Theorem mulsproplem6 27995

Description: Lemma for surreal multiplication. Show one of the inequalities involved in surreal multiplication's cuts. (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))))))
mulsproplem6.1	⊢ (𝜑 → 𝐴 ∈ No )
mulsproplem6.2	⊢ (𝜑 → 𝐵 ∈ No )
mulsproplem6.3	⊢ (𝜑 → 𝑃 ∈ ( L ‘𝐴))
mulsproplem6.4	⊢ (𝜑 → 𝑄 ∈ ( L ‘𝐵))
mulsproplem6.5	⊢ (𝜑 → 𝑉 ∈ ( R ‘𝐴))
mulsproplem6.6	⊢ (𝜑 → 𝑊 ∈ ( L ‘𝐵))

Assertion

Ref	Expression
mulsproplem6	⊢ (𝜑 → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) <s (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑉 ·s 𝑊)))

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   𝑉,𝑎,𝑏,𝑐,𝑑,𝑒,𝑓   𝑊,𝑏,𝑐,𝑑,𝑒,𝑓

Allowed substitution hints:   𝜑(𝑒,𝑓,𝑎,𝑏,𝑐,𝑑)   𝑄(𝑎)   𝑊(𝑎)

Proof of Theorem mulsproplem6

Step	Hyp	Ref	Expression
1		leftssno 27781	. . . 4 ⊢ ( L ‘𝐵) ⊆ No
2		mulsproplem6.4	. . . 4 ⊢ (𝜑 → 𝑄 ∈ ( L ‘𝐵))
3	1, 2	sselid 3976	. . 3 ⊢ (𝜑 → 𝑄 ∈ No )
4		mulsproplem6.6	. . . 4 ⊢ (𝜑 → 𝑊 ∈ ( L ‘𝐵))
5	1, 4	sselid 3976	. . 3 ⊢ (𝜑 → 𝑊 ∈ No )
6		sltlin 27656	. . 3 ⊢ ((𝑄 ∈ No ∧ 𝑊 ∈ No ) → (𝑄 <s 𝑊 ∨ 𝑄 = 𝑊 ∨ 𝑊 <s 𝑄))
7	3, 5, 6	syl2anc 583	. 2 ⊢ (𝜑 → (𝑄 <s 𝑊 ∨ 𝑄 = 𝑊 ∨ 𝑊 <s 𝑄))
8		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))))))
9		leftssold 27779	. . . . . . . . . 10 ⊢ ( L ‘𝐴) ⊆ ( O ‘( bday ‘𝐴))
10		mulsproplem6.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ ( L ‘𝐴))
11	9, 10	sselid 3976	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ( O ‘( bday ‘𝐴)))
12		mulsproplem6.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ No )
13	8, 11, 12	mulsproplem2 27991	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ·s 𝐵) ∈ No )
14		mulsproplem6.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ No )
15		leftssold 27779	. . . . . . . . . 10 ⊢ ( L ‘𝐵) ⊆ ( O ‘( bday ‘𝐵))
16	15, 2	sselid 3976	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ ( O ‘( bday ‘𝐵)))
17	8, 14, 16	mulsproplem3 27992	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ·s 𝑄) ∈ No )
18	13, 17	addscld 27871	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) ∈ No )
19	8, 11, 16	mulsproplem4 27993	. . . . . . 7 ⊢ (𝜑 → (𝑃 ·s 𝑄) ∈ No )
20	18, 19	subscld 27947	. . . . . 6 ⊢ (𝜑 → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) ∈ No )
21	20	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑄 <s 𝑊) → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) ∈ No )
22	15, 4	sselid 3976	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ ( O ‘( bday ‘𝐵)))
23	8, 14, 22	mulsproplem3 27992	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ·s 𝑊) ∈ No )
24	13, 23	addscld 27871	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑊)) ∈ No )
25	8, 11, 22	mulsproplem4 27993	. . . . . . 7 ⊢ (𝜑 → (𝑃 ·s 𝑊) ∈ No )
26	24, 25	subscld 27947	. . . . . 6 ⊢ (𝜑 → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑃 ·s 𝑊)) ∈ No )
27	26	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑄 <s 𝑊) → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑃 ·s 𝑊)) ∈ No )
28		rightssold 27780	. . . . . . . . . 10 ⊢ ( R ‘𝐴) ⊆ ( O ‘( bday ‘𝐴))
29		mulsproplem6.5	. . . . . . . . . 10 ⊢ (𝜑 → 𝑉 ∈ ( R ‘𝐴))
30	28, 29	sselid 3976	. . . . . . . . 9 ⊢ (𝜑 → 𝑉 ∈ ( O ‘( bday ‘𝐴)))
31	8, 30, 12	mulsproplem2 27991	. . . . . . . 8 ⊢ (𝜑 → (𝑉 ·s 𝐵) ∈ No )
32	31, 23	addscld 27871	. . . . . . 7 ⊢ (𝜑 → ((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) ∈ No )
33	8, 30, 22	mulsproplem4 27993	. . . . . . 7 ⊢ (𝜑 → (𝑉 ·s 𝑊) ∈ No )
34	32, 33	subscld 27947	. . . . . 6 ⊢ (𝜑 → (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑉 ·s 𝑊)) ∈ No )
35	34	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑄 <s 𝑊) → (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑉 ·s 𝑊)) ∈ No )
36		ssltleft 27771	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → ( L ‘𝐴) <<s {𝐴})
37	14, 36	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ( L ‘𝐴) <<s {𝐴})
38		snidg 4658	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → 𝐴 ∈ {𝐴})
39	14, 38	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
40	37, 10, 39	ssltsepcd 27701	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 <s 𝐴)
41		0sno 27733	. . . . . . . . . . . 12 ⊢ 0s ∈ No
42	41	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0s ∈ No )
43		leftssno 27781	. . . . . . . . . . . 12 ⊢ ( L ‘𝐴) ⊆ No
44	43, 10	sselid 3976	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ No )
45		bday0s 27735	. . . . . . . . . . . . . . . 16 ⊢ ( bday ‘ 0s ) = ∅
46	45, 45	oveq12i 7426	. . . . . . . . . . . . . . 15 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = (∅ +no ∅)
47		0elon 6417	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ On
48		naddrid 8695	. . . . . . . . . . . . . . . 16 ⊢ (∅ ∈ On → (∅ +no ∅) = ∅)
49	47, 48	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (∅ +no ∅) = ∅
50	46, 49	eqtri 2755	. . . . . . . . . . . . . 14 ⊢ (( bday ‘ 0s ) +no ( bday ‘ 0s )) = ∅
51	50	uneq1i 4155	. . . . . . . . . . . . 13 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑊))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))))) = (∅ ∪ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑊))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))))
52		0un 4388	. . . . . . . . . . . . 13 ⊢ (∅ ∪ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑊))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))))) = (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑊))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))))
53	51, 52	eqtri 2755	. . . . . . . . . . . 12 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑊))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))))) = (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑊))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))))
54		oldbdayim 27789	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑃) ∈ ( bday ‘𝐴))
55	11, 54	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ( bday ‘𝑃) ∈ ( bday ‘𝐴))
56		oldbdayim 27789	. . . . . . . . . . . . . . . . 17 ⊢ (𝑄 ∈ ( O ‘( bday ‘𝐵)) → ( bday ‘𝑄) ∈ ( bday ‘𝐵))
57	16, 56	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ( bday ‘𝑄) ∈ ( bday ‘𝐵))
58		bdayelon 27683	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝐴) ∈ On
59		bdayelon 27683	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝐵) ∈ On
60		naddel12 8712	. . . . . . . . . . . . . . . . 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))))
61	58, 59, 60	mp2an 691	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑃) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑄) ∈ ( bday ‘𝐵)) → (( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
62	55, 57, 61	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
63		oldbdayim 27789	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ ( O ‘( bday ‘𝐵)) → ( bday ‘𝑊) ∈ ( bday ‘𝐵))
64	22, 63	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ( bday ‘𝑊) ∈ ( bday ‘𝐵))
65		bdayelon 27683	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑊) ∈ On
66		naddel2 8700	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑊) ∈ On ∧ ( bday ‘𝐵) ∈ On ∧ ( bday ‘𝐴) ∈ On) → (( bday ‘𝑊) ∈ ( bday ‘𝐵) ↔ (( bday ‘𝐴) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
67	65, 59, 58, 66	mp3an 1458	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑊) ∈ ( bday ‘𝐵) ↔ (( bday ‘𝐴) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
68	64, 67	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝐴) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
69	62, 68	jca 511	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
70		naddel12 8712	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → ((( bday ‘𝑃) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑊) ∈ ( bday ‘𝐵)) → (( bday ‘𝑃) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
71	58, 59, 70	mp2an 691	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑃) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑊) ∈ ( bday ‘𝐵)) → (( bday ‘𝑃) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
72	55, 64, 71	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝑃) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
73		bdayelon 27683	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑄) ∈ On
74		naddel2 8700	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑄) ∈ On ∧ ( bday ‘𝐵) ∈ On ∧ ( bday ‘𝐴) ∈ On) → (( bday ‘𝑄) ∈ ( bday ‘𝐵) ↔ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
75	73, 59, 58, 74	mp3an 1458	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑄) ∈ ( bday ‘𝐵) ↔ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
76	57, 75	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
77	72, 76	jca 511	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
78		bdayelon 27683	. . . . . . . . . . . . . . . . . 18 ⊢ ( bday ‘𝑃) ∈ On
79		naddcl 8689	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑃) ∈ On ∧ ( bday ‘𝑄) ∈ On) → (( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ On)
80	78, 73, 79	mp2an 691	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ On
81		naddcl 8689	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝑊) ∈ On) → (( bday ‘𝐴) +no ( bday ‘𝑊)) ∈ On)
82	58, 65, 81	mp2an 691	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝐴) +no ( bday ‘𝑊)) ∈ On
83	80, 82	onun2i 6485	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑊))) ∈ On
84		naddcl 8689	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑃) ∈ On ∧ ( bday ‘𝑊) ∈ On) → (( bday ‘𝑃) +no ( bday ‘𝑊)) ∈ On)
85	78, 65, 84	mp2an 691	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑃) +no ( bday ‘𝑊)) ∈ On
86		naddcl 8689	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝑄) ∈ On) → (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ On)
87	58, 73, 86	mp2an 691	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ On
88	85, 87	onun2i 6485	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ On
89		naddcl 8689	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On)
90	58, 59, 89	mp2an 691	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On
91		onunel 6468	. . . . . . . . . . . . . . . 16 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑊))) ∈ On ∧ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑊))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑊))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
92	83, 88, 90, 91	mp3an 1458	. . . . . . . . . . . . . . 15 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑊))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑊))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
93		onunel 6468	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝑊)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑊))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
94	80, 82, 90, 93	mp3an 1458	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑊))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
95		onunel 6468	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑃) +no ( bday ‘𝑊)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → (((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
96	85, 87, 90, 95	mp3an 1458	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
97	94, 96	anbi12i 626	. . . . . . . . . . . . . . 15 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑊))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ↔ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ∧ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
98	92, 97	bitri 275	. . . . . . . . . . . . . 14 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑊))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ∧ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
99	69, 77, 98	sylanbrc 582	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑊))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
100		elun1 4172	. . . . . . . . . . . . 13 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑊))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) → (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑊))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( 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))))))
101	99, 100	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑊))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( 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))))))
102	53, 101	eqeltrid 2832	. . . . . . . . . . 11 ⊢ (𝜑 → ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑊))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( 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))))))
103	8, 42, 42, 44, 14, 3, 5, 102	mulsproplem1 27990	. . . . . . . . . 10 ⊢ (𝜑 → (( 0s ·s 0s ) ∈ No ∧ ((𝑃 <s 𝐴 ∧ 𝑄 <s 𝑊) → ((𝑃 ·s 𝑊) -s (𝑃 ·s 𝑄)) <s ((𝐴 ·s 𝑊) -s (𝐴 ·s 𝑄)))))
104	103	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 <s 𝐴 ∧ 𝑄 <s 𝑊) → ((𝑃 ·s 𝑊) -s (𝑃 ·s 𝑄)) <s ((𝐴 ·s 𝑊) -s (𝐴 ·s 𝑄))))
105	40, 104	mpand 694	. . . . . . . 8 ⊢ (𝜑 → (𝑄 <s 𝑊 → ((𝑃 ·s 𝑊) -s (𝑃 ·s 𝑄)) <s ((𝐴 ·s 𝑊) -s (𝐴 ·s 𝑄))))
106	105	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑄 <s 𝑊) → ((𝑃 ·s 𝑊) -s (𝑃 ·s 𝑄)) <s ((𝐴 ·s 𝑊) -s (𝐴 ·s 𝑄)))
107	25, 23, 19, 17	sltsubsub3bd 27967	. . . . . . . . 9 ⊢ (𝜑 → (((𝑃 ·s 𝑊) -s (𝑃 ·s 𝑄)) <s ((𝐴 ·s 𝑊) -s (𝐴 ·s 𝑄)) ↔ ((𝐴 ·s 𝑄) -s (𝑃 ·s 𝑄)) <s ((𝐴 ·s 𝑊) -s (𝑃 ·s 𝑊))))
108	17, 19	subscld 27947	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ·s 𝑄) -s (𝑃 ·s 𝑄)) ∈ No )
109	23, 25	subscld 27947	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ·s 𝑊) -s (𝑃 ·s 𝑊)) ∈ No )
110	108, 109, 13	sltadd2d 27888	. . . . . . . . 9 ⊢ (𝜑 → (((𝐴 ·s 𝑄) -s (𝑃 ·s 𝑄)) <s ((𝐴 ·s 𝑊) -s (𝑃 ·s 𝑊)) ↔ ((𝑃 ·s 𝐵) +s ((𝐴 ·s 𝑄) -s (𝑃 ·s 𝑄))) <s ((𝑃 ·s 𝐵) +s ((𝐴 ·s 𝑊) -s (𝑃 ·s 𝑊)))))
111	107, 110	bitrd 279	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 ·s 𝑊) -s (𝑃 ·s 𝑄)) <s ((𝐴 ·s 𝑊) -s (𝐴 ·s 𝑄)) ↔ ((𝑃 ·s 𝐵) +s ((𝐴 ·s 𝑄) -s (𝑃 ·s 𝑄))) <s ((𝑃 ·s 𝐵) +s ((𝐴 ·s 𝑊) -s (𝑃 ·s 𝑊)))))
112	111	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑄 <s 𝑊) → (((𝑃 ·s 𝑊) -s (𝑃 ·s 𝑄)) <s ((𝐴 ·s 𝑊) -s (𝐴 ·s 𝑄)) ↔ ((𝑃 ·s 𝐵) +s ((𝐴 ·s 𝑄) -s (𝑃 ·s 𝑄))) <s ((𝑃 ·s 𝐵) +s ((𝐴 ·s 𝑊) -s (𝑃 ·s 𝑊)))))
113	106, 112	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ 𝑄 <s 𝑊) → ((𝑃 ·s 𝐵) +s ((𝐴 ·s 𝑄) -s (𝑃 ·s 𝑄))) <s ((𝑃 ·s 𝐵) +s ((𝐴 ·s 𝑊) -s (𝑃 ·s 𝑊))))
114	13, 17, 19	addsubsassd 27963	. . . . . . 7 ⊢ (𝜑 → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) = ((𝑃 ·s 𝐵) +s ((𝐴 ·s 𝑄) -s (𝑃 ·s 𝑄))))
115	114	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑄 <s 𝑊) → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) = ((𝑃 ·s 𝐵) +s ((𝐴 ·s 𝑄) -s (𝑃 ·s 𝑄))))
116	13, 23, 25	addsubsassd 27963	. . . . . . 7 ⊢ (𝜑 → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑃 ·s 𝑊)) = ((𝑃 ·s 𝐵) +s ((𝐴 ·s 𝑊) -s (𝑃 ·s 𝑊))))
117	116	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑄 <s 𝑊) → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑃 ·s 𝑊)) = ((𝑃 ·s 𝐵) +s ((𝐴 ·s 𝑊) -s (𝑃 ·s 𝑊))))
118	113, 115, 117	3brtr4d 5174	. . . . 5 ⊢ ((𝜑 ∧ 𝑄 <s 𝑊) → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) <s (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑃 ·s 𝑊)))
119		lltropt 27773	. . . . . . . . . . 11 ⊢ ( L ‘𝐴) <<s ( R ‘𝐴)
120	119	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ( L ‘𝐴) <<s ( R ‘𝐴))
121	120, 10, 29	ssltsepcd 27701	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 <s 𝑉)
122		ssltleft 27771	. . . . . . . . . . 11 ⊢ (𝐵 ∈ No → ( L ‘𝐵) <<s {𝐵})
123	12, 122	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ( L ‘𝐵) <<s {𝐵})
124		snidg 4658	. . . . . . . . . . 11 ⊢ (𝐵 ∈ No → 𝐵 ∈ {𝐵})
125	12, 124	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ {𝐵})
126	123, 4, 125	ssltsepcd 27701	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 <s 𝐵)
127		rightssno 27782	. . . . . . . . . . . 12 ⊢ ( R ‘𝐴) ⊆ No
128	127, 29	sselid 3976	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑉 ∈ No )
129	50	uneq1i 4155	. . . . . . . . . . . . 13 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))))) = (∅ ∪ (((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊)))))
130		0un 4388	. . . . . . . . . . . . 13 ⊢ (∅ ∪ (((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))))) = (((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))))
131	129, 130	eqtri 2755	. . . . . . . . . . . 12 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))))) = (((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))))
132		oldbdayim 27789	. . . . . . . . . . . . . . . . 17 ⊢ (𝑉 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑉) ∈ ( bday ‘𝐴))
133	30, 132	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ( bday ‘𝑉) ∈ ( bday ‘𝐴))
134		bdayelon 27683	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑉) ∈ On
135		naddel1 8699	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑉) ∈ On ∧ ( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → (( bday ‘𝑉) ∈ ( bday ‘𝐴) ↔ (( bday ‘𝑉) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
136	134, 58, 59, 135	mp3an 1458	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑉) ∈ ( bday ‘𝐴) ↔ (( bday ‘𝑉) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
137	133, 136	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝑉) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
138	72, 137	jca 511	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
139		naddel1 8699	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑃) ∈ On ∧ ( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → (( bday ‘𝑃) ∈ ( bday ‘𝐴) ↔ (( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
140	78, 58, 59, 139	mp3an 1458	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑃) ∈ ( bday ‘𝐴) ↔ (( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
141	55, 140	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
142		naddel12 8712	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → ((( bday ‘𝑉) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑊) ∈ ( bday ‘𝐵)) → (( bday ‘𝑉) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
143	58, 59, 142	mp2an 691	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑉) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑊) ∈ ( bday ‘𝐵)) → (( bday ‘𝑉) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
144	133, 64, 143	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝑉) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
145	141, 144	jca 511	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
146		naddcl 8689	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑉) ∈ On ∧ ( bday ‘𝐵) ∈ On) → (( bday ‘𝑉) +no ( bday ‘𝐵)) ∈ On)
147	134, 59, 146	mp2an 691	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑉) +no ( bday ‘𝐵)) ∈ On
148	85, 147	onun2i 6485	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∈ On
149		naddcl 8689	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑃) ∈ On ∧ ( bday ‘𝐵) ∈ On) → (( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ On)
150	78, 59, 149	mp2an 691	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ On
151		naddcl 8689	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑉) ∈ On ∧ ( bday ‘𝑊) ∈ On) → (( bday ‘𝑉) +no ( bday ‘𝑊)) ∈ On)
152	134, 65, 151	mp2an 691	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑉) +no ( bday ‘𝑊)) ∈ On
153	150, 152	onun2i 6485	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))) ∈ On
154		onunel 6468	. . . . . . . . . . . . . . . 16 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∈ On ∧ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → ((((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
155	148, 153, 90, 154	mp3an 1458	. . . . . . . . . . . . . . 15 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
156		onunel 6468	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑃) +no ( bday ‘𝑊)) ∈ On ∧ (( bday ‘𝑉) +no ( bday ‘𝐵)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → (((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
157	85, 147, 90, 156	mp3an 1458	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑃) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
158		onunel 6468	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ On ∧ (( bday ‘𝑉) +no ( bday ‘𝑊)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → (((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
159	150, 152, 90, 158	mp3an 1458	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
160	157, 159	anbi12i 626	. . . . . . . . . . . . . . 15 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ↔ (((( bday ‘𝑃) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ∧ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
161	155, 160	bitri 275	. . . . . . . . . . . . . 14 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝑃) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ∧ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
162	138, 145, 161	sylanbrc 582	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
163		elun1 4172	. . . . . . . . . . . . 13 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) → (((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊)))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
164	162, 163	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊)))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
165	131, 164	eqeltrid 2832	. . . . . . . . . . 11 ⊢ (𝜑 → ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday ‘𝑃) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
166	8, 42, 42, 44, 128, 5, 12, 165	mulsproplem1 27990	. . . . . . . . . 10 ⊢ (𝜑 → (( 0s ·s 0s ) ∈ No ∧ ((𝑃 <s 𝑉 ∧ 𝑊 <s 𝐵) → ((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑊)) <s ((𝑉 ·s 𝐵) -s (𝑉 ·s 𝑊)))))
167	166	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 <s 𝑉 ∧ 𝑊 <s 𝐵) → ((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑊)) <s ((𝑉 ·s 𝐵) -s (𝑉 ·s 𝑊))))
168	121, 126, 167	mp2and 698	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑊)) <s ((𝑉 ·s 𝐵) -s (𝑉 ·s 𝑊)))
169	13, 25	subscld 27947	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑊)) ∈ No )
170	31, 33	subscld 27947	. . . . . . . . 9 ⊢ (𝜑 → ((𝑉 ·s 𝐵) -s (𝑉 ·s 𝑊)) ∈ No )
171	169, 170, 23	sltadd1d 27889	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑊)) <s ((𝑉 ·s 𝐵) -s (𝑉 ·s 𝑊)) ↔ (((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑊)) +s (𝐴 ·s 𝑊)) <s (((𝑉 ·s 𝐵) -s (𝑉 ·s 𝑊)) +s (𝐴 ·s 𝑊))))
172	168, 171	mpbid 231	. . . . . . 7 ⊢ (𝜑 → (((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑊)) +s (𝐴 ·s 𝑊)) <s (((𝑉 ·s 𝐵) -s (𝑉 ·s 𝑊)) +s (𝐴 ·s 𝑊)))
173	13, 23, 25	addsubsd 27964	. . . . . . 7 ⊢ (𝜑 → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑃 ·s 𝑊)) = (((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑊)) +s (𝐴 ·s 𝑊)))
174	31, 23, 33	addsubsd 27964	. . . . . . 7 ⊢ (𝜑 → (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑉 ·s 𝑊)) = (((𝑉 ·s 𝐵) -s (𝑉 ·s 𝑊)) +s (𝐴 ·s 𝑊)))
175	172, 173, 174	3brtr4d 5174	. . . . . 6 ⊢ (𝜑 → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑃 ·s 𝑊)) <s (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑉 ·s 𝑊)))
176	175	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑄 <s 𝑊) → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑃 ·s 𝑊)) <s (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑉 ·s 𝑊)))
177	21, 27, 35, 118, 176	slttrd 27666	. . . 4 ⊢ ((𝜑 ∧ 𝑄 <s 𝑊) → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) <s (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑉 ·s 𝑊)))
178	177	ex 412	. . 3 ⊢ (𝜑 → (𝑄 <s 𝑊 → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) <s (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑉 ·s 𝑊))))
179		oveq2 7422	. . . . . . 7 ⊢ (𝑄 = 𝑊 → (𝐴 ·s 𝑄) = (𝐴 ·s 𝑊))
180	179	oveq2d 7430	. . . . . 6 ⊢ (𝑄 = 𝑊 → ((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) = ((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑊)))
181		oveq2 7422	. . . . . 6 ⊢ (𝑄 = 𝑊 → (𝑃 ·s 𝑄) = (𝑃 ·s 𝑊))
182	180, 181	oveq12d 7432	. . . . 5 ⊢ (𝑄 = 𝑊 → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) = (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑃 ·s 𝑊)))
183	182	breq1d 5152	. . . 4 ⊢ (𝑄 = 𝑊 → ((((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) <s (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑉 ·s 𝑊)) ↔ (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑃 ·s 𝑊)) <s (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑉 ·s 𝑊))))
184	175, 183	syl5ibrcom 246	. . 3 ⊢ (𝜑 → (𝑄 = 𝑊 → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) <s (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑉 ·s 𝑊))))
185	20	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑊 <s 𝑄) → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) ∈ No )
186	31, 17	addscld 27871	. . . . . . 7 ⊢ (𝜑 → ((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑄)) ∈ No )
187	8, 30, 16	mulsproplem4 27993	. . . . . . 7 ⊢ (𝜑 → (𝑉 ·s 𝑄) ∈ No )
188	186, 187	subscld 27947	. . . . . 6 ⊢ (𝜑 → (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑉 ·s 𝑄)) ∈ No )
189	188	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑊 <s 𝑄) → (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑉 ·s 𝑄)) ∈ No )
190	34	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑊 <s 𝑄) → (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑉 ·s 𝑊)) ∈ No )
191	123, 2, 125	ssltsepcd 27701	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 <s 𝐵)
192	50	uneq1i 4155	. . . . . . . . . . . . 13 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))))) = (∅ ∪ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄)))))
193		0un 4388	. . . . . . . . . . . . 13 ⊢ (∅ ∪ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))))) = (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))))
194	192, 193	eqtri 2755	. . . . . . . . . . . 12 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))))) = (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))))
195	62, 137	jca 511	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
196		naddel12 8712	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → ((( bday ‘𝑉) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑄) ∈ ( bday ‘𝐵)) → (( bday ‘𝑉) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
197	58, 59, 196	mp2an 691	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑉) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑄) ∈ ( bday ‘𝐵)) → (( bday ‘𝑉) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
198	133, 57, 197	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝑉) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
199	141, 198	jca 511	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
200	80, 147	onun2i 6485	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∈ On
201		naddcl 8689	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑉) ∈ On ∧ ( bday ‘𝑄) ∈ On) → (( bday ‘𝑉) +no ( bday ‘𝑄)) ∈ On)
202	134, 73, 201	mp2an 691	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑉) +no ( bday ‘𝑄)) ∈ On
203	150, 202	onun2i 6485	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∈ On
204		onunel 6468	. . . . . . . . . . . . . . . 16 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∈ On ∧ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
205	200, 203, 90, 204	mp3an 1458	. . . . . . . . . . . . . . 15 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
206		onunel 6468	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ On ∧ (( bday ‘𝑉) +no ( bday ‘𝐵)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
207	80, 147, 90, 206	mp3an 1458	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
208		onunel 6468	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ On ∧ (( bday ‘𝑉) +no ( bday ‘𝑄)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → (((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
209	150, 202, 90, 208	mp3an 1458	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
210	207, 209	anbi12i 626	. . . . . . . . . . . . . . 15 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ↔ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ∧ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
211	205, 210	bitri 275	. . . . . . . . . . . . . 14 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ∧ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
212	195, 199, 211	sylanbrc 582	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
213		elun1 4172	. . . . . . . . . . . . 13 ⊢ ((((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) → (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +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))))))
214	212, 213	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +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))))))
215	194, 214	eqeltrid 2832	. . . . . . . . . . 11 ⊢ (𝜑 → ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝐵))) ∪ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑉) +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))))))
216	8, 42, 42, 44, 128, 3, 12, 215	mulsproplem1 27990	. . . . . . . . . 10 ⊢ (𝜑 → (( 0s ·s 0s ) ∈ No ∧ ((𝑃 <s 𝑉 ∧ 𝑄 <s 𝐵) → ((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑄)) <s ((𝑉 ·s 𝐵) -s (𝑉 ·s 𝑄)))))
217	216	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 <s 𝑉 ∧ 𝑄 <s 𝐵) → ((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑄)) <s ((𝑉 ·s 𝐵) -s (𝑉 ·s 𝑄))))
218	121, 191, 217	mp2and 698	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑄)) <s ((𝑉 ·s 𝐵) -s (𝑉 ·s 𝑄)))
219	13, 19	subscld 27947	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑄)) ∈ No )
220	31, 187	subscld 27947	. . . . . . . . 9 ⊢ (𝜑 → ((𝑉 ·s 𝐵) -s (𝑉 ·s 𝑄)) ∈ No )
221	219, 220, 17	sltadd1d 27889	. . . . . . . 8 ⊢ (𝜑 → (((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑄)) <s ((𝑉 ·s 𝐵) -s (𝑉 ·s 𝑄)) ↔ (((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑄)) +s (𝐴 ·s 𝑄)) <s (((𝑉 ·s 𝐵) -s (𝑉 ·s 𝑄)) +s (𝐴 ·s 𝑄))))
222	218, 221	mpbid 231	. . . . . . 7 ⊢ (𝜑 → (((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑄)) +s (𝐴 ·s 𝑄)) <s (((𝑉 ·s 𝐵) -s (𝑉 ·s 𝑄)) +s (𝐴 ·s 𝑄)))
223	13, 17, 19	addsubsd 27964	. . . . . . 7 ⊢ (𝜑 → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) = (((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑄)) +s (𝐴 ·s 𝑄)))
224	31, 17, 187	addsubsd 27964	. . . . . . 7 ⊢ (𝜑 → (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑉 ·s 𝑄)) = (((𝑉 ·s 𝐵) -s (𝑉 ·s 𝑄)) +s (𝐴 ·s 𝑄)))
225	222, 223, 224	3brtr4d 5174	. . . . . 6 ⊢ (𝜑 → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) <s (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑉 ·s 𝑄)))
226	225	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑊 <s 𝑄) → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) <s (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑉 ·s 𝑄)))
227		ssltright 27772	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → {𝐴} <<s ( R ‘𝐴))
228	14, 227	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → {𝐴} <<s ( R ‘𝐴))
229	228, 39, 29	ssltsepcd 27701	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 <s 𝑉)
230	50	uneq1i 4155	. . . . . . . . . . . . 13 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday ‘𝐴) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∪ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))))) = (∅ ∪ (((( bday ‘𝐴) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∪ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊)))))
231		0un 4388	. . . . . . . . . . . . 13 ⊢ (∅ ∪ (((( bday ‘𝐴) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∪ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))))) = (((( bday ‘𝐴) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∪ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))))
232	230, 231	eqtri 2755	. . . . . . . . . . . 12 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday ‘𝐴) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∪ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))))) = (((( bday ‘𝐴) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∪ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))))
233	68, 198	jca 511	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝐴) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
234	76, 144	jca 511	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
235	82, 202	onun2i 6485	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝐴) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∈ On
236	87, 152	onun2i 6485	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))) ∈ On
237		onunel 6468	. . . . . . . . . . . . . . . 16 ⊢ ((((( bday ‘𝐴) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∈ On ∧ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → ((((( bday ‘𝐴) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∪ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝐴) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
238	235, 236, 90, 237	mp3an 1458	. . . . . . . . . . . . . . 15 ⊢ ((((( bday ‘𝐴) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∪ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝐴) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
239		onunel 6468	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝐴) +no ( bday ‘𝑊)) ∈ On ∧ (( bday ‘𝑉) +no ( bday ‘𝑄)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → (((( bday ‘𝐴) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝐴) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
240	82, 202, 90, 239	mp3an 1458	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝐴) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝐴) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
241		onunel 6468	. . . . . . . . . . . . . . . . 17 ⊢ (((( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ On ∧ (( bday ‘𝑉) +no ( bday ‘𝑊)) ∈ On ∧ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∈ On) → (((( bday ‘𝐴) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
242	87, 152, 90, 241	mp3an 1458	. . . . . . . . . . . . . . . 16 ⊢ (((( bday ‘𝐴) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
243	240, 242	anbi12i 626	. . . . . . . . . . . . . . 15 ⊢ ((((( bday ‘𝐴) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ↔ (((( bday ‘𝐴) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ∧ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
244	238, 243	bitri 275	. . . . . . . . . . . . . 14 ⊢ ((((( bday ‘𝐴) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∪ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ↔ (((( bday ‘𝐴) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))) ∧ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑉) +no ( bday ‘𝑊)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))))
245	233, 234, 244	sylanbrc 582	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((( bday ‘𝐴) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∪ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
246		elun1 4172	. . . . . . . . . . . . 13 ⊢ ((((( bday ‘𝐴) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∪ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊)))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) → (((( bday ‘𝐴) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∪ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊)))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
247	245, 246	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (((( bday ‘𝐴) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∪ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊)))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
248	232, 247	eqeltrid 2832	. . . . . . . . . . 11 ⊢ (𝜑 → ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( bday ‘𝐴) +no ( bday ‘𝑊)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑄))) ∪ ((( bday ‘𝐴) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑉) +no ( bday ‘𝑊))))) ∈ ((( bday ‘𝐴) +no ( bday ‘𝐵)) ∪ (((( bday ‘𝐶) +no ( bday ‘𝐸)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐹))) ∪ ((( bday ‘𝐶) +no ( bday ‘𝐹)) ∪ (( bday ‘𝐷) +no ( bday ‘𝐸))))))
249	8, 42, 42, 14, 128, 5, 3, 248	mulsproplem1 27990	. . . . . . . . . 10 ⊢ (𝜑 → (( 0s ·s 0s ) ∈ No ∧ ((𝐴 <s 𝑉 ∧ 𝑊 <s 𝑄) → ((𝐴 ·s 𝑄) -s (𝐴 ·s 𝑊)) <s ((𝑉 ·s 𝑄) -s (𝑉 ·s 𝑊)))))
250	249	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → ((𝐴 <s 𝑉 ∧ 𝑊 <s 𝑄) → ((𝐴 ·s 𝑄) -s (𝐴 ·s 𝑊)) <s ((𝑉 ·s 𝑄) -s (𝑉 ·s 𝑊))))
251	229, 250	mpand 694	. . . . . . . 8 ⊢ (𝜑 → (𝑊 <s 𝑄 → ((𝐴 ·s 𝑄) -s (𝐴 ·s 𝑊)) <s ((𝑉 ·s 𝑄) -s (𝑉 ·s 𝑊))))
252	251	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑊 <s 𝑄) → ((𝐴 ·s 𝑄) -s (𝐴 ·s 𝑊)) <s ((𝑉 ·s 𝑄) -s (𝑉 ·s 𝑊)))
253	17, 187, 23, 33	sltsubsubbd 27965	. . . . . . . . 9 ⊢ (𝜑 → (((𝐴 ·s 𝑄) -s (𝐴 ·s 𝑊)) <s ((𝑉 ·s 𝑄) -s (𝑉 ·s 𝑊)) ↔ ((𝐴 ·s 𝑄) -s (𝑉 ·s 𝑄)) <s ((𝐴 ·s 𝑊) -s (𝑉 ·s 𝑊))))
254	17, 187	subscld 27947	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ·s 𝑄) -s (𝑉 ·s 𝑄)) ∈ No )
255	23, 33	subscld 27947	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ·s 𝑊) -s (𝑉 ·s 𝑊)) ∈ No )
256	254, 255, 31	sltadd2d 27888	. . . . . . . . 9 ⊢ (𝜑 → (((𝐴 ·s 𝑄) -s (𝑉 ·s 𝑄)) <s ((𝐴 ·s 𝑊) -s (𝑉 ·s 𝑊)) ↔ ((𝑉 ·s 𝐵) +s ((𝐴 ·s 𝑄) -s (𝑉 ·s 𝑄))) <s ((𝑉 ·s 𝐵) +s ((𝐴 ·s 𝑊) -s (𝑉 ·s 𝑊)))))
257	253, 256	bitrd 279	. . . . . . . 8 ⊢ (𝜑 → (((𝐴 ·s 𝑄) -s (𝐴 ·s 𝑊)) <s ((𝑉 ·s 𝑄) -s (𝑉 ·s 𝑊)) ↔ ((𝑉 ·s 𝐵) +s ((𝐴 ·s 𝑄) -s (𝑉 ·s 𝑄))) <s ((𝑉 ·s 𝐵) +s ((𝐴 ·s 𝑊) -s (𝑉 ·s 𝑊)))))
258	257	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑊 <s 𝑄) → (((𝐴 ·s 𝑄) -s (𝐴 ·s 𝑊)) <s ((𝑉 ·s 𝑄) -s (𝑉 ·s 𝑊)) ↔ ((𝑉 ·s 𝐵) +s ((𝐴 ·s 𝑄) -s (𝑉 ·s 𝑄))) <s ((𝑉 ·s 𝐵) +s ((𝐴 ·s 𝑊) -s (𝑉 ·s 𝑊)))))
259	252, 258	mpbid 231	. . . . . 6 ⊢ ((𝜑 ∧ 𝑊 <s 𝑄) → ((𝑉 ·s 𝐵) +s ((𝐴 ·s 𝑄) -s (𝑉 ·s 𝑄))) <s ((𝑉 ·s 𝐵) +s ((𝐴 ·s 𝑊) -s (𝑉 ·s 𝑊))))
260	31, 17, 187	addsubsassd 27963	. . . . . . 7 ⊢ (𝜑 → (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑉 ·s 𝑄)) = ((𝑉 ·s 𝐵) +s ((𝐴 ·s 𝑄) -s (𝑉 ·s 𝑄))))
261	260	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑊 <s 𝑄) → (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑉 ·s 𝑄)) = ((𝑉 ·s 𝐵) +s ((𝐴 ·s 𝑄) -s (𝑉 ·s 𝑄))))
262	31, 23, 33	addsubsassd 27963	. . . . . . 7 ⊢ (𝜑 → (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑉 ·s 𝑊)) = ((𝑉 ·s 𝐵) +s ((𝐴 ·s 𝑊) -s (𝑉 ·s 𝑊))))
263	262	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑊 <s 𝑄) → (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑉 ·s 𝑊)) = ((𝑉 ·s 𝐵) +s ((𝐴 ·s 𝑊) -s (𝑉 ·s 𝑊))))
264	259, 261, 263	3brtr4d 5174	. . . . 5 ⊢ ((𝜑 ∧ 𝑊 <s 𝑄) → (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑉 ·s 𝑄)) <s (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑉 ·s 𝑊)))
265	185, 189, 190, 226, 264	slttrd 27666	. . . 4 ⊢ ((𝜑 ∧ 𝑊 <s 𝑄) → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) <s (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑉 ·s 𝑊)))
266	265	ex 412	. . 3 ⊢ (𝜑 → (𝑊 <s 𝑄 → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) <s (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑉 ·s 𝑊))))
267	178, 184, 266	3jaod 1426	. 2 ⊢ (𝜑 → ((𝑄 <s 𝑊 ∨ 𝑄 = 𝑊 ∨ 𝑊 <s 𝑄) → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) <s (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑉 ·s 𝑊))))
268	7, 267	mpd 15	1 ⊢ (𝜑 → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) <s (((𝑉 ·s 𝐵) +s (𝐴 ·s 𝑊)) -s (𝑉 ·s 𝑊)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ w3o 1084   = wceq 1534   ∈ wcel 2099  ∀wral 3056   ∪ cun 3942  ∅c0 4318  {csn 4624   class class class wbr 5142  Oncon0 6363  ‘cfv 6542  (class class class)co 7414   +no cnadd 8677   No csur 27547   <s cslt 27548   bday cbday 27549   <<s csslt 27687   0_s c0s 27729   O cold 27744   L cleft 27746   R cright 27747   +_s cadds 27850   -_s csubs 27907   ·_s cmuls 27980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-ot 4633  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-1o 8478  df-2o 8479  df-nadd 8678  df-no 27550  df-slt 27551  df-bday 27552  df-sle 27652  df-sslt 27688  df-scut 27690  df-0s 27731  df-made 27748  df-old 27749  df-left 27751  df-right 27752  df-norec 27829  df-norec2 27840  df-adds 27851  df-negs 27908  df-subs 27909

This theorem is referenced by:  mulsproplem9  27998