Theorem mulsprop 27499

Description: Surreals are closed under multiplication and obey a particular ordering law. Theorem 3.4 of [Gonshor] p. 17. (Contributed by Scott Fenton, 5-Mar-2025.)

Assertion

Ref	Expression
mulsprop	⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐶 ∈ No ∧ 𝐷 ∈ No ) ∧ (𝐸 ∈ No ∧ 𝐹 ∈ No )) → ((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)))))

Proof of Theorem mulsprop

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑖 𝑗 𝑘 𝑙 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdayelon 27204	. . . . 5 ⊢ ( bday ‘𝐴) ∈ On
2		bdayelon 27204	. . . . 5 ⊢ ( bday ‘𝐵) ∈ On
3		naddcl 8659	. . . . 5 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On)
4	1, 2, 3	mp2an 690	. . . 4 ⊢ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On
5		bdayelon 27204	. . . . . . 7 ⊢ ( bday ‘𝐶) ∈ On
6		bdayelon 27204	. . . . . . 7 ⊢ ( bday ‘𝐸) ∈ On
7		naddcl 8659	. . . . . . 7 ⊢ ((( bday ‘𝐶) ∈ On ∧ ( bday ‘𝐸) ∈ On) → (( bday ‘𝐶) +no ( bday ‘𝐸)) ∈ On)
8	5, 6, 7	mp2an 690	. . . . . 6 ⊢ (( bday ‘𝐶) +no ( bday ‘𝐸)) ∈ On
9		bdayelon 27204	. . . . . . 7 ⊢ ( bday ‘𝐷) ∈ On
10		bdayelon 27204	. . . . . . 7 ⊢ ( bday ‘𝐹) ∈ On
11		naddcl 8659	. . . . . . 7 ⊢ ((( bday ‘𝐷) ∈ On ∧ ( bday ‘𝐹) ∈ On) → (( bday ‘𝐷) +no ( bday ‘𝐹)) ∈ On)
12	9, 10, 11	mp2an 690	. . . . . 6 ⊢ (( bday ‘𝐷) +no ( bday ‘𝐹)) ∈ On
13	8, 12	onun2i 6475	. . . . 5 ⊢ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∈ On
14		naddcl 8659	. . . . . . 7 ⊢ ((( bday ‘𝐶) ∈ On ∧ ( bday ‘𝐹) ∈ On) → (( bday ‘𝐶) +no ( bday ‘𝐹)) ∈ On)
15	5, 10, 14	mp2an 690	. . . . . 6 ⊢ (( bday ‘𝐶) +no ( bday ‘𝐹)) ∈ On
16		naddcl 8659	. . . . . . 7 ⊢ ((( bday ‘𝐷) ∈ On ∧ ( bday ‘𝐸) ∈ On) → (( bday ‘𝐷) +no ( bday ‘𝐸)) ∈ On)
17	9, 6, 16	mp2an 690	. . . . . 6 ⊢ (( bday ‘𝐷) +no ( bday ‘𝐸)) ∈ On
18	15, 17	onun2i 6475	. . . . 5 ⊢ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))) ∈ On
19	13, 18	onun2i 6475	. . . 4 ⊢ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))) ∈ On
20	4, 19	onun2i 6475	. . 3 ⊢ ((( 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))))) ∈ On
21		risset 3229	. . 3 ⊢ (((( 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))))) ∈ On ↔ ∃𝑥 ∈ On 𝑥 = ((( 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))))))
22	20, 21	mpbi 229	. 2 ⊢ ∃𝑥 ∈ On 𝑥 = ((( 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)))))
23		fveq2 6878	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑔 → ( bday ‘𝑎) = ( bday ‘𝑔))
24	23	oveq1d 7408	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑔 → (( bday ‘𝑎) +no ( bday ‘𝑏)) = (( bday ‘𝑔) +no ( bday ‘𝑏)))
25	24	uneq1d 4158	. . . . . . . . . 10 ⊢ (𝑎 = 𝑔 → ((( 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 ‘𝑔) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))))
26	25	eqeq2d 2742	. . . . . . . . 9 ⊢ (𝑎 = 𝑔 → (𝑥 = ((( 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))))) ↔ 𝑥 = ((( bday ‘𝑔) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒)))))))
27		oveq1 7400	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑔 → (𝑎 ·s 𝑏) = (𝑔 ·s 𝑏))
28	27	eleq1d 2817	. . . . . . . . . 10 ⊢ (𝑎 = 𝑔 → ((𝑎 ·s 𝑏) ∈ No ↔ (𝑔 ·s 𝑏) ∈ No ))
29	28	anbi1d 630	. . . . . . . . 9 ⊢ (𝑎 = 𝑔 → (((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)))) ↔ ((𝑔 ·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	26, 29	imbi12d 344	. . . . . . . 8 ⊢ (𝑎 = 𝑔 → ((𝑥 = ((( 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))))) ↔ (𝑥 = ((( bday ‘𝑔) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) → ((𝑔 ·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		fveq2 6878	. . . . . . . . . . . 12 ⊢ (𝑏 = ℎ → ( bday ‘𝑏) = ( bday ‘ℎ))
32	31	oveq2d 7409	. . . . . . . . . . 11 ⊢ (𝑏 = ℎ → (( bday ‘𝑔) +no ( bday ‘𝑏)) = (( bday ‘𝑔) +no ( bday ‘ℎ)))
33	32	uneq1d 4158	. . . . . . . . . 10 ⊢ (𝑏 = ℎ → ((( bday ‘𝑔) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))))
34	33	eqeq2d 2742	. . . . . . . . 9 ⊢ (𝑏 = ℎ → (𝑥 = ((( bday ‘𝑔) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ↔ 𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒)))))))
35		oveq2 7401	. . . . . . . . . . 11 ⊢ (𝑏 = ℎ → (𝑔 ·s 𝑏) = (𝑔 ·s ℎ))
36	35	eleq1d 2817	. . . . . . . . . 10 ⊢ (𝑏 = ℎ → ((𝑔 ·s 𝑏) ∈ No ↔ (𝑔 ·s ℎ) ∈ No ))
37	36	anbi1d 630	. . . . . . . . 9 ⊢ (𝑏 = ℎ → (((𝑔 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))) ↔ ((𝑔 ·s ℎ) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))
38	34, 37	imbi12d 344	. . . . . . . 8 ⊢ (𝑏 = ℎ → ((𝑥 = ((( bday ‘𝑔) +no ( bday ‘𝑏)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) → ((𝑔 ·s 𝑏) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))))))
39		fveq2 6878	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑖 → ( bday ‘𝑐) = ( bday ‘𝑖))
40	39	oveq1d 7408	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑖 → (( bday ‘𝑐) +no ( bday ‘𝑒)) = (( bday ‘𝑖) +no ( bday ‘𝑒)))
41	40	uneq1d 4158	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑖 → ((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) = ((( bday ‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))))
42	39	oveq1d 7408	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑖 → (( bday ‘𝑐) +no ( bday ‘𝑓)) = (( bday ‘𝑖) +no ( bday ‘𝑓)))
43	42	uneq1d 4158	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑖 → ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))) = ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))
44	41, 43	uneq12d 4160	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑖 → (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒)))) = (((( bday ‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒)))))
45	44	uneq2d 4159	. . . . . . . . . 10 ⊢ (𝑐 = 𝑖 → ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))))
46	45	eqeq2d 2742	. . . . . . . . 9 ⊢ (𝑐 = 𝑖 → (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ↔ 𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒)))))))
47		breq1 5144	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑖 → (𝑐 <s 𝑑 ↔ 𝑖 <s 𝑑))
48	47	anbi1d 630	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑖 → ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) ↔ (𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓)))
49		oveq1 7400	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑖 → (𝑐 ·s 𝑓) = (𝑖 ·s 𝑓))
50		oveq1 7400	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑖 → (𝑐 ·s 𝑒) = (𝑖 ·s 𝑒))
51	49, 50	oveq12d 7411	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑖 → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) = ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)))
52	51	breq1d 5151	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑖 → (((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)) ↔ ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))
53	48, 52	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑐 = 𝑖 → (((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))) ↔ ((𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))))
54	53	anbi2d 629	. . . . . . . . 9 ⊢ (𝑐 = 𝑖 → (((𝑔 ·s ℎ) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))) ↔ ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))))
55	46, 54	imbi12d 344	. . . . . . . 8 ⊢ (𝑐 = 𝑖 → ((𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑐) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑐) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑐 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑐 ·s 𝑓) -s (𝑐 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))))))
56		fveq2 6878	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑗 → ( bday ‘𝑑) = ( bday ‘𝑗))
57	56	oveq1d 7408	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑗 → (( bday ‘𝑑) +no ( bday ‘𝑓)) = (( bday ‘𝑗) +no ( bday ‘𝑓)))
58	57	uneq2d 4159	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑗 → ((( bday ‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) = ((( bday ‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑓))))
59	56	oveq1d 7408	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑗 → (( bday ‘𝑑) +no ( bday ‘𝑒)) = (( bday ‘𝑗) +no ( bday ‘𝑒)))
60	59	uneq2d 4159	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑗 → ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))) = ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑒))))
61	58, 60	uneq12d 4160	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑗 → (((( bday ‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒)))) = (((( bday ‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑒)))))
62	61	uneq2d 4159	. . . . . . . . . 10 ⊢ (𝑑 = 𝑗 → ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑒))))))
63	62	eqeq2d 2742	. . . . . . . . 9 ⊢ (𝑑 = 𝑗 → (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) ↔ 𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑒)))))))
64		breq2 5145	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑗 → (𝑖 <s 𝑑 ↔ 𝑖 <s 𝑗))
65	64	anbi1d 630	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑗 → ((𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓) ↔ (𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓)))
66		oveq1 7400	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑗 → (𝑑 ·s 𝑓) = (𝑗 ·s 𝑓))
67		oveq1 7400	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑗 → (𝑑 ·s 𝑒) = (𝑗 ·s 𝑒))
68	66, 67	oveq12d 7411	. . . . . . . . . . . 12 ⊢ (𝑑 = 𝑗 → ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)) = ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑒)))
69	68	breq2d 5153	. . . . . . . . . . 11 ⊢ (𝑑 = 𝑗 → (((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)) ↔ ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑒))))
70	65, 69	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑑 = 𝑗 → (((𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))) ↔ ((𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑒)))))
71	70	anbi2d 629	. . . . . . . . 9 ⊢ (𝑑 = 𝑗 → (((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒)))) ↔ ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑒))))))
72	63, 71	imbi12d 344	. . . . . . . 8 ⊢ (𝑑 = 𝑗 → ((𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑑) +no ( bday ‘𝑒))))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑑 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑑 ·s 𝑓) -s (𝑑 ·s 𝑒))))) ↔ (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑒))))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑒)))))))
73		fveq2 6878	. . . . . . . . . . . . . 14 ⊢ (𝑒 = 𝑘 → ( bday ‘𝑒) = ( bday ‘𝑘))
74	73	oveq2d 7409	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝑘 → (( bday ‘𝑖) +no ( bday ‘𝑒)) = (( bday ‘𝑖) +no ( bday ‘𝑘)))
75	74	uneq1d 4158	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝑘 → ((( bday ‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑓))) = ((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑓))))
76	73	oveq2d 7409	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝑘 → (( bday ‘𝑗) +no ( bday ‘𝑒)) = (( bday ‘𝑗) +no ( bday ‘𝑘)))
77	76	uneq2d 4159	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝑘 → ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑒))) = ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))
78	75, 77	uneq12d 4160	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑘 → (((( bday ‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑒)))) = (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))
79	78	uneq2d 4159	. . . . . . . . . 10 ⊢ (𝑒 = 𝑘 → ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑒))))) = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))))
80	79	eqeq2d 2742	. . . . . . . . 9 ⊢ (𝑒 = 𝑘 → (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑒))))) ↔ 𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))))
81		breq1 5144	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝑘 → (𝑒 <s 𝑓 ↔ 𝑘 <s 𝑓))
82	81	anbi2d 629	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑘 → ((𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓) ↔ (𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓)))
83		oveq2 7401	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝑘 → (𝑖 ·s 𝑒) = (𝑖 ·s 𝑘))
84	83	oveq2d 7409	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝑘 → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) = ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑘)))
85		oveq2 7401	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝑘 → (𝑗 ·s 𝑒) = (𝑗 ·s 𝑘))
86	85	oveq2d 7409	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝑘 → ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑒)) = ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑘)))
87	84, 86	breq12d 5154	. . . . . . . . . . 11 ⊢ (𝑒 = 𝑘 → (((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑒)) ↔ ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑘))))
88	82, 87	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑒 = 𝑘 → (((𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑒))) ↔ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑘)))))
89	88	anbi2d 629	. . . . . . . . 9 ⊢ (𝑒 = 𝑘 → (((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑒)))) ↔ ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑘))))))
90	80, 89	imbi12d 344	. . . . . . . 8 ⊢ (𝑒 = 𝑘 → ((𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑒)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑒))))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑒 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑒)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑒))))) ↔ (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑘)))))))
91		fveq2 6878	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑙 → ( bday ‘𝑓) = ( bday ‘𝑙))
92	91	oveq2d 7409	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑙 → (( bday ‘𝑗) +no ( bday ‘𝑓)) = (( bday ‘𝑗) +no ( bday ‘𝑙)))
93	92	uneq2d 4159	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑙 → ((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑓))) = ((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))))
94	91	oveq2d 7409	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑙 → (( bday ‘𝑖) +no ( bday ‘𝑓)) = (( bday ‘𝑖) +no ( bday ‘𝑙)))
95	94	uneq1d 4158	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑙 → ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))) = ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))
96	93, 95	uneq12d 4160	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑙 → (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))) = (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))
97	96	uneq2d 4159	. . . . . . . . . 10 ⊢ (𝑓 = 𝑙 → ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))))
98	97	eqeq2d 2742	. . . . . . . . 9 ⊢ (𝑓 = 𝑙 → (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) ↔ 𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))))
99		breq2 5145	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑙 → (𝑘 <s 𝑓 ↔ 𝑘 <s 𝑙))
100	99	anbi2d 629	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑙 → ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓) ↔ (𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙)))
101		oveq2 7401	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑙 → (𝑖 ·s 𝑓) = (𝑖 ·s 𝑙))
102	101	oveq1d 7408	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑙 → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑘)) = ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)))
103		oveq2 7401	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑙 → (𝑗 ·s 𝑓) = (𝑗 ·s 𝑙))
104	103	oveq1d 7408	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑙 → ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑘)) = ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))
105	102, 104	breq12d 5154	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑙 → (((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑘)) ↔ ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))
106	100, 105	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑓 = 𝑙 → (((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑘))) ↔ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))))
107	106	anbi2d 629	. . . . . . . . 9 ⊢ (𝑓 = 𝑙 → (((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑘)))) ↔ ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))))
108	98, 107	imbi12d 344	. . . . . . . 8 ⊢ (𝑓 = 𝑙 → ((𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑓))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑓)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑓) → ((𝑖 ·s 𝑓) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑓) -s (𝑗 ·s 𝑘))))) ↔ (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))))))
109	30, 38, 55, 72, 90, 108	cbvral6vw 3241	. . . . . . 7 ⊢ (∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (𝑥 = ((( 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))))) ↔ ∀𝑔 ∈ No ∀ℎ ∈ No ∀𝑖 ∈ No ∀𝑗 ∈ No ∀𝑘 ∈ No ∀𝑙 ∈ No (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))))
110		eqeq1 2735	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 = ((( 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))))) ↔ 𝑦 = ((( 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)))))))
111	110	imbi1d 341	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((𝑥 = ((( 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))))) ↔ (𝑦 = ((( 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)))))))
112	111	6ralbidv 3222	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (𝑥 = ((( 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))))) ↔ ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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)))))))
113	109, 112	bitr3id 284	. . . . . 6 ⊢ (𝑥 = 𝑦 → (∀𝑔 ∈ No ∀ℎ ∈ No ∀𝑖 ∈ No ∀𝑗 ∈ No ∀𝑘 ∈ No ∀𝑙 ∈ No (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))) ↔ ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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)))))))
114		raleq 3321	. . . . . . . . . . . . . 14 ⊢ (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → (∀𝑦 ∈ 𝑥 ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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))))) ↔ ∀𝑦 ∈ ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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)))))))
115		ralrot3 3289	. . . . . . . . . . . . . . 15 ⊢ (∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑦 ∈ ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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))))) ↔ ∀𝑦 ∈ ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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))))))
116		ralrot3 3289	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑦 ∈ ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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))))) ↔ ∀𝑦 ∈ ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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))))))
117		ralrot3 3289	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑒 ∈ No ∀𝑓 ∈ No ∀𝑦 ∈ ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))(𝑦 = ((( 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))))) ↔ ∀𝑦 ∈ ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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))))))
118		r19.23v 3181	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑦 ∈ ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))(𝑦 = ((( 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))))) ↔ (∃𝑦 ∈ ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))𝑦 = ((( 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))))))
119		risset 3229	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((( 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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) ↔ ∃𝑦 ∈ ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))𝑦 = ((( 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))))))
120	119	imbi1i 349	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((( 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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))) ↔ (∃𝑦 ∈ ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))𝑦 = ((( 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))))))
121	118, 120	bitr4i 277	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑦 ∈ ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))(𝑦 = ((( 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))))) ↔ (((( 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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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	121	2ralbii 3127	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑒 ∈ No ∀𝑓 ∈ No ∀𝑦 ∈ ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))(𝑦 = ((( 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))))) ↔ ∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))))
123	117, 122	bitr3i 276	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑦 ∈ ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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))))) ↔ ∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))))
124	123	2ralbii 3127	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑦 ∈ ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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))))) ↔ ∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))))
125	116, 124	bitr3i 276	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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))))) ↔ ∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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	125	2ralbii 3127	. . . . . . . . . . . . . . 15 ⊢ (∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑦 ∈ ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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))))) ↔ ∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))))
127	115, 126	bitr3i 276	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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))))) ↔ ∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))))
128	114, 127	bitrdi 286	. . . . . . . . . . . . 13 ⊢ (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → (∀𝑦 ∈ 𝑥 ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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))))) ↔ ∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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)))))))
129		simpl 483	. . . . . . . . . . . . . . . 16 ⊢ ((∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))) ∧ ((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ))) → ∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))))
130		simprl1 1218	. . . . . . . . . . . . . . . 16 ⊢ ((∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))) ∧ ((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ))) → 𝑔 ∈ No )
131		simprl2 1219	. . . . . . . . . . . . . . . 16 ⊢ ((∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))) ∧ ((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ))) → ℎ ∈ No )
132	129, 130, 131	mulsproplem11 27495	. . . . . . . . . . . . . . 15 ⊢ ((∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))) ∧ ((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ))) → (𝑔 ·s ℎ) ∈ No )
133	129	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))) ∧ ((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ))) ∧ (𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙)) → ∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))))
134		simprl3 1220	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))) ∧ ((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ))) → 𝑖 ∈ No )
135	134	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))) ∧ ((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ))) ∧ (𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙)) → 𝑖 ∈ No )
136		simprr1 1221	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))) ∧ ((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ))) → 𝑗 ∈ No )
137	136	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))) ∧ ((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ))) ∧ (𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙)) → 𝑗 ∈ No )
138		simprr2 1222	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))) ∧ ((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ))) → 𝑘 ∈ No )
139	138	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))) ∧ ((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ))) ∧ (𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙)) → 𝑘 ∈ No )
140		simprr3 1223	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))) ∧ ((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ))) → 𝑙 ∈ No )
141	140	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))) ∧ ((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ))) ∧ (𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙)) → 𝑙 ∈ No )
142		simprl 769	. . . . . . . . . . . . . . . . 17 ⊢ (((∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))) ∧ ((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ))) ∧ (𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙)) → 𝑖 <s 𝑗)
143		simprr 771	. . . . . . . . . . . . . . . . 17 ⊢ (((∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))) ∧ ((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ))) ∧ (𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙)) → 𝑘 <s 𝑙)
144	133, 135, 137, 139, 141, 142, 143	mulsproplem14 27498	. . . . . . . . . . . . . . . 16 ⊢ (((∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))) ∧ ((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ))) ∧ (𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙)) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))
145	144	ex 413	. . . . . . . . . . . . . . 15 ⊢ ((∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))) ∧ ((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ))) → ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))
146	132, 145	jca 512	. . . . . . . . . . . . . 14 ⊢ ((∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))) ∧ ((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))))
147	146	ex 413	. . . . . . . . . . . . 13 ⊢ (∀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 ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((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))))) → (((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No )) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))))
148	128, 147	syl6bi 252	. . . . . . . . . . . 12 ⊢ (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → (∀𝑦 ∈ 𝑥 ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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))))) → (((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No )) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))))))
149	148	impd 411	. . . . . . . . . . 11 ⊢ (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((∀𝑦 ∈ 𝑥 ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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))))) ∧ ((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))))
150	149	com12 32	. . . . . . . . . 10 ⊢ ((∀𝑦 ∈ 𝑥 ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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))))) ∧ ((𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No ) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No ))) → (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))))
151	150	anassrs 468	. . . . . . . . 9 ⊢ (((∀𝑦 ∈ 𝑥 ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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))))) ∧ (𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No )) ∧ (𝑗 ∈ No ∧ 𝑘 ∈ No ∧ 𝑙 ∈ No )) → (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))))
152	151	ralrimivvva 3202	. . . . . . . 8 ⊢ ((∀𝑦 ∈ 𝑥 ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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))))) ∧ (𝑔 ∈ No ∧ ℎ ∈ No ∧ 𝑖 ∈ No )) → ∀𝑗 ∈ No ∀𝑘 ∈ No ∀𝑙 ∈ No (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))))
153	152	ralrimivvva 3202	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝑥 ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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))))) → ∀𝑔 ∈ No ∀ℎ ∈ No ∀𝑖 ∈ No ∀𝑗 ∈ No ∀𝑘 ∈ No ∀𝑙 ∈ No (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))))
154	153	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 ∀𝑎 ∈ No ∀𝑏 ∈ No ∀𝑐 ∈ No ∀𝑑 ∈ No ∀𝑒 ∈ No ∀𝑓 ∈ No (𝑦 = ((( 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))))) → ∀𝑔 ∈ No ∀ℎ ∈ No ∀𝑖 ∈ No ∀𝑗 ∈ No ∀𝑘 ∈ No ∀𝑙 ∈ No (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))))))
155	113, 154	tfis2 7829	. . . . 5 ⊢ (𝑥 ∈ On → ∀𝑔 ∈ No ∀ℎ ∈ No ∀𝑖 ∈ No ∀𝑗 ∈ No ∀𝑘 ∈ No ∀𝑙 ∈ No (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))))
156		fveq2 6878	. . . . . . . . . 10 ⊢ (𝑔 = 𝐴 → ( bday ‘𝑔) = ( bday ‘𝐴))
157	156	oveq1d 7408	. . . . . . . . 9 ⊢ (𝑔 = 𝐴 → (( bday ‘𝑔) +no ( bday ‘ℎ)) = (( bday ‘𝐴) +no ( bday ‘ℎ)))
158	157	uneq1d 4158	. . . . . . . 8 ⊢ (𝑔 = 𝐴 → ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) = ((( bday ‘𝐴) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))))
159	158	eqeq2d 2742	. . . . . . 7 ⊢ (𝑔 = 𝐴 → (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) ↔ 𝑥 = ((( bday ‘𝐴) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))))
160		oveq1 7400	. . . . . . . . 9 ⊢ (𝑔 = 𝐴 → (𝑔 ·s ℎ) = (𝐴 ·s ℎ))
161	160	eleq1d 2817	. . . . . . . 8 ⊢ (𝑔 = 𝐴 → ((𝑔 ·s ℎ) ∈ No ↔ (𝐴 ·s ℎ) ∈ No ))
162	161	anbi1d 630	. . . . . . 7 ⊢ (𝑔 = 𝐴 → (((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))) ↔ ((𝐴 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))))
163	159, 162	imbi12d 344	. . . . . 6 ⊢ (𝑔 = 𝐴 → ((𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))) ↔ (𝑥 = ((( bday ‘𝐴) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((𝐴 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))))))
164		fveq2 6878	. . . . . . . . . 10 ⊢ (ℎ = 𝐵 → ( bday ‘ℎ) = ( bday ‘𝐵))
165	164	oveq2d 7409	. . . . . . . . 9 ⊢ (ℎ = 𝐵 → (( bday ‘𝐴) +no ( bday ‘ℎ)) = (( bday ‘𝐴) +no ( bday ‘𝐵)))
166	165	uneq1d 4158	. . . . . . . 8 ⊢ (ℎ = 𝐵 → ((( bday ‘𝐴) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) = ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))))
167	166	eqeq2d 2742	. . . . . . 7 ⊢ (ℎ = 𝐵 → (𝑥 = ((( bday ‘𝐴) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) ↔ 𝑥 = ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))))
168		oveq2 7401	. . . . . . . . 9 ⊢ (ℎ = 𝐵 → (𝐴 ·s ℎ) = (𝐴 ·s 𝐵))
169	168	eleq1d 2817	. . . . . . . 8 ⊢ (ℎ = 𝐵 → ((𝐴 ·s ℎ) ∈ No ↔ (𝐴 ·s 𝐵) ∈ No ))
170	169	anbi1d 630	. . . . . . 7 ⊢ (ℎ = 𝐵 → (((𝐴 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))) ↔ ((𝐴 ·s 𝐵) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))))
171	167, 170	imbi12d 344	. . . . . 6 ⊢ (ℎ = 𝐵 → ((𝑥 = ((( bday ‘𝐴) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((𝐴 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))) ↔ (𝑥 = ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((𝐴 ·s 𝐵) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))))))
172		fveq2 6878	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐶 → ( bday ‘𝑖) = ( bday ‘𝐶))
173	172	oveq1d 7408	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐶 → (( bday ‘𝑖) +no ( bday ‘𝑘)) = (( bday ‘𝐶) +no ( bday ‘𝑘)))
174	173	uneq1d 4158	. . . . . . . . . 10 ⊢ (𝑖 = 𝐶 → ((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) = ((( bday ‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))))
175	172	oveq1d 7408	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐶 → (( bday ‘𝑖) +no ( bday ‘𝑙)) = (( bday ‘𝐶) +no ( bday ‘𝑙)))
176	175	uneq1d 4158	. . . . . . . . . 10 ⊢ (𝑖 = 𝐶 → ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))) = ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))
177	174, 176	uneq12d 4160	. . . . . . . . 9 ⊢ (𝑖 = 𝐶 → (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))) = (((( bday ‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))
178	177	uneq2d 4159	. . . . . . . 8 ⊢ (𝑖 = 𝐶 → ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) = ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))))
179	178	eqeq2d 2742	. . . . . . 7 ⊢ (𝑖 = 𝐶 → (𝑥 = ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) ↔ 𝑥 = ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))))))
180		breq1 5144	. . . . . . . . . 10 ⊢ (𝑖 = 𝐶 → (𝑖 <s 𝑗 ↔ 𝐶 <s 𝑗))
181	180	anbi1d 630	. . . . . . . . 9 ⊢ (𝑖 = 𝐶 → ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) ↔ (𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙)))
182		oveq1 7400	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐶 → (𝑖 ·s 𝑙) = (𝐶 ·s 𝑙))
183		oveq1 7400	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐶 → (𝑖 ·s 𝑘) = (𝐶 ·s 𝑘))
184	182, 183	oveq12d 7411	. . . . . . . . . 10 ⊢ (𝑖 = 𝐶 → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) = ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)))
185	184	breq1d 5151	. . . . . . . . 9 ⊢ (𝑖 = 𝐶 → (((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)) ↔ ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))
186	181, 185	imbi12d 344	. . . . . . . 8 ⊢ (𝑖 = 𝐶 → (((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))) ↔ ((𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))))
187	186	anbi2d 629	. . . . . . 7 ⊢ (𝑖 = 𝐶 → (((𝐴 ·s 𝐵) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))) ↔ ((𝐴 ·s 𝐵) ∈ No ∧ ((𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))))
188	179, 187	imbi12d 344	. . . . . 6 ⊢ (𝑖 = 𝐶 → ((𝑥 = ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((𝐴 ·s 𝐵) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))) ↔ (𝑥 = ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((𝐴 ·s 𝐵) ∈ No ∧ ((𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))))))
189		fveq2 6878	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐷 → ( bday ‘𝑗) = ( bday ‘𝐷))
190	189	oveq1d 7408	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐷 → (( bday ‘𝑗) +no ( bday ‘𝑙)) = (( bday ‘𝐷) +no ( bday ‘𝑙)))
191	190	uneq2d 4159	. . . . . . . . . 10 ⊢ (𝑗 = 𝐷 → ((( bday ‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) = ((( bday ‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑙))))
192	189	oveq1d 7408	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐷 → (( bday ‘𝑗) +no ( bday ‘𝑘)) = (( bday ‘𝐷) +no ( bday ‘𝑘)))
193	192	uneq2d 4159	. . . . . . . . . 10 ⊢ (𝑗 = 𝐷 → ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))) = ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑘))))
194	191, 193	uneq12d 4160	. . . . . . . . 9 ⊢ (𝑗 = 𝐷 → (((( bday ‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘)))) = (((( bday ‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑘)))))
195	194	uneq2d 4159	. . . . . . . 8 ⊢ (𝑗 = 𝐷 → ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) = ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑘))))))
196	195	eqeq2d 2742	. . . . . . 7 ⊢ (𝑗 = 𝐷 → (𝑥 = ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) ↔ 𝑥 = ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑘)))))))
197		breq2 5145	. . . . . . . . . 10 ⊢ (𝑗 = 𝐷 → (𝐶 <s 𝑗 ↔ 𝐶 <s 𝐷))
198	197	anbi1d 630	. . . . . . . . 9 ⊢ (𝑗 = 𝐷 → ((𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙) ↔ (𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙)))
199		oveq1 7400	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐷 → (𝑗 ·s 𝑙) = (𝐷 ·s 𝑙))
200		oveq1 7400	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐷 → (𝑗 ·s 𝑘) = (𝐷 ·s 𝑘))
201	199, 200	oveq12d 7411	. . . . . . . . . 10 ⊢ (𝑗 = 𝐷 → ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)) = ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝑘)))
202	201	breq2d 5153	. . . . . . . . 9 ⊢ (𝑗 = 𝐷 → (((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)) ↔ ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝑘))))
203	198, 202	imbi12d 344	. . . . . . . 8 ⊢ (𝑗 = 𝐷 → (((𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))) ↔ ((𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝑘)))))
204	203	anbi2d 629	. . . . . . 7 ⊢ (𝑗 = 𝐷 → (((𝐴 ·s 𝐵) ∈ No ∧ ((𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘)))) ↔ ((𝐴 ·s 𝐵) ∈ No ∧ ((𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝑘))))))
205	196, 204	imbi12d 344	. . . . . 6 ⊢ (𝑗 = 𝐷 → ((𝑥 = ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((𝐴 ·s 𝐵) ∈ No ∧ ((𝐶 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))) ↔ (𝑥 = ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑘))))) → ((𝐴 ·s 𝐵) ∈ No ∧ ((𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝑘)))))))
206		fveq2 6878	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝐸 → ( bday ‘𝑘) = ( bday ‘𝐸))
207	206	oveq2d 7409	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐸 → (( bday ‘𝐶) +no ( bday ‘𝑘)) = (( bday ‘𝐶) +no ( bday ‘𝐸)))
208	207	uneq1d 4158	. . . . . . . . . 10 ⊢ (𝑘 = 𝐸 → ((( bday ‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑙))) = ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑙))))
209	206	oveq2d 7409	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐸 → (( bday ‘𝐷) +no ( bday ‘𝑘)) = (( bday ‘𝐷) +no ( bday ‘𝐸)))
210	209	uneq2d 4159	. . . . . . . . . 10 ⊢ (𝑘 = 𝐸 → ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑘))) = ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))
211	208, 210	uneq12d 4160	. . . . . . . . 9 ⊢ (𝑘 = 𝐸 → (((( bday ‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑘)))) = (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))))
212	211	uneq2d 4159	. . . . . . . 8 ⊢ (𝑘 = 𝐸 → ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑘))))) = ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
213	212	eqeq2d 2742	. . . . . . 7 ⊢ (𝑘 = 𝐸 → (𝑥 = ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑘))))) ↔ 𝑥 = ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))))))
214		breq1 5144	. . . . . . . . . 10 ⊢ (𝑘 = 𝐸 → (𝑘 <s 𝑙 ↔ 𝐸 <s 𝑙))
215	214	anbi2d 629	. . . . . . . . 9 ⊢ (𝑘 = 𝐸 → ((𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙) ↔ (𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙)))
216		oveq2 7401	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐸 → (𝐶 ·s 𝑘) = (𝐶 ·s 𝐸))
217	216	oveq2d 7409	. . . . . . . . . 10 ⊢ (𝑘 = 𝐸 → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) = ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝐸)))
218		oveq2 7401	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐸 → (𝐷 ·s 𝑘) = (𝐷 ·s 𝐸))
219	218	oveq2d 7409	. . . . . . . . . 10 ⊢ (𝑘 = 𝐸 → ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝑘)) = ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝐸)))
220	217, 219	breq12d 5154	. . . . . . . . 9 ⊢ (𝑘 = 𝐸 → (((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝑘)) ↔ ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝐸))))
221	215, 220	imbi12d 344	. . . . . . . 8 ⊢ (𝑘 = 𝐸 → (((𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝑘))) ↔ ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝐸)))))
222	221	anbi2d 629	. . . . . . 7 ⊢ (𝑘 = 𝐸 → (((𝐴 ·s 𝐵) ∈ No ∧ ((𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝑘)))) ↔ ((𝐴 ·s 𝐵) ∈ No ∧ ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝐸))))))
223	213, 222	imbi12d 344	. . . . . 6 ⊢ (𝑘 = 𝐸 → ((𝑥 = ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝑘)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑘))))) → ((𝐴 ·s 𝐵) ∈ No ∧ ((𝐶 <s 𝐷 ∧ 𝑘 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝑘)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝑘))))) ↔ (𝑥 = ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝐴 ·s 𝐵) ∈ No ∧ ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝐸)))))))
224		fveq2 6878	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝐹 → ( bday ‘𝑙) = ( bday ‘𝐹))
225	224	oveq2d 7409	. . . . . . . . . . 11 ⊢ (𝑙 = 𝐹 → (( bday ‘𝐷) +no ( bday ‘𝑙)) = (( bday ‘𝐷) +no ( bday ‘𝐹)))
226	225	uneq2d 4159	. . . . . . . . . 10 ⊢ (𝑙 = 𝐹 → ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑙))) = ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))))
227	224	oveq2d 7409	. . . . . . . . . . 11 ⊢ (𝑙 = 𝐹 → (( bday ‘𝐶) +no ( bday ‘𝑙)) = (( bday ‘𝐶) +no ( bday ‘𝐹)))
228	227	uneq1d 4158	. . . . . . . . . 10 ⊢ (𝑙 = 𝐹 → ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))) = ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))
229	226, 228	uneq12d 4160	. . . . . . . . 9 ⊢ (𝑙 = 𝐹 → (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))) = (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))))
230	229	uneq2d 4159	. . . . . . . 8 ⊢ (𝑙 = 𝐹 → ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( 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))))))
231	230	eqeq2d 2742	. . . . . . 7 ⊢ (𝑙 = 𝐹 → (𝑥 = ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) ↔ 𝑥 = ((( 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)))))))
232		breq2 5145	. . . . . . . . . 10 ⊢ (𝑙 = 𝐹 → (𝐸 <s 𝑙 ↔ 𝐸 <s 𝐹))
233	232	anbi2d 629	. . . . . . . . 9 ⊢ (𝑙 = 𝐹 → ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙) ↔ (𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹)))
234		oveq2 7401	. . . . . . . . . . 11 ⊢ (𝑙 = 𝐹 → (𝐶 ·s 𝑙) = (𝐶 ·s 𝐹))
235	234	oveq1d 7408	. . . . . . . . . 10 ⊢ (𝑙 = 𝐹 → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝐸)) = ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)))
236		oveq2 7401	. . . . . . . . . . 11 ⊢ (𝑙 = 𝐹 → (𝐷 ·s 𝑙) = (𝐷 ·s 𝐹))
237	236	oveq1d 7408	. . . . . . . . . 10 ⊢ (𝑙 = 𝐹 → ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝐸)) = ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))
238	235, 237	breq12d 5154	. . . . . . . . 9 ⊢ (𝑙 = 𝐹 → (((𝐶 ·s 𝑙) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝐸)) ↔ ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸))))
239	233, 238	imbi12d 344	. . . . . . . 8 ⊢ (𝑙 = 𝐹 → (((𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝐸))) ↔ ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝐹) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))))
240	239	anbi2d 629	. . . . . . 7 ⊢ (𝑙 = 𝐹 → (((𝐴 ·s 𝐵) ∈ No ∧ ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 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))))))
241	231, 240	imbi12d 344	. . . . . 6 ⊢ (𝑙 = 𝐹 → ((𝑥 = ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑙)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((𝐴 ·s 𝐵) ∈ No ∧ ((𝐶 <s 𝐷 ∧ 𝐸 <s 𝑙) → ((𝐶 ·s 𝑙) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝑙) -s (𝐷 ·s 𝐸))))) ↔ (𝑥 = ((( 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)))))))
242	163, 171, 188, 205, 223, 241	rspc6v 3627	. . . . 5 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐶 ∈ No ∧ 𝐷 ∈ No ) ∧ (𝐸 ∈ No ∧ 𝐹 ∈ No )) → (∀𝑔 ∈ No ∀ℎ ∈ No ∀𝑖 ∈ No ∀𝑗 ∈ No ∀𝑘 ∈ No ∀𝑙 ∈ No (𝑥 = ((( bday ‘𝑔) +no ( bday ‘ℎ)) ∪ (((( bday ‘𝑖) +no ( bday ‘𝑘)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑙))) ∪ ((( bday ‘𝑖) +no ( bday ‘𝑙)) ∪ (( bday ‘𝑗) +no ( bday ‘𝑘))))) → ((𝑔 ·s ℎ) ∈ No ∧ ((𝑖 <s 𝑗 ∧ 𝑘 <s 𝑙) → ((𝑖 ·s 𝑙) -s (𝑖 ·s 𝑘)) <s ((𝑗 ·s 𝑙) -s (𝑗 ·s 𝑘))))) → (𝑥 = ((( 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)))))))
243	155, 242	syl5com 31	. . . 4 ⊢ (𝑥 ∈ On → (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐶 ∈ No ∧ 𝐷 ∈ No ) ∧ (𝐸 ∈ No ∧ 𝐹 ∈ No )) → (𝑥 = ((( 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)))))))
244	243	com23 86	. . 3 ⊢ (𝑥 ∈ On → (𝑥 = ((( 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 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐶 ∈ No ∧ 𝐷 ∈ No ) ∧ (𝐸 ∈ No ∧ 𝐹 ∈ No )) → ((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)))))))
245	244	rexlimiv 3147	. 2 ⊢ (∃𝑥 ∈ On 𝑥 = ((( 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 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐶 ∈ No ∧ 𝐷 ∈ No ) ∧ (𝐸 ∈ No ∧ 𝐹 ∈ No )) → ((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))))))
246	22, 245	ax-mp 5	1 ⊢ (((𝐴 ∈ No ∧ 𝐵 ∈ No ) ∧ (𝐶 ∈ No ∧ 𝐷 ∈ No ) ∧ (𝐸 ∈ No ∧ 𝐹 ∈ No )) → ((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)))))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3060  ∃wrex 3069   ∪ cun 3942   class class class wbr 5141  Oncon0 6353  ‘cfv 6532  (class class class)co 7393   +no cnadd 8647   No csur 27070   <s cslt 27071   bday cbday 27072   -_s csubs 27411   ·_s cmuls 27476

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-tp 4627  df-op 4629  df-ot 4631  df-uni 4902  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-1o 8448  df-2o 8449  df-nadd 8648  df-no 27073  df-slt 27074  df-bday 27075  df-sle 27175  df-sslt 27209  df-scut 27211  df-0s 27251  df-made 27265  df-old 27266  df-left 27268  df-right 27269  df-norec 27338  df-norec2 27349  df-adds 27360  df-negs 27412  df-subs 27413  df-muls 27477

This theorem is referenced by:  mulscutlem  27500  mulscl  27503  sltmul  27505