Theorem mulsproplem14 27498

Description: Lemma for surreal multiplication. Finally, we remove the restriction on 𝐸 and 𝐹 from mulsproplem12 27496 and mulsproplem13 27497. This completes the induction on surreal multiplication. mulsprop 27499 brings all this together technically. (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)

Assertion

Ref	Expression
mulsproplem14	⊢ (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 mulsproplem14

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 ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘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 ‘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 ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))) → 𝐶 ∈ No )
5		mulsproplem.3	. . . 4 ⊢ (𝜑 → 𝐷 ∈ No )
6	5	adantr 481	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))) → 𝐷 ∈ No )
7		mulsproplem.4	. . . 4 ⊢ (𝜑 → 𝐸 ∈ No )
8	7	adantr 481	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))) → 𝐸 ∈ No )
9		mulsproplem.5	. . . 4 ⊢ (𝜑 → 𝐹 ∈ No )
10	9	adantr 481	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))) → 𝐹 ∈ No )
11		mulsproplem.6	. . . 4 ⊢ (𝜑 → 𝐶 <s 𝐷)
12	11	adantr 481	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))) → 𝐶 <s 𝐷)
13		mulsproplem.7	. . . 4 ⊢ (𝜑 → 𝐸 <s 𝐹)
14	13	adantr 481	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))) → 𝐸 <s 𝐹)
15		simpr 485	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))) → (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸)))
16	2, 4, 6, 8, 10, 12, 14, 15	mulsproplem13 27497	. 2 ⊢ ((𝜑 ∧ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))
17	7	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐸) = ( bday ‘𝐹)) → 𝐸 ∈ No )
18	9	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐸) = ( bday ‘𝐹)) → 𝐹 ∈ No )
19		simpr 485	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐸) = ( bday ‘𝐹)) → ( bday ‘𝐸) = ( bday ‘𝐹))
20	13	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐸) = ( bday ‘𝐹)) → 𝐸 <s 𝐹)
21		nodense 27122	. . . 4 ⊢ (((𝐸 ∈ No ∧ 𝐹 ∈ No ) ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ 𝐸 <s 𝐹)) → ∃𝑥 ∈ No (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹))
22	17, 18, 19, 20, 21	syl22anc 837	. . 3 ⊢ ((𝜑 ∧ ( bday ‘𝐸) = ( bday ‘𝐹)) → ∃𝑥 ∈ No (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹))
23		unidm 4148	. . . . . . . . . . . . . . . . 17 ⊢ (((( 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 )))
24		unidm 4148	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) = (( bday ‘ 0s ) +no ( bday ‘ 0s ))
25		bday0s 27255	. . . . . . . . . . . . . . . . . . 19 ⊢ ( bday ‘ 0s ) = ∅
26	25, 25	oveq12i 7405	. . . . . . . . . . . . . . . . . 18 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = (∅ +no ∅)
27		0elon 6407	. . . . . . . . . . . . . . . . . . 19 ⊢ ∅ ∈ On
28		naddrid 8665	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∈ On → (∅ +no ∅) = ∅)
29	27, 28	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ +no ∅) = ∅
30	26, 29	eqtri 2759	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = ∅
31	23, 24, 30	3eqtri 2763	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s )))) = ∅
32	31	uneq2i 4156	. . . . . . . . . . . . . . 15 ⊢ ((( 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)) ∪ ∅)
33		un0 4386	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝐷) +no ( bday ‘𝐸)) ∪ ∅) = (( bday ‘𝐷) +no ( bday ‘𝐸))
34	32, 33	eqtri 2759	. . . . . . . . . . . . . 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))
35		ssun2 4169	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝐷) +no ( bday ‘𝐸)) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))
36		ssun2 4169	. . . . . . . . . . . . . . . 16 ⊢ ((( 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))))
37	35, 36	sstri 3987	. . . . . . . . . . . . . . 15 ⊢ (( 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))))
38		ssun2 4169	. . . . . . . . . . . . . . 15 ⊢ (((( 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)))))
39	37, 38	sstri 3987	. . . . . . . . . . . . . 14 ⊢ (( 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)))))
40	34, 39	eqsstri 4012	. . . . . . . . . . . . 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 ‘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	40	sseli 3974	. . . . . . . . . . . 12 ⊢ (((( 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))))))
42	41	imim1i 63	. . . . . . . . . . 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 ‘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))))))
43	42	6ralimi 3126	. . . . . . . . . 10 ⊢ (∀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))))))
44	1, 43	syl 17	. . . . . . . . 9 ⊢ (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))))))
45	44, 5, 7	mulsproplem11 27495	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ·s 𝐸) ∈ No )
46	31	uneq2i 4156	. . . . . . . . . . . . . . 15 ⊢ ((( 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)) ∪ ∅)
47		un0 4386	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ ∅) = (( bday ‘𝐶) +no ( bday ‘𝐸))
48	46, 47	eqtri 2759	. . . . . . . . . . . . . 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		ssun1 4168	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝐶) +no ( bday ‘𝐸)) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹)))
50		ssun1 4168	. . . . . . . . . . . . . . . 16 ⊢ ((( 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))))
51	49, 50	sstri 3987	. . . . . . . . . . . . . . 15 ⊢ (( 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))))
52	51, 38	sstri 3987	. . . . . . . . . . . . . 14 ⊢ (( 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)))))
53	48, 52	eqsstri 4012	. . . . . . . . . . . . 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 ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))))
54	53	sseli 3974	. . . . . . . . . . . 12 ⊢ (((( 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))))))
55	54	imim1i 63	. . . . . . . . . . 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 ‘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))))))
56	55	6ralimi 3126	. . . . . . . . . 10 ⊢ (∀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))))))
57	1, 56	syl 17	. . . . . . . . 9 ⊢ (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))))))
58	57, 3, 7	mulsproplem11 27495	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ·s 𝐸) ∈ No )
59	45, 58	subscld 27449	. . . . . . 7 ⊢ (𝜑 → ((𝐷 ·s 𝐸) -s (𝐶 ·s 𝐸)) ∈ No )
60	59	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ((𝐷 ·s 𝐸) -s (𝐶 ·s 𝐸)) ∈ No )
61	44	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ∀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))))))
62	5	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → 𝐷 ∈ No )
63		simprr1 1221	. . . . . . . . . 10 ⊢ ((( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹))) → ( bday ‘𝑥) ∈ ( bday ‘𝐸))
64	63	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ( bday ‘𝑥) ∈ ( bday ‘𝐸))
65		bdayelon 27204	. . . . . . . . . 10 ⊢ ( bday ‘𝐸) ∈ On
66		simprrl 779	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → 𝑥 ∈ No )
67		oldbday 27318	. . . . . . . . . 10 ⊢ ((( bday ‘𝐸) ∈ On ∧ 𝑥 ∈ No ) → (𝑥 ∈ ( O ‘( bday ‘𝐸)) ↔ ( bday ‘𝑥) ∈ ( bday ‘𝐸)))
68	65, 66, 67	sylancr 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (𝑥 ∈ ( O ‘( bday ‘𝐸)) ↔ ( bday ‘𝑥) ∈ ( bday ‘𝐸)))
69	64, 68	mpbird 256	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → 𝑥 ∈ ( O ‘( bday ‘𝐸)))
70	61, 62, 69	mulsproplem3 27487	. . . . . . 7 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (𝐷 ·s 𝑥) ∈ No )
71	57	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ∀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))))))
72	3	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → 𝐶 ∈ No )
73	71, 72, 69	mulsproplem3 27487	. . . . . . 7 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (𝐶 ·s 𝑥) ∈ No )
74	70, 73	subscld 27449	. . . . . 6 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ((𝐷 ·s 𝑥) -s (𝐶 ·s 𝑥)) ∈ No )
75	31	uneq2i 4156	. . . . . . . . . . . . . . 15 ⊢ ((( 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)) ∪ ∅)
76		un0 4386	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝐷) +no ( bday ‘𝐹)) ∪ ∅) = (( bday ‘𝐷) +no ( bday ‘𝐹))
77	75, 76	eqtri 2759	. . . . . . . . . . . . . 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		ssun2 4169	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝐷) +no ( bday ‘𝐹)) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹)))
79	78, 50	sstri 3987	. . . . . . . . . . . . . . 15 ⊢ (( 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))))
80	79, 38	sstri 3987	. . . . . . . . . . . . . 14 ⊢ (( 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)))))
81	77, 80	eqsstri 4012	. . . . . . . . . . . . 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 ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))))
82	81	sseli 3974	. . . . . . . . . . . 12 ⊢ (((( 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))))))
83	82	imim1i 63	. . . . . . . . . . 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 ‘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))))))
84	83	6ralimi 3126	. . . . . . . . . 10 ⊢ (∀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))))))
85	1, 84	syl 17	. . . . . . . . 9 ⊢ (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))))))
86	85, 5, 9	mulsproplem11 27495	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ·s 𝐹) ∈ No )
87	31	uneq2i 4156	. . . . . . . . . . . . . . 15 ⊢ ((( 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)) ∪ ∅)
88		un0 4386	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ ∅) = (( bday ‘𝐶) +no ( bday ‘𝐹))
89	87, 88	eqtri 2759	. . . . . . . . . . . . . 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))
90		ssun1 4168	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝐶) +no ( bday ‘𝐹)) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))
91	90, 36	sstri 3987	. . . . . . . . . . . . . . 15 ⊢ (( 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))))
92	91, 38	sstri 3987	. . . . . . . . . . . . . 14 ⊢ (( 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)))))
93	89, 92	eqsstri 4012	. . . . . . . . . . . . 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 ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))))
94	93	sseli 3974	. . . . . . . . . . . 12 ⊢ (((( 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))))))
95	94	imim1i 63	. . . . . . . . . . 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 ‘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))))))
96	95	6ralimi 3126	. . . . . . . . . 10 ⊢ (∀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))))))
97	1, 96	syl 17	. . . . . . . . 9 ⊢ (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))))))
98	97, 3, 9	mulsproplem11 27495	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ·s 𝐹) ∈ No )
99	86, 98	subscld 27449	. . . . . . 7 ⊢ (𝜑 → ((𝐷 ·s 𝐹) -s (𝐶 ·s 𝐹)) ∈ No )
100	99	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ((𝐷 ·s 𝐹) -s (𝐶 ·s 𝐹)) ∈ No )
101	1	mulsproplemcbv 27484	. . . . . . . . . 10 ⊢ (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))))))
102	101	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ∀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))))))
103		onelss 6395	. . . . . . . . . . . . . . . . . 18 ⊢ (( bday ‘𝐸) ∈ On → (( bday ‘𝑥) ∈ ( bday ‘𝐸) → ( bday ‘𝑥) ⊆ ( bday ‘𝐸)))
104	65, 64, 103	mpsyl 68	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ( bday ‘𝑥) ⊆ ( bday ‘𝐸))
105		simprl 769	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ( bday ‘𝐸) = ( bday ‘𝐹))
106	104, 105	sseqtrd 4018	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ( bday ‘𝑥) ⊆ ( bday ‘𝐹))
107		bdayelon 27204	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑥) ∈ On
108		bdayelon 27204	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝐹) ∈ On
109		bdayelon 27204	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝐷) ∈ On
110		naddss2 8672	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑥) ∈ On ∧ ( bday ‘𝐹) ∈ On ∧ ( bday ‘𝐷) ∈ On) → (( bday ‘𝑥) ⊆ ( bday ‘𝐹) ↔ (( bday ‘𝐷) +no ( bday ‘𝑥)) ⊆ (( bday ‘𝐷) +no ( bday ‘𝐹))))
111	107, 108, 109, 110	mp3an 1461	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑥) ⊆ ( bday ‘𝐹) ↔ (( bday ‘𝐷) +no ( bday ‘𝑥)) ⊆ (( bday ‘𝐷) +no ( bday ‘𝐹)))
112	106, 111	sylib 217	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (( bday ‘𝐷) +no ( bday ‘𝑥)) ⊆ (( bday ‘𝐷) +no ( bday ‘𝐹)))
113		unss2 4177	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝐷) +no ( bday ‘𝑥)) ⊆ (( bday ‘𝐷) +no ( bday ‘𝐹)) → ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑥))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))))
114	112, 113	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑥))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))))
115		bdayelon 27204	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝐶) ∈ On
116		naddss2 8672	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑥) ∈ On ∧ ( bday ‘𝐹) ∈ On ∧ ( bday ‘𝐶) ∈ On) → (( bday ‘𝑥) ⊆ ( bday ‘𝐹) ↔ (( bday ‘𝐶) +no ( bday ‘𝑥)) ⊆ (( bday ‘𝐶) +no ( bday ‘𝐹))))
117	107, 108, 115, 116	mp3an 1461	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑥) ⊆ ( bday ‘𝐹) ↔ (( bday ‘𝐶) +no ( bday ‘𝑥)) ⊆ (( bday ‘𝐶) +no ( bday ‘𝐹)))
118	106, 117	sylib 217	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (( bday ‘𝐶) +no ( bday ‘𝑥)) ⊆ (( bday ‘𝐶) +no ( bday ‘𝐹)))
119		unss1 4175	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝐶) +no ( bday ‘𝑥)) ⊆ (( bday ‘𝐶) +no ( bday ‘𝐹)) → ((( bday ‘𝐶) +no ( bday ‘𝑥)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))
120	118, 119	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ((( bday ‘𝐶) +no ( bday ‘𝑥)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))
121		unss12 4178	. . . . . . . . . . . . . 14 ⊢ ((((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑥))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∧ ((( bday ‘𝐶) +no ( bday ‘𝑥)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))) → (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑥))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑥)) ∪ (( 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)))))
122	114, 120, 121	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑥))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑥)) ∪ (( 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)))))
123		unss2 4177	. . . . . . . . . . . . 13 ⊢ ((((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑥))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑥)) ∪ (( 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 ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑥))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑥)) ∪ (( 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))))))
124	122, 123	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑥))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑥)) ∪ (( 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))))))
125	124	sseld 3977	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (((( 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 ‘𝑥))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑥)) ∪ (( 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)))))))
126	125	imim1d 82	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ((((( 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 ‘𝐷) +no ( bday ‘𝑥))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑥)) ∪ (( 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)))))))
127	126	ralimd6v 3207	. . . . . . . . 9 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (∀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 ‘𝐷) +no ( bday ‘𝑥))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑥)) ∪ (( 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)))))))
128	102, 127	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ∀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 ‘𝑥))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝑥)) ∪ (( 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))))))
129	7	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → 𝐸 ∈ No )
130	11	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → 𝐶 <s 𝐷)
131		simprr2 1222	. . . . . . . . 9 ⊢ ((( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹))) → 𝐸 <s 𝑥)
132	131	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → 𝐸 <s 𝑥)
133	64	olcd 872	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (( bday ‘𝐸) ∈ ( bday ‘𝑥) ∨ ( bday ‘𝑥) ∈ ( bday ‘𝐸)))
134	128, 72, 62, 129, 66, 130, 132, 133	mulsproplem13 27497	. . . . . . 7 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ((𝐶 ·s 𝑥) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝑥) -s (𝐷 ·s 𝐸)))
135	45	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (𝐷 ·s 𝐸) ∈ No )
136	58	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (𝐶 ·s 𝐸) ∈ No )
137	135, 70, 136, 73	sltsubsub3bd 27466	. . . . . . 7 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (((𝐷 ·s 𝐸) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝑥) -s (𝐶 ·s 𝑥)) ↔ ((𝐶 ·s 𝑥) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝑥) -s (𝐷 ·s 𝐸))))
138	134, 137	mpbird 256	. . . . . 6 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ((𝐷 ·s 𝐸) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝑥) -s (𝐶 ·s 𝑥)))
139		naddss2 8672	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑥) ∈ On ∧ ( bday ‘𝐸) ∈ On ∧ ( bday ‘𝐶) ∈ On) → (( bday ‘𝑥) ⊆ ( bday ‘𝐸) ↔ (( bday ‘𝐶) +no ( bday ‘𝑥)) ⊆ (( bday ‘𝐶) +no ( bday ‘𝐸))))
140	107, 65, 115, 139	mp3an 1461	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑥) ⊆ ( bday ‘𝐸) ↔ (( bday ‘𝐶) +no ( bday ‘𝑥)) ⊆ (( bday ‘𝐶) +no ( bday ‘𝐸)))
141	104, 140	sylib 217	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (( bday ‘𝐶) +no ( bday ‘𝑥)) ⊆ (( bday ‘𝐶) +no ( bday ‘𝐸)))
142		unss1 4175	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝐶) +no ( bday ‘𝑥)) ⊆ (( bday ‘𝐶) +no ( bday ‘𝐸)) → ((( bday ‘𝐶) +no ( bday ‘𝑥)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))))
143	141, 142	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ((( bday ‘𝐶) +no ( bday ‘𝑥)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))))
144		naddss2 8672	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑥) ∈ On ∧ ( bday ‘𝐸) ∈ On ∧ ( bday ‘𝐷) ∈ On) → (( bday ‘𝑥) ⊆ ( bday ‘𝐸) ↔ (( bday ‘𝐷) +no ( bday ‘𝑥)) ⊆ (( bday ‘𝐷) +no ( bday ‘𝐸))))
145	107, 65, 109, 144	mp3an 1461	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑥) ⊆ ( bday ‘𝐸) ↔ (( bday ‘𝐷) +no ( bday ‘𝑥)) ⊆ (( bday ‘𝐷) +no ( bday ‘𝐸)))
146	104, 145	sylib 217	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (( bday ‘𝐷) +no ( bday ‘𝑥)) ⊆ (( bday ‘𝐷) +no ( bday ‘𝐸)))
147		unss2 4177	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝐷) +no ( bday ‘𝑥)) ⊆ (( bday ‘𝐷) +no ( bday ‘𝐸)) → ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑥))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))
148	146, 147	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑥))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))
149		unss12 4178	. . . . . . . . . . . . . 14 ⊢ ((((( bday ‘𝐶) +no ( bday ‘𝑥)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∧ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑥))) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))) → (((( bday ‘𝐶) +no ( bday ‘𝑥)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑥)))) ⊆ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))))
150	143, 148, 149	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (((( bday ‘𝐶) +no ( bday ‘𝑥)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑥)))) ⊆ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))))
151		unss2 4177	. . . . . . . . . . . . 13 ⊢ ((((( bday ‘𝐶) +no ( bday ‘𝑥)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑥)))) ⊆ (((( 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 ‘𝑥)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘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))))))
152	150, 151	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝑥)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘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))))))
153	152	sseld 3977	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (((( 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 ‘𝑥)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘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))))) ∈ ((( 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)))))))
154	153	imim1d 82	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ((((( 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 ‘𝑥)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑥))))) → ((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	154	ralimd6v 3207	. . . . . . . . 9 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (∀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 ‘𝑥)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑥))))) → ((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	102, 155	mpd 15	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ∀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 ‘𝑥)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝑥))))) → ((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))))))
157	9	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → 𝐹 ∈ No )
158		simprr3 1223	. . . . . . . . 9 ⊢ ((( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹))) → 𝑥 <s 𝐹)
159	158	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → 𝑥 <s 𝐹)
160	64, 105	eleqtrd 2834	. . . . . . . . 9 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ( bday ‘𝑥) ∈ ( bday ‘𝐹))
161	160	orcd 871	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (( bday ‘𝑥) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝑥)))
162	156, 72, 62, 66, 157, 130, 159, 161	mulsproplem13 27497	. . . . . . 7 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝑥)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝑥)))
163	86	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (𝐷 ·s 𝐹) ∈ No )
164	98	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (𝐶 ·s 𝐹) ∈ No )
165	70, 163, 73, 164	sltsubsub3bd 27466	. . . . . . 7 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (((𝐷 ·s 𝑥) -s (𝐶 ·s 𝑥)) <s ((𝐷 ·s 𝐹) -s (𝐶 ·s 𝐹)) ↔ ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝑥)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝑥))))
166	162, 165	mpbird 256	. . . . . 6 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ((𝐷 ·s 𝑥) -s (𝐶 ·s 𝑥)) <s ((𝐷 ·s 𝐹) -s (𝐶 ·s 𝐹)))
167	60, 74, 100, 138, 166	slttrd 27189	. . . . 5 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ((𝐷 ·s 𝐸) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐶 ·s 𝐹)))
168	45, 86, 58, 98	sltsubsub3bd 27466	. . . . . 6 ⊢ (𝜑 → (((𝐷 ·s 𝐸) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐶 ·s 𝐹)) ↔ ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸))))
169	168	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (((𝐷 ·s 𝐸) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐶 ·s 𝐹)) ↔ ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸))))
170	167, 169	mpbid 231	. . . 4 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))
171	170	anassrs 468	. . 3 ⊢ (((𝜑 ∧ ( bday ‘𝐸) = ( bday ‘𝐹)) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹))) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))
172	22, 171	rexlimddv 3160	. 2 ⊢ ((𝜑 ∧ ( bday ‘𝐸) = ( bday ‘𝐹)) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))
173	65	onordi 6464	. . . . 5 ⊢ Ord ( bday ‘𝐸)
174	108	onordi 6464	. . . . 5 ⊢ Ord ( bday ‘𝐹)
175		ordtri3or 6385	. . . . 5 ⊢ ((Ord ( bday ‘𝐸) ∧ Ord ( bday ‘𝐹)) → (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐸) = ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸)))
176	173, 174, 175	mp2an 690	. . . 4 ⊢ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐸) = ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))
177		df-3or 1088	. . . . 5 ⊢ ((( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐸) = ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸)) ↔ ((( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐸) = ( bday ‘𝐹)) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸)))
178		or32 924	. . . . 5 ⊢ (((( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐸) = ( bday ‘𝐹)) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸)) ↔ ((( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸)) ∨ ( bday ‘𝐸) = ( bday ‘𝐹)))
179	177, 178	bitri 274	. . . 4 ⊢ ((( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐸) = ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸)) ↔ ((( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸)) ∨ ( bday ‘𝐸) = ( bday ‘𝐹)))
180	176, 179	mpbi 229	. . 3 ⊢ ((( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸)) ∨ ( bday ‘𝐸) = ( bday ‘𝐹))
181	180	a1i 11	. 2 ⊢ (𝜑 → ((( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸)) ∨ ( bday ‘𝐸) = ( bday ‘𝐹)))
182	16, 172, 181	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:  mulsprop  27499