulsproplem5 28161	. . . . . . . . . . . 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)))
133		breq2 5152	. . . . . . . . . . . 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))))
134	132, 133	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))
135	134	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))
136	135	rexlimdvva 3211	. . . . . . . . 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))
137		breq1 5151	. . . . . . . . . 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))
138	137	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)))
139	136, 138	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)))
140	139	rexlimdvva 3211	. . . . . . 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)))
141	140	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))
142	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))))))
143	37	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝐴 ∈ No )
144	34	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝐵 ∈ No )
145		simprll 779	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑝 ∈ ( L ‘𝐴))
146		simprlr 780	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑞 ∈ ( L ‘𝐵))
147		simprrl 781	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑣 ∈ ( R ‘𝐴))
148		simprrr 782	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑤 ∈ ( L ‘𝐵))
149	142, 143, 144, 145, 146, 147, 148	mulsproplem6 28162	. . . . . . . . . . . 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)))
150		breq2 5152	. . . . . . . . . . . 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))))
151	149, 150	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))
152	151	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))
153	152	rexlimdvva 3211	. . . . . . . . 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))
154	137	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)))
155	153, 154	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)))
156	155	rexlimdvva 3211	. . . . . . 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)))
157	156	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))
158	141, 157	jaod 859	. . . . 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))
159	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))))))
160	37	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝐴 ∈ No )
161	34	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝐵 ∈ No )
162		simprll 779	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑟 ∈ ( R ‘𝐴))
163		simprlr 780	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑠 ∈ ( R ‘𝐵))
164		simprrl 781	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑡 ∈ ( L ‘𝐴))
165		simprrr 782	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑢 ∈ ( R ‘𝐵))
166	159, 160, 161, 162, 163, 164, 165	mulsproplem7 28163	. . . . . . . . . . . 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)))
167		breq2 5152	. . . . . . . . . . . 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))))
168	166, 167	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))
169	168	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))
170	169	rexlimdvva 3211	. . . . . . . . 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))
171		breq1 5151	. . . . . . . . . 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))
172	171	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)))
173	170, 172	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)))
174	173	rexlimdvva 3211	. . . . . . 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)))
175	174	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))
176	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))))))
177	37	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝐴 ∈ No )
178	34	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝐵 ∈ No )
179		simprll 779	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑟 ∈ ( R ‘𝐴))
180		simprlr 780	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑠 ∈ ( R ‘𝐵))
181		simprrl 781	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑣 ∈ ( R ‘𝐴))
182		simprrr 782	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑤 ∈ ( L ‘𝐵))
183	176, 177, 178, 179, 180, 181, 182	mulsproplem8 28164	. . . . . . . . . . . 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)))
184		breq2 5152	. . . . . . . . . . . 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))))
185	183, 184	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))
186	185	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))
187	186	rexlimdvva 3211	. . . . . . . . 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))
188	171	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)))
189	187, 188	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)))
190	189	rexlimdvva 3211	. . . . . . 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)))
191	190	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))
192	175, 191	jaod 859	. . . . 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))
193	158, 192	jaod 859	. . . 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))
194	124, 193	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))
195	194	3impib 1115	. 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)
196	16, 28, 68, 101, 195	ssltd 27851	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 847   = wceq 1537   ∈ wcel 2106  {cab 2712  ∀wral 3059  ∃wrex 3068  Vcvv 3478   ∪ cun 3961   class class class wbr 5148  ran crn 5690  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433   +no cnadd 8702   No csur 27699   <s cslt 27700   bday cbday 27701   <<s csslt 27840   O cold 27897   L cleft 27899   R cright 27900   +_s cadds 28007   -_s csubs 28067   ·_s cmuls 28147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068  df-subs 28069

This theorem is referenced by:  mulsproplem10  28166