Theorem mulsproplem9 28168

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 2740	. . . . . 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 7583	. . . . 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 6933	. . . . . . 7 ⊢ ( L ‘𝐴) ∈ V
4		fvex 6933	. . . . . . 7 ⊢ ( L ‘𝐵) ∈ V
5	3, 4	mpoex 8120	. . . . . 6 ⊢ (𝑝 ∈ ( L ‘𝐴), 𝑞 ∈ ( L ‘𝐵) ↦ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) ∈ V
6	5	rnex 7950	. . . . 5 ⊢ ran (𝑝 ∈ ( L ‘𝐴), 𝑞 ∈ ( L ‘𝐵) ↦ (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))) ∈ V
7	2, 6	eqeltrri 2841	. . . 4 ⊢ {𝑔 ∣ ∃𝑝 ∈ ( L ‘𝐴)∃𝑞 ∈ ( L ‘𝐵)𝑔 = (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞))} ∈ V
8		eqid 2740	. . . . . 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 7583	. . . . 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 6933	. . . . . . 7 ⊢ ( R ‘𝐴) ∈ V
11		fvex 6933	. . . . . . 7 ⊢ ( R ‘𝐵) ∈ V
12	10, 11	mpoex 8120	. . . . . 6 ⊢ (𝑟 ∈ ( R ‘𝐴), 𝑠 ∈ ( R ‘𝐵) ↦ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) ∈ V
13	12	rnex 7950	. . . . 5 ⊢ ran (𝑟 ∈ ( R ‘𝐴), 𝑠 ∈ ( R ‘𝐵) ↦ (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠))) ∈ V
14	9, 13	eqeltrri 2841	. . . 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 7779	. . 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 2740	. . . . . 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 7583	. . . . 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 8120	. . . . . 6 ⊢ (𝑡 ∈ ( L ‘𝐴), 𝑢 ∈ ( R ‘𝐵) ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) ∈ V
20	19	rnex 7950	. . . . 5 ⊢ ran (𝑡 ∈ ( L ‘𝐴), 𝑢 ∈ ( R ‘𝐵) ↦ (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))) ∈ V
21	18, 20	eqeltrri 2841	. . . 4 ⊢ {𝑖 ∣ ∃𝑡 ∈ ( L ‘𝐴)∃𝑢 ∈ ( R ‘𝐵)𝑖 = (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢))} ∈ V
22		eqid 2740	. . . . . 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 7583	. . . . 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 8120	. . . . . 6 ⊢ (𝑣 ∈ ( R ‘𝐴), 𝑤 ∈ ( L ‘𝐵) ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))) ∈ V
25	24	rnex 7950	. . . . 5 ⊢ ran (𝑣 ∈ ( R ‘𝐴), 𝑤 ∈ ( L ‘𝐵) ↦ (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤))) ∈ V
26	23, 25	eqeltrri 2841	. . . 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 7779	. . 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 27935	. . . . . . . . . 10 ⊢ ( L ‘𝐴) ⊆ ( O ‘( bday ‘𝐴))
32		simprl 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝑝 ∈ ( L ‘𝐴))
33	31, 32	sselid 4006	. . . . . . . . 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 28161	. . . . . . . 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 27935	. . . . . . . . . 10 ⊢ ( L ‘𝐵) ⊆ ( O ‘( bday ‘𝐵))
40		simprr 772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝑞 ∈ ( L ‘𝐵))
41	39, 40	sselid 4006	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → 𝑞 ∈ ( O ‘( bday ‘𝐵)))
42	30, 38, 41	mulsproplem3 28162	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (𝐴 ·s 𝑞) ∈ No )
43	36, 42	addscld 28031	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → ((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) ∈ No )
44	30, 33, 41	mulsproplem4 28163	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (𝑝 ·s 𝑞) ∈ No )
45	43, 44	subscld 28111	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵))) → (((𝑝 ·s 𝐵) +s (𝐴 ·s 𝑞)) -s (𝑝 ·s 𝑞)) ∈ No )
46		eleq1 2832	. . . . . 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 3219	. . . 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 4091	. . 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 27936	. . . . . . . . . 10 ⊢ ( R ‘𝐴) ⊆ ( O ‘( bday ‘𝐴))
52		simprl 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝑟 ∈ ( R ‘𝐴))
53	51, 52	sselid 4006	. . . . . . . . 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 28161	. . . . . . . 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 27936	. . . . . . . . . 10 ⊢ ( R ‘𝐵) ⊆ ( O ‘( bday ‘𝐵))
58		simprr 772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝑠 ∈ ( R ‘𝐵))
59	57, 58	sselid 4006	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → 𝑠 ∈ ( O ‘( bday ‘𝐵)))
60	50, 56, 59	mulsproplem3 28162	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (𝐴 ·s 𝑠) ∈ No )
61	55, 60	addscld 28031	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → ((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) ∈ No )
62	50, 53, 59	mulsproplem4 28163	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (𝑟 ·s 𝑠) ∈ No )
63	61, 62	subscld 28111	. . . . . 6 ⊢ ((𝜑 ∧ (𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵))) → (((𝑟 ·s 𝐵) +s (𝐴 ·s 𝑠)) -s (𝑟 ·s 𝑠)) ∈ No )
64		eleq1 2832	. . . . . 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 3219	. . . 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 4091	. . 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 4215	. 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 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝑡 ∈ ( L ‘𝐴))
71	31, 70	sselid 4006	. . . . . . . . 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 28161	. . . . . . . 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 772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝑢 ∈ ( R ‘𝐵))
76	57, 75	sselid 4006	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → 𝑢 ∈ ( O ‘( bday ‘𝐵)))
77	69, 74, 76	mulsproplem3 28162	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (𝐴 ·s 𝑢) ∈ No )
78	73, 77	addscld 28031	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → ((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) ∈ No )
79	69, 71, 76	mulsproplem4 28163	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (𝑡 ·s 𝑢) ∈ No )
80	78, 79	subscld 28111	. . . . . 6 ⊢ ((𝜑 ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵))) → (((𝑡 ·s 𝐵) +s (𝐴 ·s 𝑢)) -s (𝑡 ·s 𝑢)) ∈ No )
81		eleq1 2832	. . . . . 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 3219	. . . 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 4091	. . 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 770	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝑣 ∈ ( R ‘𝐴))
87	51, 86	sselid 4006	. . . . . . . . 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 28161	. . . . . . . 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 772	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝑤 ∈ ( L ‘𝐵))
92	39, 91	sselid 4006	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → 𝑤 ∈ ( O ‘( bday ‘𝐵)))
93	85, 90, 92	mulsproplem3 28162	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (𝐴 ·s 𝑤) ∈ No )
94	89, 93	addscld 28031	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → ((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) ∈ No )
95	85, 87, 92	mulsproplem4 28163	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (𝑣 ·s 𝑤) ∈ No )
96	94, 95	subscld 28111	. . . . . 6 ⊢ ((𝜑 ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵))) → (((𝑣 ·s 𝐵) +s (𝐴 ·s 𝑤)) -s (𝑣 ·s 𝑤)) ∈ No )
97		eleq1 2832	. . . . . 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 3219	. . . 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 4091	. . 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 4215	. 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 4176	. . . . . . 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 3492	. . . . . . . . 9 ⊢ 𝑥 ∈ V
104		eqeq1 2744	. . . . . . . . . 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 3228	. . . . . . . . 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 3694	. . . . . . . 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 2744	. . . . . . . . . 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 3228	. . . . . . . . 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 3694	. . . . . . . 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 913	. . . . . . 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 4176	. . . . . . 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 3492	. . . . . . . . 9 ⊢ 𝑦 ∈ V
114		eqeq1 2744	. . . . . . . . . 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 3228	. . . . . . . . 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 3694	. . . . . . . 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 2744	. . . . . . . . . 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 3228	. . . . . . . . 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 3694	. . . . . . . 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 913	. . . . . . 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 627	. . . . 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 1011	. . . . 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 778	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑝 ∈ ( L ‘𝐴))
129		simprlr 779	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑞 ∈ ( L ‘𝐵))
130		simprrl 780	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑡 ∈ ( L ‘𝐴))
131		simprrr 781	. . . . . . . . . . . . 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 28164	. . . . . . . . . . . 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 5170	. . . . . . . . . . . 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 3219	. . . . . . . . 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 5169	. . . . . . . . . 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 3219	. . . . . . 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 778	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑝 ∈ ( L ‘𝐴))
146		simprlr 779	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑞 ∈ ( L ‘𝐵))
147		simprrl 780	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑝 ∈ ( L ‘𝐴) ∧ 𝑞 ∈ ( L ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑣 ∈ ( R ‘𝐴))
148		simprrr 781	. . . . . . . . . . . . 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 28165	. . . . . . . . . . . 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 5170	. . . . . . . . . . . 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 3219	. . . . . . . . 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 3219	. . . . . . 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 858	. . . . 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 778	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑟 ∈ ( R ‘𝐴))
163		simprlr 779	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑠 ∈ ( R ‘𝐵))
164		simprrl 780	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑡 ∈ ( L ‘𝐴) ∧ 𝑢 ∈ ( R ‘𝐵)))) → 𝑡 ∈ ( L ‘𝐴))
165		simprrr 781	. . . . . . . . . . . . 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 28166	. . . . . . . . . . . 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 5170	. . . . . . . . . . . 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 3219	. . . . . . . . 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 5169	. . . . . . . . . 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 3219	. . . . . . 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 778	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑟 ∈ ( R ‘𝐴))
180		simprlr 779	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑠 ∈ ( R ‘𝐵))
181		simprrl 780	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ ((𝑟 ∈ ( R ‘𝐴) ∧ 𝑠 ∈ ( R ‘𝐵)) ∧ (𝑣 ∈ ( R ‘𝐴) ∧ 𝑤 ∈ ( L ‘𝐵)))) → 𝑣 ∈ ( R ‘𝐴))
182		simprrr 781	. . . . . . . . . . . . 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 28167	. . . . . . . . . . . 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 5170	. . . . . . . . . . . 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 3219	. . . . . . . . 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 3219	. . . . . . 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 858	. . . . 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 858	. . . 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 1116	. 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 27854	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 846   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ∪ cun 3974   class class class wbr 5166  ran crn 5701  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450   +no cnadd 8721   No csur 27702   <s cslt 27703   bday cbday 27704   <<s csslt 27843   O cold 27900   L cleft 27902   R cright 27903   +_s cadds 28010   -_s csubs 28070   ·_s cmuls 28150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec 27989  df-norec2 28000  df-adds 28011  df-negs 28071  df-subs 28072

This theorem is referenced by:  mulsproplem10  28169