Theorem mulsproplem5 27994

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		leftssno 27781	. . . 4 ⊢ ( L ‘𝐴) ⊆ No
2		mulsproplem5.3	. . . 4 ⊢ (𝜑 → 𝑃 ∈ ( L ‘𝐴))
3	1, 2	sselid 3976	. . 3 ⊢ (𝜑 → 𝑃 ∈ No )
4		mulsproplem5.5	. . . 4 ⊢ (𝜑 → 𝑇 ∈ ( L ‘𝐴))
5	1, 4	sselid 3976	. . 3 ⊢ (𝜑 → 𝑇 ∈ No )
6		sltlin 27656	. . 3 ⊢ ((𝑃 ∈ No ∧ 𝑇 ∈ No ) → (𝑃 <s 𝑇 ∨ 𝑃 = 𝑇 ∨ 𝑇 <s 𝑃))
7	3, 5, 6	syl2anc 583	. 2 ⊢ (𝜑 → (𝑃 <s 𝑇 ∨ 𝑃 = 𝑇 ∨ 𝑇 <s 𝑃))
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	9, 2	sselid 3976	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ( O ‘( bday ‘𝐴)))
11		mulsproplem5.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ No )
12	8, 10, 11	mulsproplem2 27991	. . . . . . . 8 ⊢ (𝜑 → (𝑃 ·s 𝐵) ∈ No )
13		mulsproplem5.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ No )
14		leftssold 27779	. . . . . . . . . 10 ⊢ ( L ‘𝐵) ⊆ ( O ‘( bday ‘𝐵))
15		mulsproplem5.4	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ ( L ‘𝐵))
16	14, 15	sselid 3976	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 ∈ ( O ‘( bday ‘𝐵)))
17	8, 13, 16	mulsproplem3 27992	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ·s 𝑄) ∈ No )
18	12, 17	addscld 27871	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) ∈ No )
19	8, 10, 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 ∧ 𝑃 <s 𝑇) → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) ∈ No )
22	9, 4	sselid 3976	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ ( O ‘( bday ‘𝐴)))
23	8, 22, 11	mulsproplem2 27991	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ·s 𝐵) ∈ No )
24	23, 17	addscld 27871	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑄)) ∈ No )
25	8, 22, 16	mulsproplem4 27993	. . . . . . 7 ⊢ (𝜑 → (𝑇 ·s 𝑄) ∈ No )
26	24, 25	subscld 27947	. . . . . 6 ⊢ (𝜑 → (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑇 ·s 𝑄)) ∈ No )
27	26	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 <s 𝑇) → (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑇 ·s 𝑄)) ∈ No )
28		rightssold 27780	. . . . . . . . . 10 ⊢ ( R ‘𝐵) ⊆ ( O ‘( bday ‘𝐵))
29		mulsproplem5.6	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ∈ ( R ‘𝐵))
30	28, 29	sselid 3976	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ ( O ‘( bday ‘𝐵)))
31	8, 13, 30	mulsproplem3 27992	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ·s 𝑈) ∈ No )
32	23, 31	addscld 27871	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) ∈ No )
33	8, 22, 30	mulsproplem4 27993	. . . . . . 7 ⊢ (𝜑 → (𝑇 ·s 𝑈) ∈ No )
34	32, 33	subscld 27947	. . . . . 6 ⊢ (𝜑 → (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈)) ∈ No )
35	34	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑃 <s 𝑇) → (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈)) ∈ No )
36		ssltleft 27771	. . . . . . . . . . 11 ⊢ (𝐵 ∈ No → ( L ‘𝐵) <<s {𝐵})
37	11, 36	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ( L ‘𝐵) <<s {𝐵})
38		snidg 4658	. . . . . . . . . . 11 ⊢ (𝐵 ∈ No → 𝐵 ∈ {𝐵})
39	11, 38	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ {𝐵})
40	37, 15, 39	ssltsepcd 27701	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 <s 𝐵)
41		0sno 27733	. . . . . . . . . . . 12 ⊢ 0s ∈ No
42	41	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → 0s ∈ No )
43		leftssno 27781	. . . . . . . . . . . 12 ⊢ ( L ‘𝐵) ⊆ No
44	43, 15	sselid 3976	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ 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 ‘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)))))
52		0un 4388	. . . . . . . . . . . . 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))))
53	51, 52	eqtri 2755	. . . . . . . . . . . 12 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( 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))))
54		oldbdayim 27789	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑃) ∈ ( bday ‘𝐴))
55	10, 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 ⊢ (𝑇 ∈ ( O ‘( bday ‘𝐴)) → ( bday ‘𝑇) ∈ ( bday ‘𝐴))
64	22, 63	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ( bday ‘𝑇) ∈ ( bday ‘𝐴))
65		bdayelon 27683	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑇) ∈ On
66		naddel1 8699	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑇) ∈ On ∧ ( bday ‘𝐴) ∈ On ∧ ( bday ‘𝐵) ∈ On) → (( bday ‘𝑇) ∈ ( bday ‘𝐴) ↔ (( bday ‘𝑇) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
67	65, 58, 59, 66	mp3an 1458	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑇) ∈ ( bday ‘𝐴) ↔ (( bday ‘𝑇) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
68	64, 67	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝑇) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
69	62, 68	jca 511	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑇) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
70		bdayelon 27683	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑃) ∈ On
71		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))))
72	70, 58, 59, 71	mp3an 1458	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑃) ∈ ( bday ‘𝐴) ↔ (( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
73	55, 72	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
74		naddel12 8712	. . . . . . . . . . . . . . . . 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))))
75	58, 59, 74	mp2an 691	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑇) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑄) ∈ ( bday ‘𝐵)) → (( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
76	64, 57, 75	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
77	73, 76	jca 511	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
78		bdayelon 27683	. . . . . . . . . . . . . . . . . 18 ⊢ ( bday ‘𝑄) ∈ On
79		naddcl 8689	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑃) ∈ On ∧ ( bday ‘𝑄) ∈ On) → (( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ On)
80	70, 78, 79	mp2an 691	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ On
81		naddcl 8689	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑇) ∈ On ∧ ( bday ‘𝐵) ∈ On) → (( bday ‘𝑇) +no ( bday ‘𝐵)) ∈ On)
82	65, 59, 81	mp2an 691	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑇) +no ( bday ‘𝐵)) ∈ On
83	80, 82	onun2i 6485	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝑇) +no ( bday ‘𝐵))) ∈ On
84		naddcl 8689	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑃) ∈ On ∧ ( bday ‘𝐵) ∈ On) → (( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ On)
85	70, 59, 84	mp2an 691	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ On
86		naddcl 8689	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑇) ∈ On ∧ ( bday ‘𝑄) ∈ On) → (( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ On)
87	65, 78, 86	mp2an 691	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ On
88	85, 87	onun2i 6485	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑃) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑇) +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 ‘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)))))
92	83, 88, 90, 91	mp3an 1458	. . . . . . . . . . . . . . 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))))
93		onunel 6468	. . . . . . . . . . . . . . . . 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)))))
94	80, 82, 90, 93	mp3an 1458	. . . . . . . . . . . . . . . 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))))
95		onunel 6468	. . . . . . . . . . . . . . . . 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)))))
96	85, 87, 90, 95	mp3an 1458	. . . . . . . . . . . . . . . 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))))
97	94, 96	anbi12i 626	. . . . . . . . . . . . . . 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)))))
98	92, 97	bitri 275	. . . . . . . . . . . . . 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)))))
99	69, 77, 98	sylanbrc 582	. . . . . . . . . . . . 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)))
100		elun1 4172	. . . . . . . . . . . . 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))))))
101	99, 100	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))))))
102	53, 101	eqeltrid 2832	. . . . . . . . . . 11 ⊢ (𝜑 → ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( 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))))))
103	8, 42, 42, 3, 5, 44, 11, 102	mulsproplem1 27990	. . . . . . . . . 10 ⊢ (𝜑 → (( 0s ·s 0s ) ∈ No ∧ ((𝑃 <s 𝑇 ∧ 𝑄 <s 𝐵) → ((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑄)) <s ((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑄)))))
104	103	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 <s 𝑇 ∧ 𝑄 <s 𝐵) → ((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑄)) <s ((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑄))))
105	40, 104	mpan2d 693	. . . . . . . 8 ⊢ (𝜑 → (𝑃 <s 𝑇 → ((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑄)) <s ((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑄))))
106	105	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑃 <s 𝑇) → ((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑄)) <s ((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑄)))
107	12, 19	subscld 27947	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑄)) ∈ No )
108	23, 25	subscld 27947	. . . . . . . . 9 ⊢ (𝜑 → ((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑄)) ∈ No )
109	107, 108, 17	sltadd1d 27889	. . . . . . . 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))))
110	109	adantr 480	. . . . . . 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))))
111	106, 110	mpbid 231	. . . . . 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)))
112	12, 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)))
113	112	adantr 480	. . . . . 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)))
114	23, 17, 25	addsubsd 27964	. . . . . . 7 ⊢ (𝜑 → (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑇 ·s 𝑄)) = (((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑄)) +s (𝐴 ·s 𝑄)))
115	114	adantr 480	. . . . . 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)))
116	111, 113, 115	3brtr4d 5174	. . . . 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)))
117		ssltleft 27771	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → ( L ‘𝐴) <<s {𝐴})
118	13, 117	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ( L ‘𝐴) <<s {𝐴})
119		snidg 4658	. . . . . . . . . . 11 ⊢ (𝐴 ∈ No → 𝐴 ∈ {𝐴})
120	13, 119	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
121	118, 4, 120	ssltsepcd 27701	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 <s 𝐴)
122		lltropt 27773	. . . . . . . . . . 11 ⊢ ( L ‘𝐵) <<s ( R ‘𝐵)
123	122	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ( L ‘𝐵) <<s ( R ‘𝐵))
124	123, 15, 29	ssltsepcd 27701	. . . . . . . . 9 ⊢ (𝜑 → 𝑄 <s 𝑈)
125		rightssno 27782	. . . . . . . . . . . 12 ⊢ ( R ‘𝐵) ⊆ No
126	125, 29	sselid 3976	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ∈ No )
127	50	uneq1i 4155	. . . . . . . . . . . . 13 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( 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)))))
128		0un 4388	. . . . . . . . . . . . 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))))
129	127, 128	eqtri 2755	. . . . . . . . . . . 12 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( 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))))
130		oldbdayim 27789	. . . . . . . . . . . . . . . . 17 ⊢ (𝑈 ∈ ( O ‘( bday ‘𝐵)) → ( bday ‘𝑈) ∈ ( bday ‘𝐵))
131	30, 130	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ( bday ‘𝑈) ∈ ( bday ‘𝐵))
132		bdayelon 27683	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘𝑈) ∈ On
133		naddel2 8700	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘𝑈) ∈ On ∧ ( bday ‘𝐵) ∈ On ∧ ( bday ‘𝐴) ∈ On) → (( bday ‘𝑈) ∈ ( bday ‘𝐵) ↔ (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
134	132, 59, 58, 133	mp3an 1458	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑈) ∈ ( bday ‘𝐵) ↔ (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
135	131, 134	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
136	76, 135	jca 511	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑇) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
137		naddel12 8712	. . . . . . . . . . . . . . . . 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))))
138	58, 59, 137	mp2an 691	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑇) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑈) ∈ ( bday ‘𝐵)) → (( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
139	64, 131, 138	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
140		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))))
141	78, 59, 58, 140	mp3an 1458	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘𝑄) ∈ ( bday ‘𝐵) ↔ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
142	57, 141	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
143	139, 142	jca 511	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
144		naddcl 8689	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝑈) ∈ On) → (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ On)
145	58, 132, 144	mp2an 691	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ On
146	87, 145	onun2i 6485	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑇) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∈ On
147		naddcl 8689	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑇) ∈ On ∧ ( bday ‘𝑈) ∈ On) → (( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ On)
148	65, 132, 147	mp2an 691	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ On
149		naddcl 8689	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝑄) ∈ On) → (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ On)
150	58, 78, 149	mp2an 691	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ On
151	148, 150	onun2i 6485	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ On
152		onunel 6468	. . . . . . . . . . . . . . . 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)))))
153	146, 151, 90, 152	mp3an 1458	. . . . . . . . . . . . . . 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))))
154		onunel 6468	. . . . . . . . . . . . . . . . 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)))))
155	87, 145, 90, 154	mp3an 1458	. . . . . . . . . . . . . . . 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))))
156		onunel 6468	. . . . . . . . . . . . . . . . 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)))))
157	148, 150, 90, 156	mp3an 1458	. . . . . . . . . . . . . . . 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))))
158	155, 157	anbi12i 626	. . . . . . . . . . . . . . 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)))))
159	153, 158	bitri 275	. . . . . . . . . . . . . 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)))))
160	136, 143, 159	sylanbrc 582	. . . . . . . . . . . . 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)))
161		elun1 4172	. . . . . . . . . . . . 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))))))
162	160, 161	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))))))
163	129, 162	eqeltrid 2832	. . . . . . . . . . 11 ⊢ (𝜑 → ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( 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))))))
164	8, 42, 42, 5, 13, 44, 126, 163	mulsproplem1 27990	. . . . . . . . . 10 ⊢ (𝜑 → (( 0s ·s 0s ) ∈ No ∧ ((𝑇 <s 𝐴 ∧ 𝑄 <s 𝑈) → ((𝑇 ·s 𝑈) -s (𝑇 ·s 𝑄)) <s ((𝐴 ·s 𝑈) -s (𝐴 ·s 𝑄)))))
165	164	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → ((𝑇 <s 𝐴 ∧ 𝑄 <s 𝑈) → ((𝑇 ·s 𝑈) -s (𝑇 ·s 𝑄)) <s ((𝐴 ·s 𝑈) -s (𝐴 ·s 𝑄))))
166	121, 124, 165	mp2and 698	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ·s 𝑈) -s (𝑇 ·s 𝑄)) <s ((𝐴 ·s 𝑈) -s (𝐴 ·s 𝑄)))
167	33, 31, 25, 17	sltsubsub3bd 27967	. . . . . . . . 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))))
168	17, 25	subscld 27947	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ·s 𝑄) -s (𝑇 ·s 𝑄)) ∈ No )
169	31, 33	subscld 27947	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ·s 𝑈) -s (𝑇 ·s 𝑈)) ∈ No )
170	168, 169, 23	sltadd2d 27888	. . . . . . . . 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)))))
171	167, 170	bitrd 279	. . . . . . . 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)))))
172	166, 171	mpbid 231	. . . . . . 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))))
173	23, 17, 25	addsubsassd 27963	. . . . . . 7 ⊢ (𝜑 → (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑇 ·s 𝑄)) = ((𝑇 ·s 𝐵) +s ((𝐴 ·s 𝑄) -s (𝑇 ·s 𝑄))))
174	23, 31, 33	addsubsassd 27963	. . . . . . 7 ⊢ (𝜑 → (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈)) = ((𝑇 ·s 𝐵) +s ((𝐴 ·s 𝑈) -s (𝑇 ·s 𝑈))))
175	172, 173, 174	3brtr4d 5174	. . . . . 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)))
176	175	adantr 480	. . . . 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)))
177	21, 27, 35, 116, 176	slttrd 27666	. . . 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)))
178	177	ex 412	. . 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))))
179		oveq1 7421	. . . . . . 7 ⊢ (𝑃 = 𝑇 → (𝑃 ·s 𝐵) = (𝑇 ·s 𝐵))
180	179	oveq1d 7429	. . . . . 6 ⊢ (𝑃 = 𝑇 → ((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) = ((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑄)))
181		oveq1 7421	. . . . . 6 ⊢ (𝑃 = 𝑇 → (𝑃 ·s 𝑄) = (𝑇 ·s 𝑄))
182	180, 181	oveq12d 7432	. . . . 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)))
183	182	breq1d 5152	. . . 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))))
184	175, 183	syl5ibrcom 246	. . 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))))
185	20	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 <s 𝑃) → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑄)) -s (𝑃 ·s 𝑄)) ∈ No )
186	12, 31	addscld 27871	. . . . . . 7 ⊢ (𝜑 → ((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑈)) ∈ No )
187	8, 10, 30	mulsproplem4 27993	. . . . . . 7 ⊢ (𝜑 → (𝑃 ·s 𝑈) ∈ No )
188	186, 187	subscld 27947	. . . . . 6 ⊢ (𝜑 → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑃 ·s 𝑈)) ∈ No )
189	188	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 <s 𝑃) → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑃 ·s 𝑈)) ∈ No )
190	34	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑇 <s 𝑃) → (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈)) ∈ No )
191	118, 2, 120	ssltsepcd 27701	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 <s 𝐴)
192	50	uneq1i 4155	. . . . . . . . . . . . 13 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( 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)))))
193		0un 4388	. . . . . . . . . . . . 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))))
194	192, 193	eqtri 2755	. . . . . . . . . . . 12 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( 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))))
195	62, 135	jca 511	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑃) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
196		naddel12 8712	. . . . . . . . . . . . . . . . 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))))
197	58, 59, 196	mp2an 691	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑃) ∈ ( bday ‘𝐴) ∧ ( bday ‘𝑈) ∈ ( bday ‘𝐵)) → (( bday ‘𝑃) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
198	55, 131, 197	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (( bday ‘𝑃) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
199	198, 142	jca 511	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝐴) +no ( bday ‘𝑄)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
200	80, 145	onun2i 6485	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑃) +no ( bday ‘𝑄)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑈))) ∈ On
201		naddcl 8689	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑃) ∈ On ∧ ( bday ‘𝑈) ∈ On) → (( bday ‘𝑃) +no ( bday ‘𝑈)) ∈ On)
202	70, 132, 201	mp2an 691	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑃) +no ( bday ‘𝑈)) ∈ On
203	202, 150	onun2i 6485	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑃) +no ( bday ‘𝑈)) ∪ (( bday ‘𝐴) +no ( bday ‘𝑄))) ∈ On
204		onunel 6468	. . . . . . . . . . . . . . . 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)))))
205	200, 203, 90, 204	mp3an 1458	. . . . . . . . . . . . . . 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))))
206		onunel 6468	. . . . . . . . . . . . . . . . 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)))))
207	80, 145, 90, 206	mp3an 1458	. . . . . . . . . . . . . . . 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))))
208		onunel 6468	. . . . . . . . . . . . . . . . 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)))))
209	202, 150, 90, 208	mp3an 1458	. . . . . . . . . . . . . . . 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))))
210	207, 209	anbi12i 626	. . . . . . . . . . . . . . 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)))))
211	205, 210	bitri 275	. . . . . . . . . . . . . 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)))))
212	195, 199, 211	sylanbrc 582	. . . . . . . . . . . . 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)))
213		elun1 4172	. . . . . . . . . . . . 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))))))
214	212, 213	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))))))
215	194, 214	eqeltrid 2832	. . . . . . . . . . 11 ⊢ (𝜑 → ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( 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))))))
216	8, 42, 42, 3, 13, 44, 126, 215	mulsproplem1 27990	. . . . . . . . . 10 ⊢ (𝜑 → (( 0s ·s 0s ) ∈ No ∧ ((𝑃 <s 𝐴 ∧ 𝑄 <s 𝑈) → ((𝑃 ·s 𝑈) -s (𝑃 ·s 𝑄)) <s ((𝐴 ·s 𝑈) -s (𝐴 ·s 𝑄)))))
217	216	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 <s 𝐴 ∧ 𝑄 <s 𝑈) → ((𝑃 ·s 𝑈) -s (𝑃 ·s 𝑄)) <s ((𝐴 ·s 𝑈) -s (𝐴 ·s 𝑄))))
218	191, 124, 217	mp2and 698	. . . . . . . 8 ⊢ (𝜑 → ((𝑃 ·s 𝑈) -s (𝑃 ·s 𝑄)) <s ((𝐴 ·s 𝑈) -s (𝐴 ·s 𝑄)))
219	187, 31, 19, 17	sltsubsub3bd 27967	. . . . . . . . 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))))
220	17, 19	subscld 27947	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ·s 𝑄) -s (𝑃 ·s 𝑄)) ∈ No )
221	31, 187	subscld 27947	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐴 ·s 𝑈) -s (𝑃 ·s 𝑈)) ∈ No )
222	220, 221, 12	sltadd2d 27888	. . . . . . . . 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)))))
223	219, 222	bitrd 279	. . . . . . . 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)))))
224	218, 223	mpbid 231	. . . . . . 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))))
225	12, 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))))
226	12, 31, 187	addsubsassd 27963	. . . . . . 7 ⊢ (𝜑 → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑃 ·s 𝑈)) = ((𝑃 ·s 𝐵) +s ((𝐴 ·s 𝑈) -s (𝑃 ·s 𝑈))))
227	224, 225, 226	3brtr4d 5174	. . . . . 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)))
228	227	adantr 480	. . . . 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)))
229		ssltright 27772	. . . . . . . . . . 11 ⊢ (𝐵 ∈ No → {𝐵} <<s ( R ‘𝐵))
230	11, 229	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → {𝐵} <<s ( R ‘𝐵))
231	230, 39, 29	ssltsepcd 27701	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 <s 𝑈)
232	50	uneq1i 4155	. . . . . . . . . . . . 13 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( 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)))))
233		0un 4388	. . . . . . . . . . . . 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))))
234	232, 233	eqtri 2755	. . . . . . . . . . . 12 ⊢ ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( 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))))
235		onunel 6468	. . . . . . . . . . . . . . . 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)))))
236	82, 202, 90, 235	mp3an 1458	. . . . . . . . . . . . . . 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))))
237	68, 198, 236	sylanbrc 582	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)))
238	139, 73	jca 511	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵)) ∧ (( bday ‘𝑃) +no ( bday ‘𝐵)) ∈ (( bday ‘𝐴) +no ( bday ‘𝐵))))
239	82, 202	onun2i 6485	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑇) +no ( bday ‘𝐵)) ∪ (( bday ‘𝑃) +no ( bday ‘𝑈))) ∈ On
240	148, 85	onun2i 6485	. . . . . . . . . . . . . . . 16 ⊢ ((( bday ‘𝑇) +no ( bday ‘𝑈)) ∪ (( bday ‘𝑃) +no ( bday ‘𝐵))) ∈ On
241		onunel 6468	. . . . . . . . . . . . . . . 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)))))
242	239, 240, 90, 241	mp3an 1458	. . . . . . . . . . . . . . 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))))
243		onunel 6468	. . . . . . . . . . . . . . . . 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)))))
244	148, 85, 90, 243	mp3an 1458	. . . . . . . . . . . . . . . 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))))
245	244	anbi2i 622	. . . . . . . . . . . . . . 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)))))
246	242, 245	bitri 275	. . . . . . . . . . . . . 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)))))
247	237, 238, 246	sylanbrc 582	. . . . . . . . . . . . 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)))
248		elun1 4172	. . . . . . . . . . . . 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))))))
249	247, 248	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))))))
250	234, 249	eqeltrid 2832	. . . . . . . . . . 11 ⊢ (𝜑 → ((( bday ‘ 0s ) +no ( bday ‘ 0s )) ∪ (((( 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))))))
251	8, 42, 42, 5, 3, 11, 126, 250	mulsproplem1 27990	. . . . . . . . . 10 ⊢ (𝜑 → (( 0s ·s 0s ) ∈ No ∧ ((𝑇 <s 𝑃 ∧ 𝐵 <s 𝑈) → ((𝑇 ·s 𝑈) -s (𝑇 ·s 𝐵)) <s ((𝑃 ·s 𝑈) -s (𝑃 ·s 𝐵)))))
252	251	simprd 495	. . . . . . . . 9 ⊢ (𝜑 → ((𝑇 <s 𝑃 ∧ 𝐵 <s 𝑈) → ((𝑇 ·s 𝑈) -s (𝑇 ·s 𝐵)) <s ((𝑃 ·s 𝑈) -s (𝑃 ·s 𝐵))))
253	231, 252	mpan2d 693	. . . . . . . 8 ⊢ (𝜑 → (𝑇 <s 𝑃 → ((𝑇 ·s 𝑈) -s (𝑇 ·s 𝐵)) <s ((𝑃 ·s 𝑈) -s (𝑃 ·s 𝐵))))
254	253	imp 406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑇 <s 𝑃) → ((𝑇 ·s 𝑈) -s (𝑇 ·s 𝐵)) <s ((𝑃 ·s 𝑈) -s (𝑃 ·s 𝐵)))
255	33, 23, 187, 12	sltsubsub2bd 27966	. . . . . . . . 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))))
256	12, 187	subscld 27947	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑈)) ∈ No )
257	23, 33	subscld 27947	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑈)) ∈ No )
258	256, 257, 31	sltadd1d 27889	. . . . . . . . 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))))
259	255, 258	bitrd 279	. . . . . . . 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))))
260	259	adantr 480	. . . . . . 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))))
261	254, 260	mpbid 231	. . . . . 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)))
262	12, 31, 187	addsubsd 27964	. . . . . . 7 ⊢ (𝜑 → (((𝑃 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑃 ·s 𝑈)) = (((𝑃 ·s 𝐵) -s (𝑃 ·s 𝑈)) +s (𝐴 ·s 𝑈)))
263	262	adantr 480	. . . . . 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)))
264	23, 31, 33	addsubsd 27964	. . . . . . 7 ⊢ (𝜑 → (((𝑇 ·s 𝐵) +s (𝐴 ·s 𝑈)) -s (𝑇 ·s 𝑈)) = (((𝑇 ·s 𝐵) -s (𝑇 ·s 𝑈)) +s (𝐴 ·s 𝑈)))
265	264	adantr 480	. . . . . 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)))
266	261, 263, 265	3brtr4d 5174	. . . . 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)))
267	185, 189, 190, 228, 266	slttrd 27666	. . . 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)))
268	267	ex 412	. . 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))))
269	178, 184, 268	3jaod 1426	. 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))))
270	7, 269	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)))

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