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Theorem mulsproplem5 28210

Description: Lemma for surreal multiplication. Show one of the inequalities involved in surreal multiplication's cuts. (Contributed by Scott Fenton, 4-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))))))
mulsproplem5.1	⊢ (𝜑 → 𝐴 ∈ No )
mulsproplem5.2	⊢ (𝜑 → 𝐵 ∈ No )
mulsproplem5.3	⊢ (𝜑 → 𝑃 ∈ ( L ‘𝐴))
mulsproplem5.4	⊢ (𝜑 → 𝑄 ∈ ( L ‘𝐵))
mulsproplem5.5	⊢ (𝜑 → 𝑇 ∈ ( L ‘𝐴))
mulsproplem5.6	⊢ (𝜑 → 𝑈 ∈ ( R ‘𝐵))

**Assertion**
Ref	Expression
mulsproplem5	⊢ (𝜑 → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑃 ·_s 𝑄)) <s (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑇 ·_s 𝑈)))

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 𝑃,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓 𝑄,𝑏,𝑐,𝑑,𝑒,𝑓 𝑇,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓 𝑈,𝑏,𝑐,𝑑,𝑒,𝑓 Allowed substitution hints: 𝜑(𝑒,𝑓,𝑎,𝑏,𝑐,𝑑) 𝑄(𝑎) 𝑈(𝑎)

**Proof of Theorem mulsproplem5**
Step	Hyp	Ref	Expression
1		mulsproplem5.3	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ( L ‘𝐴))
2	1	leftnod 27970	. . 3 ⊢ (𝜑 → 𝑃 ∈ No )
3		mulsproplem5.5	. . . 4 ⊢ (𝜑 → 𝑇 ∈ ( L ‘𝐴))
4	3	leftnod 27970	. . 3 ⊢ (𝜑 → 𝑇 ∈ No )
5		ltslin 27810	. . 3 ⊢ ((𝑃 ∈ No ∧ 𝑇 ∈ No ) → (𝑃 <s 𝑇 ∨ 𝑃 = 𝑇 ∨ 𝑇 <s 𝑃))
6	2, 4, 5	syl2anc 593	. 2 ⊢ (𝜑 → (𝑃 <s 𝑇 ∨ 𝑃 = 𝑇 ∨ 𝑇 <s 𝑃))
7		mulsproplem.1	. . . . . . . . 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 ‘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))))))
8	1	leftoldd 27969	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ( O ‘( bday ‘𝐴)))
9		mulsproplem5.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ No )
10	7, 8, 9	mulsproplem2 28207	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ·_s 𝐵) ∈ No )
11		mulsproplem5.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ No )
12		mulsproplem5.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ ( L ‘𝐵))
13	12	leftoldd 27969	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ ( O ‘( bday ‘𝐵)))
14	7, 11, 13	mulsproplem3 28208	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ·_s 𝑄) ∈ No )
15	10, 14	addscld 28070	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) ∈ No )
16	7, 8, 13	mulsproplem4 28209	. . . . . . 7 ⊢ (𝜑 → (𝑃 ·_s 𝑄) ∈ No )
17	15, 16	subscld 28153	. . . . . 6 ⊢ (𝜑 → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑃 ·_s 𝑄)) ∈ No )
18	17	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 <s 𝑇) → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑃 ·_s 𝑄)) ∈ No )
19	3	leftoldd 27969	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ ( O ‘( bday ‘𝐴)))
20	7, 19, 9	mulsproplem2 28207	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ·_s 𝐵) ∈ No )
21	20, 14	addscld 28070	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) ∈ No )
22	7, 19, 13	mulsproplem4 28209	. . . . . . 7 ⊢ (𝜑 → (𝑇 ·_s 𝑄) ∈ No )
23	21, 22	subscld 28153	. . . . . 6 ⊢ (𝜑 → (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑇 ·_s 𝑄)) ∈ No )
24	23	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 <s 𝑇) → (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑇 ·_s 𝑄)) ∈ No )
25		mulsproplem5.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ ( R ‘𝐵))
26	25	rightoldd 27971	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ ( O ‘( bday ‘𝐵)))
27	7, 11, 26	mulsproplem3 28208	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ·_s 𝑈) ∈ No )
28	20, 27	addscld 28070	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) ∈ No )
29	7, 19, 26	mulsproplem4 28209	. . . . . . 7 ⊢ (𝜑 → (𝑇 ·_s 𝑈) ∈ No )
30	28, 29	subscld 28153	. . . . . 6 ⊢ (𝜑 → (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑇 ·_s 𝑈)) ∈ No )
31	30	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 <s 𝑇) → (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑇 ·_s 𝑈)) ∈ No )
32		sltsleft 27950	. . . . . . . . . . 11 ⊢ (𝐵 ∈ No → ( L ‘𝐵) <<s {𝐵})
33	9, 32	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ( L ‘𝐵) <<s {𝐵})
34		snidg 4619	. . . . . . . . . . 11 ⊢ (𝐵 ∈ No → 𝐵 ∈ {𝐵})
35	9, 34	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ {𝐵})
36	33, 12, 35	sltssepcd 27862	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 <s 𝐵)
37		0no 27899	. . . . . . . . . . . 12 ⊢ 0_s ∈ No
38	37	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0_s ∈ No )
39	12	leftnod 27970	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ No )
40		bday0 27901	. . . . . . . . . . . . . . . 16 ⊢ ( bday ‘ 0_s ) = ∅
41	40, 40	oveq12i 7408	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘ 0_s ) +no ( bday ‘ 0_s )) = (∅ +no ∅)
42		0elon 6401	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ On
43		naddrid 8654	. . . . . . . . . . . . . . . 16 ⊢ (∅ ∈ On → (∅ +no ∅) = ∅)
44	42, 43	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (∅ +no ∅) = ∅
45	41, 44	eqtri 2785	. . . . . . . . . . . . . 14 ⊢ (( bday ‘ 0_s ) +no ( bday ‘ 0_s )) = ∅
46	45	uneq1i 4117	. . . . . . . . . . . . 13 ⊢ ((( bday ‘ 0_s ) +no ( bday ‘ 0_s )) ∪ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +no ( bday ‘𝑄))))) = (∅ ∪ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +no ( bday ‘𝑄)))))
47		0un 4350	. . . . . . . . . . . . 13 ⊢ (∅ ∪ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +no ( bday ‘𝑄))))) = (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +no ( bday ‘𝑄))))
48	46, 47	eqtri 2785	. . . . . . . . . . . 12 ⊢ ((( bday ‘ 0_s ) +no ( bday ‘ 0_s )) ∪ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +no ( bday ‘𝑄))))) = (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +no ( bday ‘𝑄))))
49		oldbdayim 27979	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑃) ∈ ( bday ‘𝐴))
50	8, 49	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ( bday ‘𝑃) ∈ ( bday ‘𝐴))
51		oldbdayim 27979	. . . . . . . . . . . . . . . . 17 ⊢ (𝑄 ∈ ( O ‘( bday ‘𝐵)) → ( bday ‘𝑄) ∈ ( bday ‘𝐵))
52	13, 51	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ( bday ‘𝑄) ∈ ( bday ‘𝐵))
53		bdayon 27842	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝐴) ∈ On
54		bdayon 27842	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝐵) ∈ On
55		naddel12 8671	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → ((( bday ‘𝑃) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑄) ∈ ( bday ‘𝐵)) → (( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
56	53, 54, 55	mp2an 702	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑃) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑄) ∈ ( bday ‘𝐵)) → (( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
57	50, 52, 56	syl2anc 593	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
58		oldbdayim 27979	. . . . . . . . . . . . . . . . 17 ⊢ (𝑇 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑇) ∈ ( bday ‘𝐴))
59	19, 58	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ( bday ‘𝑇) ∈ ( bday ‘𝐴))
60		bdayon 27842	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑇) ∈ On
61		naddel1 8658	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑇) ∈ On ∧ ( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → (( bday ‘𝑇) ∈ ( bday ‘𝐴) ↔ (( bday ‘𝑇) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
62	60, 53, 54, 61	mp3an 1482	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑇) ∈ ( bday ‘𝐴) ↔ (( bday ‘𝑇) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
63	59, 62	sylib 220	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝑇) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
64	57, 63	jca 519	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑇) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
65		bdayon 27842	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑃) ∈ On
66		naddel1 8658	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑃) ∈ On ∧ ( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → (( bday ‘𝑃) ∈ ( bday ‘𝐴) ↔ (( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
67	65, 53, 54, 66	mp3an 1482	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑃) ∈ ( bday ‘𝐴) ↔ (( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
68	50, 67	sylib 220	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
69		naddel12 8671	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → ((( bday ‘𝑇) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑄) ∈ ( bday ‘𝐵)) → (( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
70	53, 54, 69	mp2an 702	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑇) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑄) ∈ ( bday ‘𝐵)) → (( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
71	59, 52, 70	syl2anc 593	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
72	68, 71	jca 519	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
73		bdayon 27842	. . . . . . . . . . . . . . . . . 18 ⊢ ( bday ‘𝑄) ∈ On
74		naddcl 8647	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑃) ∈ On ∧ ( bday ‘𝑄) ∈ On) → (( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ On)
75	65, 73, 74	mp2an 702	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ On
76		naddcl 8647	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑇) ∈ On ∧ ( bday ‘𝐵) ∈ On) → (( bday ‘𝑇) +no ( bday ‘𝐵)) ∈ On)
77	60, 54, 76	mp2an 702	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑇) +no ( bday ‘𝐵)) ∈ On
78	75, 77	onun2i 6469	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∈ On
79		naddcl 8647	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑃) ∈ On ∧ ( bday ‘𝐵) ∈ On) → (( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ On)
80	65, 54, 79	mp2an 702	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ On
81		naddcl 8647	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑇) ∈ On ∧ ( bday ‘𝑄) ∈ On) → (( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ On)
82	60, 73, 81	mp2an 702	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ On
83	80, 82	onun2i 6469	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +no ( bday ‘𝑄))) ∈ On
84		naddcl 8647	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On)
85	53, 54, 84	mp2an 702	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On
86		onunel 6453	. . . . . . . . . . . . . . . 16 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∈ On ∧ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +no ( bday ‘𝑄))) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
87	78, 83, 85, 86	mp3an 1482	. . . . . . . . . . . . . . 15 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
88		onunel 6453	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ On ∧ (( bday ‘𝑇) +no ( bday ‘𝐵)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑇) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
89	75, 77, 85, 88	mp3an 1482	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑇) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
90		onunel 6453	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ On ∧ (( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → (((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
91	80, 82, 85, 90	mp3an 1482	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
92	89, 91	anbi12i 637	. . . . . . . . . . . . . . 15 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ↔ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑇) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ∧ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
93	87, 92	bitri 277	. . . . . . . . . . . . . 14 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑇) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ∧ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
94	64, 72, 93	sylanbrc 592	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
95		elun1 4134	. . . . . . . . . . . . 13 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) → (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +no ( bday ‘𝑄)))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
96	94, 95	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +no ( bday ‘𝑄)))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
97	48, 96	eqeltrid 2866	. . . . . . . . . . 11 ⊢ (𝜑 → ((( bday ‘ 0_s ) +no ( bday ‘ 0_s )) ∪ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +no ( bday ‘𝑄))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
98	7, 38, 38, 2, 4, 39, 9, 97	mulsproplem1 28206	. . . . . . . . . 10 ⊢ (𝜑 → (( 0_s ·_s 0_s ) ∈ No ∧ ((𝑃 <s 𝑇 ∧ 𝑄 <s 𝐵) → ((𝑃 ·_s 𝐵) -_s (𝑃 ·_s 𝑄)) <s ((𝑇 ·_s 𝐵) -_s (𝑇 ·_s 𝑄)))))
99	98	simprd 499	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 <s 𝑇 ∧ 𝑄 <s 𝐵) → ((𝑃 ·_s 𝐵) -_s (𝑃 ·_s 𝑄)) <s ((𝑇 ·_s 𝐵) -_s (𝑇 ·_s 𝑄))))
100	36, 99	mpan2d 704	. . . . . . . 8 ⊢ (𝜑 → (𝑃 <s 𝑇 → ((𝑃 ·_s 𝐵) -_s (𝑃 ·_s 𝑄)) <s ((𝑇 ·_s 𝐵) -_s (𝑇 ·_s 𝑄))))
101	100	imp 410	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 <s 𝑇) → ((𝑃 ·_s 𝐵) -_s (𝑃 ·_s 𝑄)) <s ((𝑇 ·_s 𝐵) -_s (𝑇 ·_s 𝑄)))
102	10, 16	subscld 28153	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ·_s 𝐵) -_s (𝑃 ·_s 𝑄)) ∈ No )
103	20, 22	subscld 28153	. . . . . . . . 9 ⊢ (𝜑 → ((𝑇 ·_s 𝐵) -_s (𝑇 ·_s 𝑄)) ∈ No )
104	102, 103, 14	ltadds1d 28088	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 ·_s 𝐵) -_s (𝑃 ·_s 𝑄)) <s ((𝑇 ·_s 𝐵) -_s (𝑇 ·_s 𝑄)) ↔ (((𝑃 ·_s 𝐵) -_s (𝑃 ·_s 𝑄)) +_s (𝐴 ·_s 𝑄)) <s (((𝑇 ·_s 𝐵) -_s (𝑇 ·_s 𝑄)) +_s (𝐴 ·_s 𝑄))))
105	104	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 <s 𝑇) → (((𝑃 ·_s 𝐵) -_s (𝑃 ·_s 𝑄)) <s ((𝑇 ·_s 𝐵) -_s (𝑇 ·_s 𝑄)) ↔ (((𝑃 ·_s 𝐵) -_s (𝑃 ·_s 𝑄)) +_s (𝐴 ·_s 𝑄)) <s (((𝑇 ·_s 𝐵) -_s (𝑇 ·_s 𝑄)) +_s (𝐴 ·_s 𝑄))))
106	101, 105	mpbid 234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 <s 𝑇) → (((𝑃 ·_s 𝐵) -_s (𝑃 ·_s 𝑄)) +_s (𝐴 ·_s 𝑄)) <s (((𝑇 ·_s 𝐵) -_s (𝑇 ·_s 𝑄)) +_s (𝐴 ·_s 𝑄)))
107	10, 14, 16	addsubsd 28172	. . . . . . 7 ⊢ (𝜑 → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑃 ·_s 𝑄)) = (((𝑃 ·_s 𝐵) -_s (𝑃 ·_s 𝑄)) +_s (𝐴 ·_s 𝑄)))
108	107	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 <s 𝑇) → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑃 ·_s 𝑄)) = (((𝑃 ·_s 𝐵) -_s (𝑃 ·_s 𝑄)) +_s (𝐴 ·_s 𝑄)))
109	20, 14, 22	addsubsd 28172	. . . . . . 7 ⊢ (𝜑 → (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑇 ·_s 𝑄)) = (((𝑇 ·_s 𝐵) -_s (𝑇 ·_s 𝑄)) +_s (𝐴 ·_s 𝑄)))
110	109	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑃 <s 𝑇) → (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑇 ·_s 𝑄)) = (((𝑇 ·_s 𝐵) -_s (𝑇 ·_s 𝑄)) +_s (𝐴 ·_s 𝑄)))
111	106, 108, 110	3brtr4d 5132	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 <s 𝑇) → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑃 ·_s 𝑄)) <s (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑇 ·_s 𝑄)))
112		sltsleft 27950	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → ( L ‘𝐴) <<s {𝐴})
113	11, 112	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ( L ‘𝐴) <<s {𝐴})
114		snidg 4619	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → 𝐴 ∈ {𝐴})
115	11, 114	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
116	113, 3, 115	sltssepcd 27862	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 <s 𝐴)
117		lltr 27952	. . . . . . . . . . 11 ⊢ ( L ‘𝐵) <<s ( R ‘𝐵)
118	117	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ( L ‘𝐵) <<s ( R ‘𝐵))
119	118, 12, 25	sltssepcd 27862	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 <s 𝑈)
120	25	rightnod 27972	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ No )
121	45	uneq1i 4117	. . . . . . . . . . . . 13 ⊢ ((( bday ‘ 0_s ) +no ( bday ‘ 0_s )) ∪ (((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))))) = (∅ ∪ (((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))))
122		0un 4350	. . . . . . . . . . . . 13 ⊢ (∅ ∪ (((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))))) = (((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))))
123	121, 122	eqtri 2785	. . . . . . . . . . . 12 ⊢ ((( bday ‘ 0_s ) +no ( bday ‘ 0_s )) ∪ (((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))))) = (((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))))
124		oldbdayim 27979	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ ( O ‘( bday ‘𝐵)) → ( bday ‘𝑈) ∈ ( bday ‘𝐵))
125	26, 124	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ( bday ‘𝑈) ∈ ( bday ‘𝐵))
126		bdayon 27842	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑈) ∈ On
127		naddel2 8659	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑈) ∈ On ∧ ( bday ‘𝐵) ∈ On ∧ ( bday ‘𝐴) ∈ On) → (( bday ‘𝑈) ∈ ( bday ‘𝐵) ↔ (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
128	126, 54, 53, 127	mp3an 1482	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑈) ∈ ( bday ‘𝐵) ↔ (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
129	125, 128	sylib 220	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
130	71, 129	jca 519	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
131		naddel12 8671	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → ((( bday ‘𝑇) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑈) ∈ ( bday ‘𝐵)) → (( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
132	53, 54, 131	mp2an 702	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑇) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑈) ∈ ( bday ‘𝐵)) → (( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
133	59, 125, 132	syl2anc 593	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
134		naddel2 8659	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑄) ∈ On ∧ ( bday ‘𝐵) ∈ On ∧ ( bday ‘𝐴) ∈ On) → (( bday ‘𝑄) ∈ ( bday ‘𝐵) ↔ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
135	73, 54, 53, 134	mp3an 1482	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑄) ∈ ( bday ‘𝐵) ↔ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
136	52, 135	sylib 220	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
137	133, 136	jca 519	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
138		naddcl 8647	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝑈) ∈ On) → (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ On)
139	53, 126, 138	mp2an 702	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ On
140	82, 139	onun2i 6469	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∈ On
141		naddcl 8647	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑇) ∈ On ∧ ( bday ‘𝑈) ∈ On) → (( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ On)
142	60, 126, 141	mp2an 702	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ On
143		naddcl 8647	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝑄) ∈ On) → (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ On)
144	53, 73, 143	mp2an 702	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ On
145	142, 144	onun2i 6469	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ On
146		onunel 6453	. . . . . . . . . . . . . . . 16 ⊢ ((((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∈ On ∧ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → ((((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
147	140, 145, 85, 146	mp3an 1482	. . . . . . . . . . . . . . 15 ⊢ ((((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
148		onunel 6453	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → (((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
149	82, 139, 85, 148	mp3an 1482	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
150		onunel 6453	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → (((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
151	142, 144, 85, 150	mp3an 1482	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
152	149, 151	anbi12i 637	. . . . . . . . . . . . . . 15 ⊢ ((((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ↔ (((( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ∧ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
153	147, 152	bitri 277	. . . . . . . . . . . . . 14 ⊢ ((((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ∧ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
154	130, 137, 153	sylanbrc 592	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
155		elun1 4134	. . . . . . . . . . . . 13 ⊢ ((((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) → (((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
156	154, 155	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
157	123, 156	eqeltrid 2866	. . . . . . . . . . 11 ⊢ (𝜑 → ((( bday ‘ 0_s ) +no ( bday ‘ 0_s )) ∪ (((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
158	7, 38, 38, 4, 11, 39, 120, 157	mulsproplem1 28206	. . . . . . . . . 10 ⊢ (𝜑 → (( 0_s ·_s 0_s ) ∈ No ∧ ((𝑇 <s 𝐴 ∧ 𝑄 <s 𝑈) → ((𝑇 ·_s 𝑈) -_s (𝑇 ·_s 𝑄)) <s ((𝐴 ·_s 𝑈) -_s (𝐴 ·_s 𝑄)))))
159	158	simprd 499	. . . . . . . . 9 ⊢ (𝜑 → ((𝑇 <s 𝐴 ∧ 𝑄 <s 𝑈) → ((𝑇 ·_s 𝑈) -_s (𝑇 ·_s 𝑄)) <s ((𝐴 ·_s 𝑈) -_s (𝐴 ·_s 𝑄))))
160	116, 119, 159	mp2and 709	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ·_s 𝑈) -_s (𝑇 ·_s 𝑄)) <s ((𝐴 ·_s 𝑈) -_s (𝐴 ·_s 𝑄)))
161	29, 27, 22, 14	ltsubsubs3bd 28175	. . . . . . . . 9 ⊢ (𝜑 → (((𝑇 ·_s 𝑈) -_s (𝑇 ·_s 𝑄)) <s ((𝐴 ·_s 𝑈) -_s (𝐴 ·_s 𝑄)) ↔ ((𝐴 ·_s 𝑄) -_s (𝑇 ·_s 𝑄)) <s ((𝐴 ·_s 𝑈) -_s (𝑇 ·_s 𝑈))))
162	14, 22	subscld 28153	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ·_s 𝑄) -_s (𝑇 ·_s 𝑄)) ∈ No )
163	27, 29	subscld 28153	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ·_s 𝑈) -_s (𝑇 ·_s 𝑈)) ∈ No )
164	162, 163, 20	ltadds2d 28087	. . . . . . . . 9 ⊢ (𝜑 → (((𝐴 ·_s 𝑄) -_s (𝑇 ·_s 𝑄)) <s ((𝐴 ·_s 𝑈) -_s (𝑇 ·_s 𝑈)) ↔ ((𝑇 ·_s 𝐵) +_s ((𝐴 ·_s 𝑄) -_s (𝑇 ·_s 𝑄))) <s ((𝑇 ·_s 𝐵) +_s ((𝐴 ·_s 𝑈) -_s (𝑇 ·_s 𝑈)))))
165	161, 164	bitrd 281	. . . . . . . 8 ⊢ (𝜑 → (((𝑇 ·_s 𝑈) -_s (𝑇 ·_s 𝑄)) <s ((𝐴 ·_s 𝑈) -_s (𝐴 ·_s 𝑄)) ↔ ((𝑇 ·_s 𝐵) +_s ((𝐴 ·_s 𝑄) -_s (𝑇 ·_s 𝑄))) <s ((𝑇 ·_s 𝐵) +_s ((𝐴 ·_s 𝑈) -_s (𝑇 ·_s 𝑈)))))
166	160, 165	mpbid 234	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ·_s 𝐵) +_s ((𝐴 ·_s 𝑄) -_s (𝑇 ·_s 𝑄))) <s ((𝑇 ·_s 𝐵) +_s ((𝐴 ·_s 𝑈) -_s (𝑇 ·_s 𝑈))))
167	20, 14, 22	addsubsassd 28171	. . . . . . 7 ⊢ (𝜑 → (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑇 ·_s 𝑄)) = ((𝑇 ·_s 𝐵) +_s ((𝐴 ·_s 𝑄) -_s (𝑇 ·_s 𝑄))))
168	20, 27, 29	addsubsassd 28171	. . . . . . 7 ⊢ (𝜑 → (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑇 ·_s 𝑈)) = ((𝑇 ·_s 𝐵) +_s ((𝐴 ·_s 𝑈) -_s (𝑇 ·_s 𝑈))))
169	166, 167, 168	3brtr4d 5132	. . . . . 6 ⊢ (𝜑 → (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑇 ·_s 𝑄)) <s (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑇 ·_s 𝑈)))
170	169	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 <s 𝑇) → (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑇 ·_s 𝑄)) <s (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑇 ·_s 𝑈)))
171	18, 24, 31, 111, 170	ltstrd 27824	. . . 4 ⊢ ((𝜑 ∧ 𝑃 <s 𝑇) → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑃 ·_s 𝑄)) <s (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑇 ·_s 𝑈)))
172	171	ex 416	. . 3 ⊢ (𝜑 → (𝑃 <s 𝑇 → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑃 ·_s 𝑄)) <s (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑇 ·_s 𝑈))))
173		oveq1 7403	. . . . . . 7 ⊢ (𝑃 = 𝑇 → (𝑃 ·_s 𝐵) = (𝑇 ·_s 𝐵))
174	173	oveq1d 7411	. . . . . 6 ⊢ (𝑃 = 𝑇 → ((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) = ((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)))
175		oveq1 7403	. . . . . 6 ⊢ (𝑃 = 𝑇 → (𝑃 ·_s 𝑄) = (𝑇 ·_s 𝑄))
176	174, 175	oveq12d 7414	. . . . 5 ⊢ (𝑃 = 𝑇 → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑃 ·_s 𝑄)) = (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑇 ·_s 𝑄)))
177	176	breq1d 5110	. . . 4 ⊢ (𝑃 = 𝑇 → ((((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑃 ·_s 𝑄)) <s (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑇 ·_s 𝑈)) ↔ (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑇 ·_s 𝑄)) <s (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑇 ·_s 𝑈))))
178	169, 177	syl5ibrcom 249	. . 3 ⊢ (𝜑 → (𝑃 = 𝑇 → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑃 ·_s 𝑄)) <s (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑇 ·_s 𝑈))))
179	17	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 <s 𝑃) → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑃 ·_s 𝑄)) ∈ No )
180	10, 27	addscld 28070	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) ∈ No )
181	7, 8, 26	mulsproplem4 28209	. . . . . . 7 ⊢ (𝜑 → (𝑃 ·_s 𝑈) ∈ No )
182	180, 181	subscld 28153	. . . . . 6 ⊢ (𝜑 → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑃 ·_s 𝑈)) ∈ No )
183	182	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 <s 𝑃) → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑃 ·_s 𝑈)) ∈ No )
184	30	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 <s 𝑃) → (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑇 ·_s 𝑈)) ∈ No )
185	113, 1, 115	sltssepcd 27862	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 <s 𝐴)
186	45	uneq1i 4117	. . . . . . . . . . . . 13 ⊢ ((( bday ‘ 0_s ) +no ( bday ‘ 0_s )) ∪ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))))) = (∅ ∪ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))))
187		0un 4350	. . . . . . . . . . . . 13 ⊢ (∅ ∪ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))))) = (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))))
188	186, 187	eqtri 2785	. . . . . . . . . . . 12 ⊢ ((( bday ‘ 0_s ) +no ( bday ‘ 0_s )) ∪ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))))) = (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))))
189	57, 129	jca 519	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
190		naddel12 8671	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → ((( bday ‘𝑃) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑈) ∈ ( bday ‘𝐵)) → (( bday ‘𝑃) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
191	53, 54, 190	mp2an 702	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑃) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑈) ∈ ( bday ‘𝐵)) → (( bday ‘𝑃) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
192	50, 125, 191	syl2anc 593	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝑃) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
193	192, 136	jca 519	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
194	75, 139	onun2i 6469	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∈ On
195		naddcl 8647	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑃) ∈ On ∧ ( bday ‘𝑈) ∈ On) → (( bday ‘𝑃) +no ( bday ‘𝑈)) ∈ On)
196	65, 126, 195	mp2an 702	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑃) +no ( bday ‘𝑈)) ∈ On
197	196, 144	onun2i 6469	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ On
198		onunel 6453	. . . . . . . . . . . . . . . 16 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∈ On ∧ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
199	194, 197, 85, 198	mp3an 1482	. . . . . . . . . . . . . . 15 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
200		onunel 6453	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
201	75, 139, 85, 200	mp3an 1482	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
202		onunel 6453	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑃) +no ( bday ‘𝑈)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → (((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
203	196, 144, 85, 202	mp3an 1482	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
204	201, 203	anbi12i 637	. . . . . . . . . . . . . . 15 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ↔ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ∧ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
205	199, 204	bitri 277	. . . . . . . . . . . . . 14 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ∧ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
206	189, 193, 205	sylanbrc 592	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
207		elun1 4134	. . . . . . . . . . . . 13 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) → (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
208	206, 207	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
209	188, 208	eqeltrid 2866	. . . . . . . . . . 11 ⊢ (𝜑 → ((( bday ‘ 0_s ) +no ( bday ‘ 0_s )) ∪ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
210	7, 38, 38, 2, 11, 39, 120, 209	mulsproplem1 28206	. . . . . . . . . 10 ⊢ (𝜑 → (( 0_s ·_s 0_s ) ∈ No ∧ ((𝑃 <s 𝐴 ∧ 𝑄 <s 𝑈) → ((𝑃 ·_s 𝑈) -_s (𝑃 ·_s 𝑄)) <s ((𝐴 ·_s 𝑈) -_s (𝐴 ·_s 𝑄)))))
211	210	simprd 499	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 <s 𝐴 ∧ 𝑄 <s 𝑈) → ((𝑃 ·_s 𝑈) -_s (𝑃 ·_s 𝑄)) <s ((𝐴 ·_s 𝑈) -_s (𝐴 ·_s 𝑄))))
212	185, 119, 211	mp2and 709	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 ·_s 𝑈) -_s (𝑃 ·_s 𝑄)) <s ((𝐴 ·_s 𝑈) -_s (𝐴 ·_s 𝑄)))
213	181, 27, 16, 14	ltsubsubs3bd 28175	. . . . . . . . 9 ⊢ (𝜑 → (((𝑃 ·_s 𝑈) -_s (𝑃 ·_s 𝑄)) <s ((𝐴 ·_s 𝑈) -_s (𝐴 ·_s 𝑄)) ↔ ((𝐴 ·_s 𝑄) -_s (𝑃 ·_s 𝑄)) <s ((𝐴 ·_s 𝑈) -_s (𝑃 ·_s 𝑈))))
214	14, 16	subscld 28153	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ·_s 𝑄) -_s (𝑃 ·_s 𝑄)) ∈ No )
215	27, 181	subscld 28153	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ·_s 𝑈) -_s (𝑃 ·_s 𝑈)) ∈ No )
216	214, 215, 10	ltadds2d 28087	. . . . . . . . 9 ⊢ (𝜑 → (((𝐴 ·_s 𝑄) -_s (𝑃 ·_s 𝑄)) <s ((𝐴 ·_s 𝑈) -_s (𝑃 ·_s 𝑈)) ↔ ((𝑃 ·_s 𝐵) +_s ((𝐴 ·_s 𝑄) -_s (𝑃 ·_s 𝑄))) <s ((𝑃 ·_s 𝐵) +_s ((𝐴 ·_s 𝑈) -_s (𝑃 ·_s 𝑈)))))
217	213, 216	bitrd 281	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 ·_s 𝑈) -_s (𝑃 ·_s 𝑄)) <s ((𝐴 ·_s 𝑈) -_s (𝐴 ·_s 𝑄)) ↔ ((𝑃 ·_s 𝐵) +_s ((𝐴 ·_s 𝑄) -_s (𝑃 ·_s 𝑄))) <s ((𝑃 ·_s 𝐵) +_s ((𝐴 ·_s 𝑈) -_s (𝑃 ·_s 𝑈)))))
218	212, 217	mpbid 234	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ·_s 𝐵) +_s ((𝐴 ·_s 𝑄) -_s (𝑃 ·_s 𝑄))) <s ((𝑃 ·_s 𝐵) +_s ((𝐴 ·_s 𝑈) -_s (𝑃 ·_s 𝑈))))
219	10, 14, 16	addsubsassd 28171	. . . . . . 7 ⊢ (𝜑 → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑃 ·_s 𝑄)) = ((𝑃 ·_s 𝐵) +_s ((𝐴 ·_s 𝑄) -_s (𝑃 ·_s 𝑄))))
220	10, 27, 181	addsubsassd 28171	. . . . . . 7 ⊢ (𝜑 → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑃 ·_s 𝑈)) = ((𝑃 ·_s 𝐵) +_s ((𝐴 ·_s 𝑈) -_s (𝑃 ·_s 𝑈))))
221	218, 219, 220	3brtr4d 5132	. . . . . 6 ⊢ (𝜑 → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑃 ·_s 𝑄)) <s (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑃 ·_s 𝑈)))
222	221	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 <s 𝑃) → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑃 ·_s 𝑄)) <s (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑃 ·_s 𝑈)))
223		sltsright 27951	. . . . . . . . . . 11 ⊢ (𝐵 ∈ No → {𝐵} <<s ( R ‘𝐵))
224	9, 223	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → {𝐵} <<s ( R ‘𝐵))
225	224, 35, 25	sltssepcd 27862	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 <s 𝑈)
226	45	uneq1i 4117	. . . . . . . . . . . . 13 ⊢ ((( bday ‘ 0_s ) +no ( bday ‘ 0_s )) ∪ (((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵))))) = (∅ ∪ (((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵)))))
227		0un 4350	. . . . . . . . . . . . 13 ⊢ (∅ ∪ (((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵))))) = (((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵))))
228	226, 227	eqtri 2785	. . . . . . . . . . . 12 ⊢ ((( bday ‘ 0_s ) +no ( bday ‘ 0_s )) ∪ (((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵))))) = (((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵))))
229		onunel 6453	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑇) +no ( bday ‘𝐵)) ∈ On ∧ (( bday ‘𝑃) +no ( bday ‘𝑈)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → (((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑇) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑃) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
230	77, 196, 85, 229	mp3an 1482	. . . . . . . . . . . . . . 15 ⊢ (((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑇) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑃) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
231	63, 192, 230	sylanbrc 592	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
232	133, 68	jca 519	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
233	77, 196	onun2i 6469	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∈ On
234	142, 80	onun2i 6469	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵))) ∈ On
235		onunel 6453	. . . . . . . . . . . . . . . 16 ⊢ ((((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∈ On ∧ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵))) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → ((((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
236	233, 234, 85, 235	mp3an 1482	. . . . . . . . . . . . . . 15 ⊢ ((((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
237		onunel 6453	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ On ∧ (( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → (((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
238	142, 80, 85, 237	mp3an 1482	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
239	238	anbi2i 632	. . . . . . . . . . . . . . 15 ⊢ ((((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ↔ (((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
240	236, 239	bitri 277	. . . . . . . . . . . . . 14 ⊢ ((((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
241	231, 232, 240	sylanbrc 592	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
242		elun1 4134	. . . . . . . . . . . . 13 ⊢ ((((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) → (((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵)))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
243	241, 242	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵)))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
244	228, 243	eqeltrid 2866	. . . . . . . . . . 11 ⊢ (𝜑 → ((( bday ‘ 0_s ) +no ( bday ‘ 0_s )) ∪ (((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∪ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
245	7, 38, 38, 4, 2, 9, 120, 244	mulsproplem1 28206	. . . . . . . . . 10 ⊢ (𝜑 → (( 0_s ·_s 0_s ) ∈ No ∧ ((𝑇 <s 𝑃 ∧ 𝐵 <s 𝑈) → ((𝑇 ·_s 𝑈) -_s (𝑇 ·_s 𝐵)) <s ((𝑃 ·_s 𝑈) -_s (𝑃 ·_s 𝐵)))))
246	245	simprd 499	. . . . . . . . 9 ⊢ (𝜑 → ((𝑇 <s 𝑃 ∧ 𝐵 <s 𝑈) → ((𝑇 ·_s 𝑈) -_s (𝑇 ·_s 𝐵)) <s ((𝑃 ·_s 𝑈) -_s (𝑃 ·_s 𝐵))))
247	225, 246	mpan2d 704	. . . . . . . 8 ⊢ (𝜑 → (𝑇 <s 𝑃 → ((𝑇 ·_s 𝑈) -_s (𝑇 ·_s 𝐵)) <s ((𝑃 ·_s 𝑈) -_s (𝑃 ·_s 𝐵))))
248	247	imp 410	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 <s 𝑃) → ((𝑇 ·_s 𝑈) -_s (𝑇 ·_s 𝐵)) <s ((𝑃 ·_s 𝑈) -_s (𝑃 ·_s 𝐵)))
249	29, 20, 181, 10	ltsubsubs2bd 28174	. . . . . . . . 9 ⊢ (𝜑 → (((𝑇 ·_s 𝑈) -_s (𝑇 ·_s 𝐵)) <s ((𝑃 ·_s 𝑈) -_s (𝑃 ·_s 𝐵)) ↔ ((𝑃 ·_s 𝐵) -_s (𝑃 ·_s 𝑈)) <s ((𝑇 ·_s 𝐵) -_s (𝑇 ·_s 𝑈))))
250	10, 181	subscld 28153	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 ·_s 𝐵) -_s (𝑃 ·_s 𝑈)) ∈ No )
251	20, 29	subscld 28153	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑇 ·_s 𝐵) -_s (𝑇 ·_s 𝑈)) ∈ No )
252	250, 251, 27	ltadds1d 28088	. . . . . . . . 9 ⊢ (𝜑 → (((𝑃 ·_s 𝐵) -_s (𝑃 ·_s 𝑈)) <s ((𝑇 ·_s 𝐵) -_s (𝑇 ·_s 𝑈)) ↔ (((𝑃 ·_s 𝐵) -_s (𝑃 ·_s 𝑈)) +_s (𝐴 ·_s 𝑈)) <s (((𝑇 ·_s 𝐵) -_s (𝑇 ·_s 𝑈)) +_s (𝐴 ·_s 𝑈))))
253	249, 252	bitrd 281	. . . . . . . 8 ⊢ (𝜑 → (((𝑇 ·_s 𝑈) -_s (𝑇 ·_s 𝐵)) <s ((𝑃 ·_s 𝑈) -_s (𝑃 ·_s 𝐵)) ↔ (((𝑃 ·_s 𝐵) -_s (𝑃 ·_s 𝑈)) +_s (𝐴 ·_s 𝑈)) <s (((𝑇 ·_s 𝐵) -_s (𝑇 ·_s 𝑈)) +_s (𝐴 ·_s 𝑈))))
254	253	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 <s 𝑃) → (((𝑇 ·_s 𝑈) -_s (𝑇 ·_s 𝐵)) <s ((𝑃 ·_s 𝑈) -_s (𝑃 ·_s 𝐵)) ↔ (((𝑃 ·_s 𝐵) -_s (𝑃 ·_s 𝑈)) +_s (𝐴 ·_s 𝑈)) <s (((𝑇 ·_s 𝐵) -_s (𝑇 ·_s 𝑈)) +_s (𝐴 ·_s 𝑈))))
255	248, 254	mpbid 234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 <s 𝑃) → (((𝑃 ·_s 𝐵) -_s (𝑃 ·_s 𝑈)) +_s (𝐴 ·_s 𝑈)) <s (((𝑇 ·_s 𝐵) -_s (𝑇 ·_s 𝑈)) +_s (𝐴 ·_s 𝑈)))
256	10, 27, 181	addsubsd 28172	. . . . . . 7 ⊢ (𝜑 → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑃 ·_s 𝑈)) = (((𝑃 ·_s 𝐵) -_s (𝑃 ·_s 𝑈)) +_s (𝐴 ·_s 𝑈)))
257	256	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 <s 𝑃) → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑃 ·_s 𝑈)) = (((𝑃 ·_s 𝐵) -_s (𝑃 ·_s 𝑈)) +_s (𝐴 ·_s 𝑈)))
258	20, 27, 29	addsubsd 28172	. . . . . . 7 ⊢ (𝜑 → (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑇 ·_s 𝑈)) = (((𝑇 ·_s 𝐵) -_s (𝑇 ·_s 𝑈)) +_s (𝐴 ·_s 𝑈)))
259	258	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑇 <s 𝑃) → (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑇 ·_s 𝑈)) = (((𝑇 ·_s 𝐵) -_s (𝑇 ·_s 𝑈)) +_s (𝐴 ·_s 𝑈)))
260	255, 257, 259	3brtr4d 5132	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 <s 𝑃) → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑃 ·_s 𝑈)) <s (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑇 ·_s 𝑈)))
261	179, 183, 184, 222, 260	ltstrd 27824	. . . 4 ⊢ ((𝜑 ∧ 𝑇 <s 𝑃) → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑃 ·_s 𝑄)) <s (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑇 ·_s 𝑈)))
262	261	ex 416	. . 3 ⊢ (𝜑 → (𝑇 <s 𝑃 → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑃 ·_s 𝑄)) <s (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑇 ·_s 𝑈))))
263	172, 178, 262	3jaod 1449	. 2 ⊢ (𝜑 → ((𝑃 <s 𝑇 ∨ 𝑃 = 𝑇 ∨ 𝑇 <s 𝑃) → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑃 ·_s 𝑄)) <s (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑇 ·_s 𝑈))))
264	6, 263	mpd 15	1 ⊢ (𝜑 → (((𝑃 ·_s 𝐵) +_s (𝐴 ·_s 𝑄)) -_s (𝑃 ·_s 𝑄)) <s (((𝑇 ·_s 𝐵) +_s (𝐴 ·_s 𝑈)) -_s (𝑇 ·_s 𝑈)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∨ w3o 1097 = wceq 1560 ∈ wcel 2142 ∀wral 3076 ∪ cun 3902 ∅c0 4285 {csn 4582 class class class wbr 5100 Oncon0 6346 ‘cfv 6521 (class class class)co 7396 +no cnadd 8635 No csur 27701 <s clts 27702 bday cbday 27703 <<s cslts 27847 0_s c0s 27895 O cold 27913 L cleft 27915 R cright 27916 +_s cadds 28049 -_s csubs 28110 ·_s cmuls 28196

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-ot 4591 df-uni 4866 df-int 4906 df-iun 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-1st 7970 df-2nd 7971 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-1o 8437 df-2o 8438 df-nadd 8636 df-no 27704 df-lts 27705 df-bday 27706 df-les 27806 df-slts 27848 df-cuts 27850 df-0s 27897 df-made 27917 df-old 27918 df-left 27920 df-right 27921 df-norec 28028 df-norec2 28039 df-adds 28050 df-negs 28111 df-subs 28112

This theorem is referenced by: mulsproplem9 28214

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