Theorem mulsproplem13 27497

Description: Lemma for surreal multiplication. Remove the restriction on 𝐶 and 𝐷 from mulsproplem12 27496. (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))))))
mulsproplem.2	⊢ (𝜑 → 𝐶 ∈ No )
mulsproplem.3	⊢ (𝜑 → 𝐷 ∈ No )
mulsproplem.4	⊢ (𝜑 → 𝐸 ∈ No )
mulsproplem.5	⊢ (𝜑 → 𝐹 ∈ No )
mulsproplem.6	⊢ (𝜑 → 𝐶 <s 𝐷)
mulsproplem.7	⊢ (𝜑 → 𝐸 <s 𝐹)
mulsproplem13.1	⊢ (𝜑 → (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸)))

Assertion

Ref	Expression
mulsproplem13	⊢ (𝜑 → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))

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

Allowed substitution hints:   𝜑(𝑒,𝑓,𝑎,𝑏,𝑐,𝑑)

Proof of Theorem mulsproplem13

Dummy variables 𝑔 ℎ 𝑖 𝑗 𝑘 𝑙 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mulsproplem.1	. . . 4 ⊢ (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))))))
2	1	adantr 481	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶))) → ∀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))))))
3		mulsproplem.2	. . . 4 ⊢ (𝜑 → 𝐶 ∈ No )
4	3	adantr 481	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶))) → 𝐶 ∈ No )
5		mulsproplem.3	. . . 4 ⊢ (𝜑 → 𝐷 ∈ No )
6	5	adantr 481	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶))) → 𝐷 ∈ No )
7		mulsproplem.4	. . . 4 ⊢ (𝜑 → 𝐸 ∈ No )
8	7	adantr 481	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶))) → 𝐸 ∈ No )
9		mulsproplem.5	. . . 4 ⊢ (𝜑 → 𝐹 ∈ No )
10	9	adantr 481	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶))) → 𝐹 ∈ No )
11		mulsproplem.6	. . . 4 ⊢ (𝜑 → 𝐶 <s 𝐷)
12	11	adantr 481	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶))) → 𝐶 <s 𝐷)
13		mulsproplem.7	. . . 4 ⊢ (𝜑 → 𝐸 <s 𝐹)
14	13	adantr 481	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶))) → 𝐸 <s 𝐹)
15		simpr 485	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶))) → (( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶)))
16		mulsproplem13.1	. . . 4 ⊢ (𝜑 → (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸)))
17	16	adantr 481	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶))) → (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸)))
18	2, 4, 6, 8, 10, 12, 14, 15, 17	mulsproplem12 27496	. 2 ⊢ ((𝜑 ∧ (( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶))) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))
19	3	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐶) = ( bday ‘𝐷)) → 𝐶 ∈ No )
20	5	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐶) = ( bday ‘𝐷)) → 𝐷 ∈ No )
21		simpr 485	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐶) = ( bday ‘𝐷)) → ( bday ‘𝐶) = ( bday ‘𝐷))
22	11	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐶) = ( bday ‘𝐷)) → 𝐶 <s 𝐷)
23		nodense 27122	. . . 4 ⊢ (((𝐶 ∈ No ∧ 𝐷 ∈ No ) ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ 𝐶 <s 𝐷)) → ∃𝑥 ∈ No (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷))
24	19, 20, 21, 22, 23	syl22anc 837	. . 3 ⊢ ((𝜑 ∧ ( bday ‘𝐶) = ( bday ‘𝐷)) → ∃𝑥 ∈ No (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷))
25		unidm 4148	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s )))) = ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s )))
26		unidm 4148	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) = (( bday ‘ 0s ) +no ( bday ‘ 0s ))
27		bday0s 27255	. . . . . . . . . . . . . . . . . 18 ⊢ ( bday ‘ 0s ) = ∅
28	27, 27	oveq12i 7405	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = (∅ +no ∅)
29		0elon 6407	. . . . . . . . . . . . . . . . . 18 ⊢ ∅ ∈ On
30		naddrid 8665	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ ∈ On → (∅ +no ∅) = ∅)
31	29, 30	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ (∅ +no ∅) = ∅
32	28, 31	eqtri 2759	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = ∅
33	25, 26, 32	3eqtri 2763	. . . . . . . . . . . . . . 15 ⊢ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s )))) = ∅
34	33	uneq2i 4156	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) = ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ ∅)
35		un0 4386	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ ∅) = (( bday ‘𝐶) +no ( bday ‘𝐹))
36	34, 35	eqtri 2759	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) = (( bday ‘𝐶) +no ( bday ‘𝐹))
37		ssun1 4168	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝐶) +no ( bday ‘𝐹)) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))
38		ssun2 4169	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( 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))))
39	37, 38	sstri 3987	. . . . . . . . . . . . . 14 ⊢ (( bday ‘𝐶) +no ( bday ‘𝐹)) ⊆ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))
40		ssun2 4169	. . . . . . . . . . . . . 14 ⊢ (((( 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)))))
41	39, 40	sstri 3987	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝐶) +no ( bday ‘𝐹)) ⊆ ((( 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)))))
42	36, 41	eqsstri 4012	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) ⊆ ((( 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)))))
43	42	sseli 3974	. . . . . . . . . . 11 ⊢ (((( 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 ‘𝐶) +no ( bday ‘𝐹)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((( 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))))))
44	43	imim1i 63	. . . . . . . . . 10 ⊢ ((((( 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))))) → (((( 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 ‘𝐶) +no ( bday ‘𝐹)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((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))))))
45	44	6ralimi 3126	. . . . . . . . 9 ⊢ (∀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))))) → ∀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 ‘𝐶) +no ( bday ‘𝐹)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((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))))))
46	1, 45	syl 17	. . . . . . . 8 ⊢ (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 ‘𝐶) +no ( bday ‘𝐹)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((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))))))
47	46, 3, 9	mulsproplem11 27495	. . . . . . 7 ⊢ (𝜑 → (𝐶 ·s 𝐹) ∈ No )
48	33	uneq2i 4156	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) = ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ ∅)
49		un0 4386	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ ∅) = (( bday ‘𝐶) +no ( bday ‘𝐸))
50	48, 49	eqtri 2759	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) = (( bday ‘𝐶) +no ( bday ‘𝐸))
51		ssun1 4168	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝐶) +no ( bday ‘𝐸)) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹)))
52		ssun1 4168	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ⊆ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))
53	51, 52	sstri 3987	. . . . . . . . . . . . . 14 ⊢ (( bday ‘𝐶) +no ( bday ‘𝐸)) ⊆ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))
54	53, 40	sstri 3987	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝐶) +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)))))
55	50, 54	eqsstri 4012	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) ⊆ ((( 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)))))
56	55	sseli 3974	. . . . . . . . . . 11 ⊢ (((( 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 ‘𝐶) +no ( bday ‘𝐸)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((( 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))))))
57	56	imim1i 63	. . . . . . . . . 10 ⊢ ((((( 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))))) → (((( 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 ‘𝐶) +no ( bday ‘𝐸)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((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))))))
58	57	6ralimi 3126	. . . . . . . . 9 ⊢ (∀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))))) → ∀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 ‘𝐶) +no ( bday ‘𝐸)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((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))))))
59	1, 58	syl 17	. . . . . . . 8 ⊢ (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 ‘𝐶) +no ( bday ‘𝐸)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((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))))))
60	59, 3, 7	mulsproplem11 27495	. . . . . . 7 ⊢ (𝜑 → (𝐶 ·s 𝐸) ∈ No )
61	47, 60	subscld 27449	. . . . . 6 ⊢ (𝜑 → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) ∈ No )
62	61	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) ∈ No )
63	46	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ∀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 ‘𝐶) +no ( bday ‘𝐹)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((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))))))
64		simprr1 1221	. . . . . . . . 9 ⊢ ((( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷))) → ( bday ‘𝑥) ∈ ( bday ‘𝐶))
65	64	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ( bday ‘𝑥) ∈ ( bday ‘𝐶))
66		bdayelon 27204	. . . . . . . . 9 ⊢ ( bday ‘𝐶) ∈ On
67		simprrl 779	. . . . . . . . 9 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → 𝑥 ∈ No )
68		oldbday 27318	. . . . . . . . 9 ⊢ ((( bday ‘𝐶) ∈ On ∧ 𝑥 ∈ No ) → (𝑥 ∈ ( O ‘( bday ‘𝐶)) ↔ ( bday ‘𝑥) ∈ ( bday ‘𝐶)))
69	66, 67, 68	sylancr 587	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → (𝑥 ∈ ( O ‘( bday ‘𝐶)) ↔ ( bday ‘𝑥) ∈ ( bday ‘𝐶)))
70	65, 69	mpbird 256	. . . . . . 7 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → 𝑥 ∈ ( O ‘( bday ‘𝐶)))
71	9	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → 𝐹 ∈ No )
72	63, 70, 71	mulsproplem2 27486	. . . . . 6 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → (𝑥 ·s 𝐹) ∈ No )
73	59	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ∀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 ‘𝐶) +no ( bday ‘𝐸)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((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))))))
74	7	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → 𝐸 ∈ No )
75	73, 70, 74	mulsproplem2 27486	. . . . . 6 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → (𝑥 ·s 𝐸) ∈ No )
76	72, 75	subscld 27449	. . . . 5 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ((𝑥 ·s 𝐹) -s (𝑥 ·s 𝐸)) ∈ No )
77	33	uneq2i 4156	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝐷) +no ( bday ‘𝐹)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) = ((( bday ‘𝐷) +no ( bday ‘𝐹)) ∪ ∅)
78		un0 4386	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝐷) +no ( bday ‘𝐹)) ∪ ∅) = (( bday ‘𝐷) +no ( bday ‘𝐹))
79	77, 78	eqtri 2759	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝐷) +no ( bday ‘𝐹)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) = (( bday ‘𝐷) +no ( bday ‘𝐹))
80		ssun2 4169	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝐷) +no ( bday ‘𝐹)) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹)))
81	80, 52	sstri 3987	. . . . . . . . . . . . . 14 ⊢ (( bday ‘𝐷) +no ( bday ‘𝐹)) ⊆ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))
82	81, 40	sstri 3987	. . . . . . . . . . . . 13 ⊢ (( bday ‘𝐷) +no ( bday ‘𝐹)) ⊆ ((( 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)))))
83	79, 82	eqsstri 4012	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝐷) +no ( bday ‘𝐹)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) ⊆ ((( 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)))))
84	83	sseli 3974	. . . . . . . . . . 11 ⊢ (((( 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 ‘𝐷) +no ( bday ‘𝐹)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((( 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))))))
85	84	imim1i 63	. . . . . . . . . 10 ⊢ ((((( 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))))) → (((( 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 ‘𝐷) +no ( bday ‘𝐹)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((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	85	6ralimi 3126	. . . . . . . . 9 ⊢ (∀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))))) → ∀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 ‘𝐷) +no ( bday ‘𝐹)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((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))))))
87	1, 86	syl 17	. . . . . . . 8 ⊢ (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 ‘𝐷) +no ( bday ‘𝐹)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((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))))))
88	87, 5, 9	mulsproplem11 27495	. . . . . . 7 ⊢ (𝜑 → (𝐷 ·s 𝐹) ∈ No )
89	33	uneq2i 4156	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝐷) +no ( bday ‘𝐸)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) = ((( bday ‘𝐷) +no ( bday ‘𝐸)) ∪ ∅)
90		un0 4386	. . . . . . . . . . . . . 14 ⊢ ((( bday ‘𝐷) +no ( bday ‘𝐸)) ∪ ∅) = (( bday ‘𝐷) +no ( bday ‘𝐸))
91	89, 90	eqtri 2759	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝐷) +no ( bday ‘𝐸)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) = (( bday ‘𝐷) +no ( bday ‘𝐸))
92		ssun2 4169	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘𝐷) +no ( bday ‘𝐸)) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))
93	92, 38	sstri 3987	. . . . . . . . . . . . . 14 ⊢ (( 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))))
94	93, 40	sstri 3987	. . . . . . . . . . . . 13 ⊢ (( 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)))))
95	91, 94	eqsstri 4012	. . . . . . . . . . . 12 ⊢ ((( bday ‘𝐷) +no ( bday ‘𝐸)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) ⊆ ((( 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)))))
96	95	sseli 3974	. . . . . . . . . . 11 ⊢ (((( 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 ‘𝐷) +no ( bday ‘𝐸)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((( 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))))))
97	96	imim1i 63	. . . . . . . . . 10 ⊢ ((((( 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))))) → (((( 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 ‘𝐷) +no ( bday ‘𝐸)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((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))))))
98	97	6ralimi 3126	. . . . . . . . 9 ⊢ (∀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))))) → ∀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 ‘𝐷) +no ( bday ‘𝐸)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((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))))))
99	1, 98	syl 17	. . . . . . . 8 ⊢ (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 ‘𝐷) +no ( bday ‘𝐸)) ∪ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))))) → ((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))))))
100	99, 5, 7	mulsproplem11 27495	. . . . . . 7 ⊢ (𝜑 → (𝐷 ·s 𝐸) ∈ No )
101	88, 100	subscld 27449	. . . . . 6 ⊢ (𝜑 → ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)) ∈ No )
102	101	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)) ∈ No )
103	1	mulsproplemcbv 27484	. . . . . . . 8 ⊢ (𝜑 → ∀𝑔 ∈ No ∀ℎ ∈ No ∀𝑖 ∈ No ∀𝑗 ∈ No ∀𝑘 ∈ No ∀𝑙 ∈ No (((( 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 ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((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))))))
104	103	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ∀𝑔 ∈ No ∀ℎ ∈ No ∀𝑖 ∈ No ∀𝑗 ∈ No ∀𝑘 ∈ No ∀𝑙 ∈ No (((( 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 ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((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))))))
105		onelss 6395	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝐶) ∈ On → (( bday ‘𝑥) ∈ ( bday ‘𝐶) → ( bday ‘𝑥) ⊆ ( bday ‘𝐶)))
106	66, 65, 105	mpsyl 68	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ( bday ‘𝑥) ⊆ ( bday ‘𝐶))
107		simprl 769	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ( bday ‘𝐶) = ( bday ‘𝐷))
108	106, 107	sseqtrd 4018	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ( bday ‘𝑥) ⊆ ( bday ‘𝐷))
109		bdayelon 27204	. . . . . . . . . . . . . . 15 ⊢ ( bday ‘𝑥) ∈ On
110		bdayelon 27204	. . . . . . . . . . . . . . 15 ⊢ ( bday ‘𝐷) ∈ On
111		bdayelon 27204	. . . . . . . . . . . . . . 15 ⊢ ( bday ‘𝐹) ∈ On
112		naddss1 8671	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝑥) ∈ On ∧ ( bday ‘𝐷) ∈ On ∧ ( bday ‘𝐹) ∈ On) → (( bday ‘𝑥) ⊆ ( bday ‘𝐷) ↔ (( bday ‘𝑥) +no ( bday ‘𝐹)) ⊆ (( bday ‘𝐷) +no ( bday ‘𝐹))))
113	109, 110, 111, 112	mp3an 1461	. . . . . . . . . . . . . 14 ⊢ (( bday ‘𝑥) ⊆ ( bday ‘𝐷) ↔ (( bday ‘𝑥) +no ( bday ‘𝐹)) ⊆ (( bday ‘𝐷) +no ( bday ‘𝐹)))
114	108, 113	sylib 217	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → (( bday ‘𝑥) +no ( bday ‘𝐹)) ⊆ (( bday ‘𝐷) +no ( bday ‘𝐹)))
115		unss2 4177	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑥) +no ( bday ‘𝐹)) ⊆ (( bday ‘𝐷) +no ( bday ‘𝐹)) → ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝑥) +no ( bday ‘𝐹))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))))
116	114, 115	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝑥) +no ( bday ‘𝐹))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))))
117		bdayelon 27204	. . . . . . . . . . . . . . 15 ⊢ ( bday ‘𝐸) ∈ On
118		naddss1 8671	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝑥) ∈ On ∧ ( bday ‘𝐷) ∈ On ∧ ( bday ‘𝐸) ∈ On) → (( bday ‘𝑥) ⊆ ( bday ‘𝐷) ↔ (( bday ‘𝑥) +no ( bday ‘𝐸)) ⊆ (( bday ‘𝐷) +no ( bday ‘𝐸))))
119	109, 110, 117, 118	mp3an 1461	. . . . . . . . . . . . . 14 ⊢ (( bday ‘𝑥) ⊆ ( bday ‘𝐷) ↔ (( bday ‘𝑥) +no ( bday ‘𝐸)) ⊆ (( bday ‘𝐷) +no ( bday ‘𝐸)))
120	108, 119	sylib 217	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → (( bday ‘𝑥) +no ( bday ‘𝐸)) ⊆ (( bday ‘𝐷) +no ( bday ‘𝐸)))
121		unss2 4177	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑥) +no ( bday ‘𝐸)) ⊆ (( bday ‘𝐷) +no ( bday ‘𝐸)) → ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝑥) +no ( bday ‘𝐸))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))
122	120, 121	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝑥) +no ( bday ‘𝐸))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))
123		unss12 4178	. . . . . . . . . . . 12 ⊢ ((((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝑥) +no ( bday ‘𝐹))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∧ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝑥) +no ( bday ‘𝐸))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))) → (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝑥) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝑥) +no ( bday ‘𝐸)))) ⊆ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))))
124	116, 122, 123	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝑥) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝑥) +no ( bday ‘𝐸)))) ⊆ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))))
125		unss2 4177	. . . . . . . . . . 11 ⊢ ((((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝑥) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝑥) +no ( bday ‘𝐸)))) ⊆ (((( 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 ‘𝑥) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝑥) +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))))))
126	124, 125	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝑥) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝑥) +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))))))
127	126	sseld 3977	. . . . . . . . 9 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → (((( 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 ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝑥) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝑥) +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))))) ∈ ((( 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)))))))
128	127	imim1d 82	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ((((( 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 ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((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))))) → (((( 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 ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝑥) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝑥) +no ( bday ‘𝐸))))) → ((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)))))))
129	128	ralimd6v 3207	. . . . . . 7 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → (∀𝑔 ∈ No ∀ℎ ∈ No ∀𝑖 ∈ No ∀𝑗 ∈ No ∀𝑘 ∈ No ∀𝑙 ∈ No (((( 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 ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((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))))) → ∀𝑔 ∈ No ∀ℎ ∈ No ∀𝑖 ∈ No ∀𝑗 ∈ No ∀𝑘 ∈ No ∀𝑙 ∈ No (((( 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 ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝑥) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝑥) +no ( bday ‘𝐸))))) → ((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)))))))
130	104, 129	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ∀𝑔 ∈ No ∀ℎ ∈ No ∀𝑖 ∈ No ∀𝑗 ∈ No ∀𝑘 ∈ No ∀𝑙 ∈ No (((( 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 ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝑥) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝑥) +no ( bday ‘𝐸))))) → ((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))))))
131	3	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → 𝐶 ∈ No )
132		simprr2 1222	. . . . . . 7 ⊢ ((( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷))) → 𝐶 <s 𝑥)
133	132	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → 𝐶 <s 𝑥)
134	13	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → 𝐸 <s 𝐹)
135	65	olcd 872	. . . . . 6 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → (( bday ‘𝐶) ∈ ( bday ‘𝑥) ∨ ( bday ‘𝑥) ∈ ( bday ‘𝐶)))
136	16	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸)))
137	130, 131, 67, 74, 71, 133, 134, 135, 136	mulsproplem12 27496	. . . . 5 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝑥 ·s 𝐹) -s (𝑥 ·s 𝐸)))
138		naddss1 8671	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝑥) ∈ On ∧ ( bday ‘𝐶) ∈ On ∧ ( bday ‘𝐸) ∈ On) → (( bday ‘𝑥) ⊆ ( bday ‘𝐶) ↔ (( bday ‘𝑥) +no ( bday ‘𝐸)) ⊆ (( bday ‘𝐶) +no ( bday ‘𝐸))))
139	109, 66, 117, 138	mp3an 1461	. . . . . . . . . . . . . 14 ⊢ (( bday ‘𝑥) ⊆ ( bday ‘𝐶) ↔ (( bday ‘𝑥) +no ( bday ‘𝐸)) ⊆ (( bday ‘𝐶) +no ( bday ‘𝐸)))
140	106, 139	sylib 217	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → (( bday ‘𝑥) +no ( bday ‘𝐸)) ⊆ (( bday ‘𝐶) +no ( bday ‘𝐸)))
141		unss1 4175	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑥) +no ( bday ‘𝐸)) ⊆ (( bday ‘𝐶) +no ( bday ‘𝐸)) → ((( bday ‘𝑥) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))))
142	140, 141	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ((( bday ‘𝑥) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))))
143		naddss1 8671	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝑥) ∈ On ∧ ( bday ‘𝐶) ∈ On ∧ ( bday ‘𝐹) ∈ On) → (( bday ‘𝑥) ⊆ ( bday ‘𝐶) ↔ (( bday ‘𝑥) +no ( bday ‘𝐹)) ⊆ (( bday ‘𝐶) +no ( bday ‘𝐹))))
144	109, 66, 111, 143	mp3an 1461	. . . . . . . . . . . . . 14 ⊢ (( bday ‘𝑥) ⊆ ( bday ‘𝐶) ↔ (( bday ‘𝑥) +no ( bday ‘𝐹)) ⊆ (( bday ‘𝐶) +no ( bday ‘𝐹)))
145	106, 144	sylib 217	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → (( bday ‘𝑥) +no ( bday ‘𝐹)) ⊆ (( bday ‘𝐶) +no ( bday ‘𝐹)))
146		unss1 4175	. . . . . . . . . . . . 13 ⊢ ((( bday ‘𝑥) +no ( bday ‘𝐹)) ⊆ (( bday ‘𝐶) +no ( bday ‘𝐹)) → ((( bday ‘𝑥) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))
147	145, 146	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ((( bday ‘𝑥) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))
148		unss12 4178	. . . . . . . . . . . 12 ⊢ ((((( bday ‘𝑥) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∧ ((( bday ‘𝑥) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))) → (((( bday ‘𝑥) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝑥) +no ( bday ‘𝐹)) ∪ (( 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)))))
149	142, 147, 148	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → (((( bday ‘𝑥) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝑥) +no ( bday ‘𝐹)) ∪ (( 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)))))
150		unss2 4177	. . . . . . . . . . 11 ⊢ ((((( bday ‘𝑥) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝑥) +no ( bday ‘𝐹)) ∪ (( 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)))) → ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝑥) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝑥) +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))))))
151	149, 150	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝑥) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝑥) +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))))))
152	151	sseld 3977	. . . . . . . . 9 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → (((( 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 ‘𝐵)) ∪ (((( bday ‘𝑥) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝑥) +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))))) ∈ ((( 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)))))))
153	152	imim1d 82	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ((((( 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 ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((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))))) → (((( 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 ‘𝐵)) ∪ (((( bday ‘𝑥) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝑥) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((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	ralimd6v 3207	. . . . . . 7 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → (∀𝑔 ∈ No ∀ℎ ∈ No ∀𝑖 ∈ No ∀𝑗 ∈ No ∀𝑘 ∈ No ∀𝑙 ∈ No (((( 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 ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((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))))) → ∀𝑔 ∈ No ∀ℎ ∈ No ∀𝑖 ∈ No ∀𝑗 ∈ No ∀𝑘 ∈ No ∀𝑙 ∈ No (((( 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 ‘𝐵)) ∪ (((( bday ‘𝑥) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝑥) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((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	104, 154	mpd 15	. . . . . 6 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ∀𝑔 ∈ No ∀ℎ ∈ No ∀𝑖 ∈ No ∀𝑗 ∈ No ∀𝑘 ∈ No ∀𝑙 ∈ No (((( 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 ‘𝐵)) ∪ (((( bday ‘𝑥) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝑥) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))) → ((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	5	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → 𝐷 ∈ No )
157		simprr3 1223	. . . . . . 7 ⊢ ((( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷))) → 𝑥 <s 𝐷)
158	157	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → 𝑥 <s 𝐷)
159	65, 107	eleqtrd 2834	. . . . . . 7 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ( bday ‘𝑥) ∈ ( bday ‘𝐷))
160	159	orcd 871	. . . . . 6 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → (( bday ‘𝑥) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝑥)))
161	155, 67, 156, 74, 71, 158, 134, 160, 136	mulsproplem12 27496	. . . . 5 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ((𝑥 ·s 𝐹) -s (𝑥 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))
162	62, 76, 102, 137, 161	slttrd 27189	. . . 4 ⊢ ((𝜑 ∧ (( bday ‘𝐶) = ( bday ‘𝐷) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷)))) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))
163	162	anassrs 468	. . 3 ⊢ (((𝜑 ∧ ( bday ‘𝐶) = ( bday ‘𝐷)) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐶) ∧ 𝐶 <s 𝑥 ∧ 𝑥 <s 𝐷))) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))
164	24, 163	rexlimddv 3160	. 2 ⊢ ((𝜑 ∧ ( bday ‘𝐶) = ( bday ‘𝐷)) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))
165	66	onordi 6464	. . . . 5 ⊢ Ord ( bday ‘𝐶)
166	110	onordi 6464	. . . . 5 ⊢ Ord ( bday ‘𝐷)
167		ordtri3or 6385	. . . . 5 ⊢ ((Ord ( bday ‘𝐶) ∧ Ord ( bday ‘𝐷)) → (( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐶) = ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶)))
168	165, 166, 167	mp2an 690	. . . 4 ⊢ (( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐶) = ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶))
169		df-3or 1088	. . . . 5 ⊢ ((( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐶) = ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶)) ↔ ((( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐶) = ( bday ‘𝐷)) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶)))
170		or32 924	. . . . 5 ⊢ (((( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐶) = ( bday ‘𝐷)) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶)) ↔ ((( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶)) ∨ ( bday ‘𝐶) = ( bday ‘𝐷)))
171	169, 170	bitri 274	. . . 4 ⊢ ((( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐶) = ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶)) ↔ ((( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶)) ∨ ( bday ‘𝐶) = ( bday ‘𝐷)))
172	168, 171	mpbi 229	. . 3 ⊢ ((( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶)) ∨ ( bday ‘𝐶) = ( bday ‘𝐷))
173	172	a1i 11	. 2 ⊢ (𝜑 → ((( bday ‘𝐶) ∈ ( bday ‘𝐷) ∨ ( bday ‘𝐷) ∈ ( bday ‘𝐶)) ∨ ( bday ‘𝐶) = ( bday ‘𝐷)))
174	18, 164, 173	mpjaodan 957	1 ⊢ (𝜑 → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∨ w3o 1086   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3060  ∃wrex 3069   ∪ cun 3942   ⊆ wss 3944  ∅c0 4318   class class class wbr 5141  Ord word 6352  Oncon0 6353  ‘cfv 6532  (class class class)co 7393   +no cnadd 8647   No csur 27070   <s cslt 27071   bday cbday 27072   0_s c0s 27249   O cold 27261   -_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:  mulsproplem14  27498