Theorem mulsproplem14 28287

Description: Lemma for surreal multiplication. Finally, we remove the restriction on 𝐸 and 𝐹 from mulsproplem12 28285 and mulsproplem13 28286. This completes the induction on surreal multiplication. mulsprop 28288 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 485	. . 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 485	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))) → 𝐶 ∈ No )
5		mulsproplem.3	. . . 4 ⊢ (𝜑 → 𝐷 ∈ No )
6	5	adantr 485	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))) → 𝐷 ∈ No )
7		mulsproplem.4	. . . 4 ⊢ (𝜑 → 𝐸 ∈ No )
8	7	adantr 485	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))) → 𝐸 ∈ No )
9		mulsproplem.5	. . . 4 ⊢ (𝜑 → 𝐹 ∈ No )
10	9	adantr 485	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))) → 𝐹 ∈ No )
11		mulsproplem.6	. . . 4 ⊢ (𝜑 → 𝐶 <s 𝐷)
12	11	adantr 485	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))) → 𝐶 <s 𝐷)
13		mulsproplem.7	. . . 4 ⊢ (𝜑 → 𝐸 <s 𝐹)
14	13	adantr 485	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))) → 𝐸 <s 𝐹)
15		simpr 489	. . 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 28286	. 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 485	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐸) = ( bday ‘𝐹)) → 𝐸 ∈ No )
18	9	adantr 485	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐸) = ( bday ‘𝐹)) → 𝐹 ∈ No )
19		simpr 489	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐸) = ( bday ‘𝐹)) → ( bday ‘𝐸) = ( bday ‘𝐹))
20	13	adantr 485	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐸) = ( bday ‘𝐹)) → 𝐸 <s 𝐹)
21		nodense 27821	. . . 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 851	. . 3 ⊢ ((𝜑 ∧ ( bday ‘𝐸) = ( bday ‘𝐹)) → ∃𝑥 ∈ No (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹))
23		unidm 4119	. . . . . . . . . . . . . . . . 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 4119	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) = (( bday ‘ 0s ) +no ( bday ‘ 0s ))
25		bday0 27969	. . . . . . . . . . . . . . . . . . 19 ⊢ ( bday ‘ 0s ) = ∅
26	25, 25	oveq12i 7423	. . . . . . . . . . . . . . . . . 18 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = (∅ +no ∅)
27		0elon 6417	. . . . . . . . . . . . . . . . . . 19 ⊢ ∅ ∈ On
28		naddrid 8669	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∈ On → (∅ +no ∅) = ∅)
29	27, 28	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ +no ∅) = ∅
30	26, 29	eqtri 2792	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = ∅
31	23, 24, 30	3eqtri 2796	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s )))) = ∅
32	31	uneq2i 4127	. . . . . . . . . . . . . . 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 4358	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝐷) +no ( bday ‘𝐸)) ∪ ∅) = (( bday ‘𝐷) +no ( bday ‘𝐸))
34	32, 33	eqtri 2792	. . . . . . . . . . . . . 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 4140	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝐷) +no ( bday ‘𝐸)) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))
36		ssun2 4140	. . . . . . . . . . . . . . . 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 3954	. . . . . . . . . . . . . . 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 4140	. . . . . . . . . . . . . . 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 3954	. . . . . . . . . . . . . 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 3991	. . . . . . . . . . . . 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 3941	. . . . . . . . . . . 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 64	. . . . . . . . . . 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 3145	. . . . . . . . . 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 18	. . . . . . . . 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 28284	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ·s 𝐸) ∈ No )
46	31	uneq2i 4127	. . . . . . . . . . . . . . 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 4358	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ ∅) = (( bday ‘𝐶) +no ( bday ‘𝐸))
48	46, 47	eqtri 2792	. . . . . . . . . . . . . 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 4139	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝐶) +no ( bday ‘𝐸)) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹)))
50		ssun1 4139	. . . . . . . . . . . . . . . 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 3954	. . . . . . . . . . . . . . 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 3954	. . . . . . . . . . . . . 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 3991	. . . . . . . . . . . . 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 3941	. . . . . . . . . . . 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 64	. . . . . . . . . . 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 3145	. . . . . . . . . 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 18	. . . . . . . . 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 28284	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ·s 𝐸) ∈ No )
59	45, 58	subscld 28221	. . . . . . 7 ⊢ (𝜑 → ((𝐷 ·s 𝐸) -s (𝐶 ·s 𝐸)) ∈ No )
60	59	adantr 485	. . . . . 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 485	. . . . . . . 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 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → 𝐷 ∈ No )
63		simprr1 1238	. . . . . . . . . 10 ⊢ ((( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹))) → ( bday ‘𝑥) ∈ ( bday ‘𝐸))
64	63	adantl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ( bday ‘𝑥) ∈ ( bday ‘𝐸))
65		bdayon 27910	. . . . . . . . . 10 ⊢ ( bday ‘𝐸) ∈ On
66		simprrl 792	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → 𝑥 ∈ No )
67		oldbday 28059	. . . . . . . . . 10 ⊢ ((( bday ‘𝐸) ∈ On ∧ 𝑥 ∈ No ) → (𝑥 ∈ ( O ‘( bday ‘𝐸)) ↔ ( bday ‘𝑥) ∈ ( bday ‘𝐸)))
68	65, 66, 67	sylancr 598	. . . . . . . . 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 260	. . . . . . . 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 28276	. . . . . . 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 485	. . . . . . . 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 485	. . . . . . . 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 28276	. . . . . . 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 28221	. . . . . 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 4127	. . . . . . . . . . . . . . 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 4358	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝐷) +no ( bday ‘𝐹)) ∪ ∅) = (( bday ‘𝐷) +no ( bday ‘𝐹))
77	75, 76	eqtri 2792	. . . . . . . . . . . . . 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 4140	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝐷) +no ( bday ‘𝐹)) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹)))
79	78, 50	sstri 3954	. . . . . . . . . . . . . . 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 3954	. . . . . . . . . . . . . 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 3991	. . . . . . . . . . . . 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 3941	. . . . . . . . . . . 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 64	. . . . . . . . . . 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 3145	. . . . . . . . . 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 18	. . . . . . . . 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 28284	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ·s 𝐹) ∈ No )
87	31	uneq2i 4127	. . . . . . . . . . . . . . 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 4358	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ ∅) = (( bday ‘𝐶) +no ( bday ‘𝐹))
89	87, 88	eqtri 2792	. . . . . . . . . . . . . 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 4139	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝐶) +no ( bday ‘𝐹)) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))
91	90, 36	sstri 3954	. . . . . . . . . . . . . . 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 3954	. . . . . . . . . . . . . 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 3991	. . . . . . . . . . . . 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 3941	. . . . . . . . . . . 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 64	. . . . . . . . . . 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 3145	. . . . . . . . . 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 18	. . . . . . . . 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 28284	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ·s 𝐹) ∈ No )
99	86, 98	subscld 28221	. . . . . . 7 ⊢ (𝜑 → ((𝐷 ·s 𝐹) -s (𝐶 ·s 𝐹)) ∈ No )
100	99	adantr 485	. . . . . 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 28273	. . . . . . . . . 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 485	. . . . . . . . 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 6404	. . . . . . . . . . . . . . . . . 18 ⊢ (( bday ‘𝐸) ∈ On → (( bday ‘𝑥) ∈ ( bday ‘𝐸) → ( bday ‘𝑥) ⊆ ( bday ‘𝐸)))
104	65, 64, 103	mpsyl 69	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ( bday ‘𝑥) ⊆ ( bday ‘𝐸))
105		simprl 782	. . . . . . . . . . . . . . . . 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 3981	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ( bday ‘𝑥) ⊆ ( bday ‘𝐹))
107		bdayon 27910	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑥) ∈ On
108		bdayon 27910	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝐹) ∈ On
109		bdayon 27910	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝐷) ∈ On
110		naddss2 8676	. . . . . . . . . . . . . . . . 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 1487	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑥) ⊆ ( bday ‘𝐹) ↔ (( bday ‘𝐷) +no ( bday ‘𝑥)) ⊆ (( bday ‘𝐷) +no ( bday ‘𝐹)))
112	106, 111	sylib 221	. . . . . . . . . . . . . . 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 4148	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 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		bdayon 27910	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝐶) ∈ On
116		naddss2 8676	. . . . . . . . . . . . . . . . 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 1487	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑥) ⊆ ( bday ‘𝐹) ↔ (( bday ‘𝐶) +no ( bday ‘𝑥)) ⊆ (( bday ‘𝐶) +no ( bday ‘𝐹)))
118	106, 117	sylib 221	. . . . . . . . . . . . . . 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 4146	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 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 4149	. . . . . . . . . . . . . 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 595	. . . . . . . . . . . . 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 4148	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 3944	. . . . . . . . . . 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 83	. . . . . . . . . 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 3224	. . . . . . . . 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 16	. . . . . . . 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 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → 𝐸 ∈ No )
130	11	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → 𝐶 <s 𝐷)
131		simprr2 1239	. . . . . . . . 9 ⊢ ((( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹))) → 𝐸 <s 𝑥)
132	131	adantl 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → 𝐸 <s 𝑥)
133	64	olcd 887	. . . . . . . 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 28286	. . . . . . 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 485	. . . . . . . 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 485	. . . . . . . 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	ltsubsubs3bd 28243	. . . . . . 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 260	. . . . . 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 8676	. . . . . . . . . . . . . . . . 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 1487	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑥) ⊆ ( bday ‘𝐸) ↔ (( bday ‘𝐶) +no ( bday ‘𝑥)) ⊆ (( bday ‘𝐶) +no ( bday ‘𝐸)))
141	104, 140	sylib 221	. . . . . . . . . . . . . . 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 4146	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 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 8676	. . . . . . . . . . . . . . . . 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 1487	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑥) ⊆ ( bday ‘𝐸) ↔ (( bday ‘𝐷) +no ( bday ‘𝑥)) ⊆ (( bday ‘𝐷) +no ( bday ‘𝐸)))
146	104, 145	sylib 221	. . . . . . . . . . . . . . 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 4148	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 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 4149	. . . . . . . . . . . . . 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 595	. . . . . . . . . . . . 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 4148	. . . . . . . . . . . . 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 18	. . . . . . . . . . . 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 3944	. . . . . . . . . . 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 83	. . . . . . . . . 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 3224	. . . . . . . . 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 16	. . . . . . . 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 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → 𝐹 ∈ No )
158		simprr3 1240	. . . . . . . . 9 ⊢ ((( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹))) → 𝑥 <s 𝐹)
159	158	adantl 486	. . . . . . . 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 2871	. . . . . . . . 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 886	. . . . . . . 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 28286	. . . . . . 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 485	. . . . . . . 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 485	. . . . . . . 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	ltsubsubs3bd 28243	. . . . . . 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 260	. . . . . 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	ltstrd 27892	. . . . 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	ltsubsubs3bd 28243	. . . . . 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 485	. . . . 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 235	. . . 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 472	. . 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 3178	. 2 ⊢ ((𝜑 ∧ ( bday ‘𝐸) = ( bday ‘𝐹)) → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))
173	65	onordi 6475	. . . . 5 ⊢ Ord ( bday ‘𝐸)
174	108	onordi 6475	. . . . 5 ⊢ Ord ( bday ‘𝐹)
175		ordtri3or 6394	. . . . 5 ⊢ ((Ord ( bday ‘𝐸) ∧ Ord ( bday ‘𝐹)) → (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐸) = ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸)))
176	173, 174, 175	mp2an 704	. . . 4 ⊢ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐸) = ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))
177		df-3or 1102	. . . . 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 938	. . . . 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 278	. . . 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 233	. . 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 973	1 ⊢ (𝜑 → ((𝐶 ·s 𝐹) -s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) -s (𝐷 ·s 𝐸)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∨ w3o 1100   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095   ∪ cun 3911   ⊆ wss 3913  ∅c0 4294   class class class wbr 5113  Ord word 6360  Oncon0 6361  ‘cfv 6537  (class class class)co 7411   +no cnadd 8650   No csur 27769   <s clts 27770   bday cbday 27771   0_s c0s 27963   O cold 27981   -_s csubs 28178   ·_s cmuls 28264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-ot 4603  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7985  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-1o 8452  df-2o 8453  df-nadd 8651  df-no 27772  df-lts 27773  df-bday 27774  df-les 27874  df-slts 27916  df-cuts 27918  df-0s 27965  df-made 27985  df-old 27986  df-left 27988  df-right 27989  df-norec 28096  df-norec2 28107  df-adds 28118  df-negs 28179  df-subs 28180  df-muls 28265

This theorem is referenced by:  mulsprop  28288