Theorem mulsproplem14 28524

Description: Lemma for surreal multiplication. Finally, we remove the restriction on 𝐸 and 𝐹 from mulsproplem12 28522 and mulsproplem13 28523. This completes the induction on surreal multiplication. mulsprop 28525 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 28523	. 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 28058	. . . 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 4155	. . . . . . . . . . . . . . . . 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 4155	. . . . . . . . . . . . . . . . 17 ⊢ (((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s ))) = ((bday‘ 0s ) +no (bday‘ 0s ))
25		bday0 28206	. . . . . . . . . . . . . . . . . . 19 ⊢ (bday‘ 0s ) = ∅
26	25, 25	oveq12i 7436	. . . . . . . . . . . . . . . . . 18 ⊢ ((bday‘ 0s ) +no (bday‘ 0s )) = (∅ +no ∅)
27		0elon 6421	. . . . . . . . . . . . . . . . . . 19 ⊢ ∅ ∈ On
28		naddrid 8700	. . . . . . . . . . . . . . . . . . 19 ⊢ (∅ ∈ On → (∅ +no ∅) = ∅)
29	27, 28	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (∅ +no ∅) = ∅
30	26, 29	eqtri 2783	. . . . . . . . . . . . . . . . 17 ⊢ ((bday‘ 0s ) +no (bday‘ 0s )) = ∅
31	23, 24, 30	3eqtri 2787	. . . . . . . . . . . . . . . 16 ⊢ ((((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s ))) ∪ (((bday‘ 0s ) +no (bday‘ 0s )) ∪ ((bday‘ 0s ) +no (bday‘ 0s )))) = ∅
32	31	uneq2i 4163	. . . . . . . . . . . . . . 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 4343	. . . . . . . . . . . . . . 15 ⊢ (((bday‘𝐷) +no (bday‘𝐸)) ∪ ∅) = ((bday‘𝐷) +no (bday‘𝐸))
34	32, 33	eqtri 2783	. . . . . . . . . . . . . 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 4176	. . . . . . . . . . . . . . . 16 ⊢ ((bday‘𝐷) +no (bday‘𝐸)) ⊆ (((bday‘𝐶) +no (bday‘𝐹)) ∪ ((bday‘𝐷) +no (bday‘𝐸)))
36		ssun2 4176	. . . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . . 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 4176	. . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . 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 3955	. . . . . . . . . . . . 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 3905	. . . . . . . . . . . 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 3136	. . . . . . . . . 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 28521	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ·s 𝐸) ∈ No)
46	31	uneq2i 4163	. . . . . . . . . . . . . . 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 4343	. . . . . . . . . . . . . . 15 ⊢ (((bday‘𝐶) +no (bday‘𝐸)) ∪ ∅) = ((bday‘𝐶) +no (bday‘𝐸))
48	46, 47	eqtri 2783	. . . . . . . . . . . . . 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 4175	. . . . . . . . . . . . . . . 16 ⊢ ((bday‘𝐶) +no (bday‘𝐸)) ⊆ (((bday‘𝐶) +no (bday‘𝐸)) ∪ ((bday‘𝐷) +no (bday‘𝐹)))
50		ssun1 4175	. . . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . 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 3955	. . . . . . . . . . . . 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 3905	. . . . . . . . . . . 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 3136	. . . . . . . . . 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 28521	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ·s 𝐸) ∈ No)
59	45, 58	subscld 28458	. . . . . . 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 28147	. . . . . . . . . 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 28296	. . . . . . . . . 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 28513	. . . . . . 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 28513	. . . . . . 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 28458	. . . . . 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 4163	. . . . . . . . . . . . . . 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 4343	. . . . . . . . . . . . . . 15 ⊢ (((bday‘𝐷) +no (bday‘𝐹)) ∪ ∅) = ((bday‘𝐷) +no (bday‘𝐹))
77	75, 76	eqtri 2783	. . . . . . . . . . . . . 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 4176	. . . . . . . . . . . . . . . 16 ⊢ ((bday‘𝐷) +no (bday‘𝐹)) ⊆ (((bday‘𝐶) +no (bday‘𝐸)) ∪ ((bday‘𝐷) +no (bday‘𝐹)))
79	78, 50	sstri 3918	. . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . 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 3955	. . . . . . . . . . . . 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 3905	. . . . . . . . . . . 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 3136	. . . . . . . . . 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 28521	. . . . . . . 8 ⊢ (𝜑 → (𝐷 ·s 𝐹) ∈ No)
87	31	uneq2i 4163	. . . . . . . . . . . . . . 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 4343	. . . . . . . . . . . . . . 15 ⊢ (((bday‘𝐶) +no (bday‘𝐹)) ∪ ∅) = ((bday‘𝐶) +no (bday‘𝐹))
89	87, 88	eqtri 2783	. . . . . . . . . . . . . 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 4175	. . . . . . . . . . . . . . . 16 ⊢ ((bday‘𝐶) +no (bday‘𝐹)) ⊆ (((bday‘𝐶) +no (bday‘𝐹)) ∪ ((bday‘𝐷) +no (bday‘𝐸)))
91	90, 36	sstri 3918	. . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . 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 3955	. . . . . . . . . . . . 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 3905	. . . . . . . . . . . 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 3136	. . . . . . . . . 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 28521	. . . . . . . 8 ⊢ (𝜑 → (𝐶 ·s 𝐹) ∈ No)
99	86, 98	subscld 28458	. . . . . . 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 28510	. . . . . . . . . 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 6408	. . . . . . . . . . . . . . . . . 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 3945	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ ((bday‘𝐸) = (bday‘𝐹) ∧ (𝑥 ∈ No ∧ ((bday‘𝑥) ∈ (bday‘𝐸) ∧ 𝐸 <s 𝑥 ∧ 𝑥 <s 𝐹)))) → (bday‘𝑥) ⊆ (bday‘𝐹))
107		bdayon 28147	. . . . . . . . . . . . . . . . 17 ⊢ (bday‘𝑥) ∈ On
108		bdayon 28147	. . . . . . . . . . . . . . . . 17 ⊢ (bday‘𝐹) ∈ On
109		bdayon 28147	. . . . . . . . . . . . . . . . 17 ⊢ (bday‘𝐷) ∈ On
110		naddss2 8707	. . . . . . . . . . . . . . . . 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 4184	. . . . . . . . . . . . . . 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 28147	. . . . . . . . . . . . . . . . 17 ⊢ (bday‘𝐶) ∈ On
116		naddss2 8707	. . . . . . . . . . . . . . . . 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 4182	. . . . . . . . . . . . . . 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 4185	. . . . . . . . . . . . . 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 4184	. . . . . . . . . . . . 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 3908	. . . . . . . . . . 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 3215	. . . . . . . . 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 28523	. . . . . . 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 28480	. . . . . . 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 8707	. . . . . . . . . . . . . . . . 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 4182	. . . . . . . . . . . . . . 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 8707	. . . . . . . . . . . . . . . . 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 4184	. . . . . . . . . . . . . . 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 4185	. . . . . . . . . . . . . 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 4184	. . . . . . . . . . . . 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 3908	. . . . . . . . . . 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 3215	. . . . . . . . 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 2862	. . . . . . . . 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 28523	. . . . . . 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 28480	. . . . . . 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 28129	. . . . 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 28480	. . . . . 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 3169	. 2 ⊢ ((𝜑 ∧ (bday‘𝐸) = (bday‘𝐹)) → ((𝐶 ·s 𝐹) −s (𝐶 ·s 𝐸)) <s ((𝐷 ·s 𝐹) −s (𝐷 ·s 𝐸)))
173	65	onordi 6479	. . . . 5 ⊢ Ord (bday‘𝐸)
174	108	onordi 6479	. . . . 5 ⊢ Ord (bday‘𝐹)
175		ordtri3or 6398	. . . . 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 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3896   ∪ cun 4148  ∅c0 4277   class class class wbr 5102  Ord word 6364  Oncon0 6365  ‘cfv 6543  (class class class)co 7424   +no cnadd 8681  Nocsur 28006   <s clts 28007  bdaycbday 28008   0_s c0s 28200  Ocold 28218   −_s csubs 28415   ·_s cmuls 28501

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5327  ax-pr 5391  ax-un 7755

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  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 6307  df-ord 6368  df-on 6369  df-suc 6371  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7381  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8005  df-2nd 8006  df-frecs 8306  df-wrecs 8337  df-recs 8386  df-1o 8483  df-2o 8484  df-nadd 8682  df-no 28009  df-lts 28010  df-bday 28011  df-les 28111  df-slts 28153  df-cuts 28155  df-0s 28202  df-made 28222  df-old 28223  df-left 28225  df-right 28226  df-norec 28333  df-norec2 28344  df-adds 28355  df-negs 28416  df-subs 28417  df-muls 28502

This theorem is used by:  mulsprop  28525