Theorem mulsproplem14 28508

Description: Lemma for surreal multiplication. Finally, we remove the restriction on 𝐸 and 𝐹 from mulsproplem12 28506 and mulsproplem13 28507. This completes the induction on surreal multiplication. mulsprop 28509 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 486	. . 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 486	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))) → 𝐶 ∈ No )
5		mulsproplem.3	. . . 4 ⊢ (𝜑 → 𝐷 ∈ No )
6	5	adantr 486	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))) → 𝐷 ∈ No )
7		mulsproplem.4	. . . 4 ⊢ (𝜑 → 𝐸 ∈ No )
8	7	adantr 486	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))) → 𝐸 ∈ No )
9		mulsproplem.5	. . . 4 ⊢ (𝜑 → 𝐹 ∈ No )
10	9	adantr 486	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))) → 𝐹 ∈ No )
11		mulsproplem.6	. . . 4 ⊢ (𝜑 → 𝐶 <s 𝐷)
12	11	adantr 486	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))) → 𝐶 <s 𝐷)
13		mulsproplem.7	. . . 4 ⊢ (𝜑 → 𝐸 <s 𝐹)
14	13	adantr 486	. . 3 ⊢ ((𝜑 ∧ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))) → 𝐸 <s 𝐹)
15		simpr 490	. . 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 28507	. 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 486	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐸) = ( bday ‘𝐹)) → 𝐸 ∈ No )
18	9	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐸) = ( bday ‘𝐹)) → 𝐹 ∈ No )
19		simpr 490	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐸) = ( bday ‘𝐹)) → ( bday ‘𝐸) = ( bday ‘𝐹))
20	13	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ( bday ‘𝐸) = ( bday ‘𝐹)) → 𝐸 <s 𝐹)
21		nodense 28042	. . . 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 852	. . 3 ⊢ ((𝜑 ∧ ( bday ‘𝐸) = ( bday ‘𝐹)) → ∃𝑥 ∈ No (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹))
23		unidm 4104	. . . . . . . . . . . . . . . . 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 4104	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) = (( bday ‘ 0s ) +no ( bday ‘ 0s ))
25		bday0 28190	. . . . . . . . . . . . . . . . . . 19 ⊢ ( bday ‘ 0s ) = ∅
26	25, 25	oveq12i 7430	. . . . . . . . . . . . . . . . . 18 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = (∅ +no ∅)
27		0elon 6417	. . . . . . . . . . . . . . . . . . 19 ⊢ ∅ ∈ On
28		naddrid 8686	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∈ On → (∅ +no ∅) = ∅)
29	27, 28	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ +no ∅) = ∅
30	26, 29	eqtri 2784	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = ∅
31	23, 24, 30	3eqtri 2788	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s ))) ∪ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (( bday ‘ 0s ) +no ( bday ‘ 0s )))) = ∅
32	31	uneq2i 4112	. . . . . . . . . . . . . . 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 4344	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝐷) +no ( bday ‘𝐸)) ∪ ∅) = (( bday ‘𝐷) +no ( bday ‘𝐸))
34	32, 33	eqtri 2784	. . . . . . . . . . . . . 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 4125	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝐷) +no ( bday ‘𝐸)) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))
36		ssun2 4125	. . . . . . . . . . . . . . . 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 3940	. . . . . . . . . . . . . . 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 4125	. . . . . . . . . . . . . . 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 3940	. . . . . . . . . . . . . 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 3977	. . . . . . . . . . . . 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 3927	. . . . . . . . . . . 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 3137	. . . . . . . . . 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 28505	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ·s 𝐸) ∈ No )
46	31	uneq2i 4112	. . . . . . . . . . . . . . 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 4344	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ ∅) = (( bday ‘𝐶) +no ( bday ‘𝐸))
48	46, 47	eqtri 2784	. . . . . . . . . . . . . 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 4124	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝐶) +no ( bday ‘𝐸)) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹)))
50		ssun1 4124	. . . . . . . . . . . . . . . 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 3940	. . . . . . . . . . . . . . 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 3940	. . . . . . . . . . . . . 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 3977	. . . . . . . . . . . . 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 3927	. . . . . . . . . . . 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 3137	. . . . . . . . . 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 28505	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ·s 𝐸) ∈ No )
59	45, 58	subscld 28442	. . . . . . 7 ⊢ (𝜑 → ((𝐷 ·s 𝐸) -s (𝐶 ·s 𝐸)) ∈ No )
60	59	adantr 486	. . . . . 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 486	. . . . . . . 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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → 𝐷 ∈ No )
63		simprr1 1240	. . . . . . . . . 10 ⊢ ((( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹))) → ( bday ‘𝑥) ∈ ( bday ‘𝐸))
64	63	adantl 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ( bday ‘𝑥) ∈ ( bday ‘𝐸))
65		bdayon 28131	. . . . . . . . . 10 ⊢ ( bday ‘𝐸) ∈ On
66		simprrl 793	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → 𝑥 ∈ No )
67		oldbday 28280	. . . . . . . . . 10 ⊢ ((( bday ‘𝐸) ∈ On ∧ 𝑥 ∈ No ) → (𝑥 ∈ ( O ‘( bday ‘𝐸)) ↔ ( bday ‘𝑥) ∈ ( bday ‘𝐸)))
68	65, 66, 67	sylancr 599	. . . . . . . . 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 28497	. . . . . . 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 486	. . . . . . . 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 486	. . . . . . . 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 28497	. . . . . . 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 28442	. . . . . 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 4112	. . . . . . . . . . . . . . 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 4344	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝐷) +no ( bday ‘𝐹)) ∪ ∅) = (( bday ‘𝐷) +no ( bday ‘𝐹))
77	75, 76	eqtri 2784	. . . . . . . . . . . . . 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 4125	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝐷) +no ( bday ‘𝐹)) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹)))
79	78, 50	sstri 3940	. . . . . . . . . . . . . . 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 3940	. . . . . . . . . . . . . 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 3977	. . . . . . . . . . . . 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 3927	. . . . . . . . . . . 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 3137	. . . . . . . . . 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 28505	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ·s 𝐹) ∈ No )
87	31	uneq2i 4112	. . . . . . . . . . . . . . 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 4344	. . . . . . . . . . . . . . 15 ⊢ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ ∅) = (( bday ‘𝐶) +no ( bday ‘𝐹))
89	87, 88	eqtri 2784	. . . . . . . . . . . . . 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 4124	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝐶) +no ( bday ‘𝐹)) ⊆ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸)))
91	90, 36	sstri 3940	. . . . . . . . . . . . . . 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 3940	. . . . . . . . . . . . . 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 3977	. . . . . . . . . . . . 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 3927	. . . . . . . . . . . 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 3137	. . . . . . . . . 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 28505	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ·s 𝐹) ∈ No )
99	86, 98	subscld 28442	. . . . . . 7 ⊢ (𝜑 → ((𝐷 ·s 𝐹) -s (𝐶 ·s 𝐹)) ∈ No )
100	99	adantr 486	. . . . . 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 28494	. . . . . . . . . 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 486	. . . . . . . . 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 783	. . . . . . . . . . . . . . . . 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 3967	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → ( bday ‘𝑥) ⊆ ( bday ‘𝐹))
107		bdayon 28131	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑥) ∈ On
108		bdayon 28131	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝐹) ∈ On
109		bdayon 28131	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝐷) ∈ On
110		naddss2 8693	. . . . . . . . . . . . . . . . 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 1490	. . . . . . . . . . . . . . . 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 4133	. . . . . . . . . . . . . . 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 28131	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝐶) ∈ On
116		naddss2 8693	. . . . . . . . . . . . . . . . 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 1490	. . . . . . . . . . . . . . . 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 4131	. . . . . . . . . . . . . . 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 4134	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 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 4133	. . . . . . . . . . . . 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 3930	. . . . . . . . . . 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 3216	. . . . . . . . 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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → 𝐸 ∈ No )
130	11	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → 𝐶 <s 𝐷)
131		simprr2 1241	. . . . . . . . 9 ⊢ ((( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹))) → 𝐸 <s 𝑥)
132	131	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → 𝐸 <s 𝑥)
133	64	olcd 888	. . . . . . . 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 28507	. . . . . . 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 486	. . . . . . . 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 486	. . . . . . . 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 28464	. . . . . . 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 8693	. . . . . . . . . . . . . . . . 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 1490	. . . . . . . . . . . . . . . 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 4131	. . . . . . . . . . . . . . 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 8693	. . . . . . . . . . . . . . . . 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 1490	. . . . . . . . . . . . . . . 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 4133	. . . . . . . . . . . . . . 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 4134	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 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 4133	. . . . . . . . . . . . 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 3930	. . . . . . . . . . 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 3216	. . . . . . . . 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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → 𝐹 ∈ No )
158		simprr3 1242	. . . . . . . . 9 ⊢ ((( bday ‘𝐸) = ( bday ‘𝐹) ∧ (𝑥 ∈ No ∧ (( bday ‘𝑥) ∈ ( bday ‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹))) → 𝑥 <s 𝐹)
159	158	adantl 487	. . . . . . . 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 2863	. . . . . . . . 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 887	. . . . . . . 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 28507	. . . . . . 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 486	. . . . . . . 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 486	. . . . . . . 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 28464	. . . . . . 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 28113	. . . . 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 28464	. . . . . 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 486	. . . . 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 473	. . 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 3170	. 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 705	. . . 4 ⊢ (( bday ‘𝐸) ∈ ( bday ‘𝐹) ∨ ( bday ‘𝐸) = ( bday ‘𝐹) ∨ ( bday ‘𝐹) ∈ ( bday ‘𝐸))
177		df-3or 1104	. . . . 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 939	. . . . 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)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   ∪ cun 3897   ⊆ wss 3899  ∅c0 4279   class class class wbr 5103  Ord word 6360  Oncon0 6361  ‘cfv 6537  (class class class)co 7418   +no cnadd 8667   No csur 27990   <s clts 27991   bday cbday 27992   0_s c0s 28184   O cold 28202   -_s csubs 28399   ·_s cmuls 28485

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  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 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-1o 8469  df-2o 8470  df-nadd 8668  df-no 27993  df-lts 27994  df-bday 27995  df-les 28095  df-slts 28137  df-cuts 28139  df-0s 28186  df-made 28206  df-old 28207  df-left 28209  df-right 28210  df-norec 28317  df-norec2 28328  df-adds 28339  df-negs 28400  df-subs 28401  df-muls 28486

This theorem is used by:  mulsprop  28509