Theorem addsprop 28363

Description: Inductively show that surreal addition is closed and compatible with less-than. This proof follows from induction on the birthdays of the surreal numbers involved. This pattern occurs throughout surreal development. Theorem 3.1 of [Gonshor] p. 14. (Contributed by Scott Fenton, 21-Jan-2025.)

Assertion

Ref	Expression
addsprop	⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑍 ∈ No) → ((𝑋 +s 𝑌) ∈ No ∧ (𝑌 <s 𝑍 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))))

Proof of Theorem addsprop

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑝 𝑞 𝑟 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdayon 28139	. . . . 5 ⊢ (bday‘𝑋) ∈ On
2		bdayon 28139	. . . . 5 ⊢ (bday‘𝑌) ∈ On
3		naddcl 8687	. . . . 5 ⊢ (((bday‘𝑋) ∈ On ∧ (bday‘𝑌) ∈ On) → ((bday‘𝑋) +no (bday‘𝑌)) ∈ On)
4	1, 2, 3	mp2an 705	. . . 4 ⊢ ((bday‘𝑋) +no (bday‘𝑌)) ∈ On
5		bdayon 28139	. . . . 5 ⊢ (bday‘𝑍) ∈ On
6		naddcl 8687	. . . . 5 ⊢ (((bday‘𝑋) ∈ On ∧ (bday‘𝑍) ∈ On) → ((bday‘𝑋) +no (bday‘𝑍)) ∈ On)
7	1, 5, 6	mp2an 705	. . . 4 ⊢ ((bday‘𝑋) +no (bday‘𝑍)) ∈ On
8	4, 7	onun2i 6486	. . 3 ⊢ (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) ∈ On
9		risset 3237	. . 3 ⊢ ((((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) ∈ On ↔ ∃𝑎 ∈ On 𝑎 = (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))))
10	8, 9	mpbi 233	. 2 ⊢ ∃𝑎 ∈ On 𝑎 = (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍)))
11		eqeq1 2764	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (𝑎 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ↔ 𝑏 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧)))))
12	11	imbi1d 344	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → ((𝑎 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ (𝑏 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))))
13	12	ralbidv 3185	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (∀𝑧 ∈ No (𝑎 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ ∀𝑧 ∈ No (𝑏 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))))
14	13	2ralbidv 3226	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (𝑎 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (𝑏 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))))
15		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑝 → (bday‘𝑥) = (bday‘𝑝))
16	15	oveq1d 7435	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑝 → ((bday‘𝑥) +no (bday‘𝑦)) = ((bday‘𝑝) +no (bday‘𝑦)))
17	15	oveq1d 7435	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑝 → ((bday‘𝑥) +no (bday‘𝑧)) = ((bday‘𝑝) +no (bday‘𝑧)))
18	16, 17	uneq12d 4168	. . . . . . . . . 10 ⊢ (𝑥 = 𝑝 → (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) = (((bday‘𝑝) +no (bday‘𝑦)) ∪ ((bday‘𝑝) +no (bday‘𝑧))))
19	18	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑥 = 𝑝 → (𝑏 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ↔ 𝑏 = (((bday‘𝑝) +no (bday‘𝑦)) ∪ ((bday‘𝑝) +no (bday‘𝑧)))))
20		oveq1 7427	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑝 → (𝑥 +s 𝑦) = (𝑝 +s 𝑦))
21	20	eleq1d 2845	. . . . . . . . . 10 ⊢ (𝑥 = 𝑝 → ((𝑥 +s 𝑦) ∈ No ↔ (𝑝 +s 𝑦) ∈ No))
22		oveq2 7428	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑝 → (𝑦 +s 𝑥) = (𝑦 +s 𝑝))
23		oveq2 7428	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑝 → (𝑧 +s 𝑥) = (𝑧 +s 𝑝))
24	22, 23	breq12d 5115	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑝 → ((𝑦 +s 𝑥) <s (𝑧 +s 𝑥) ↔ (𝑦 +s 𝑝) <s (𝑧 +s 𝑝)))
25	24	imbi2d 343	. . . . . . . . . 10 ⊢ (𝑥 = 𝑝 → ((𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)) ↔ (𝑦 <s 𝑧 → (𝑦 +s 𝑝) <s (𝑧 +s 𝑝))))
26	21, 25	anbi12d 644	. . . . . . . . 9 ⊢ (𝑥 = 𝑝 → (((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))) ↔ ((𝑝 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑝) <s (𝑧 +s 𝑝)))))
27	19, 26	imbi12d 347	. . . . . . . 8 ⊢ (𝑥 = 𝑝 → ((𝑏 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ (𝑏 = (((bday‘𝑝) +no (bday‘𝑦)) ∪ ((bday‘𝑝) +no (bday‘𝑧))) → ((𝑝 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑝) <s (𝑧 +s 𝑝))))))
28		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑞 → (bday‘𝑦) = (bday‘𝑞))
29	28	oveq2d 7436	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑞 → ((bday‘𝑝) +no (bday‘𝑦)) = ((bday‘𝑝) +no (bday‘𝑞)))
30	29	uneq1d 4166	. . . . . . . . . 10 ⊢ (𝑦 = 𝑞 → (((bday‘𝑝) +no (bday‘𝑦)) ∪ ((bday‘𝑝) +no (bday‘𝑧))) = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑧))))
31	30	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑦 = 𝑞 → (𝑏 = (((bday‘𝑝) +no (bday‘𝑦)) ∪ ((bday‘𝑝) +no (bday‘𝑧))) ↔ 𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑧)))))
32		oveq2 7428	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑞 → (𝑝 +s 𝑦) = (𝑝 +s 𝑞))
33	32	eleq1d 2845	. . . . . . . . . 10 ⊢ (𝑦 = 𝑞 → ((𝑝 +s 𝑦) ∈ No ↔ (𝑝 +s 𝑞) ∈ No))
34		breq1 5105	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑞 → (𝑦 <s 𝑧 ↔ 𝑞 <s 𝑧))
35		oveq1 7427	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑞 → (𝑦 +s 𝑝) = (𝑞 +s 𝑝))
36	35	breq1d 5112	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑞 → ((𝑦 +s 𝑝) <s (𝑧 +s 𝑝) ↔ (𝑞 +s 𝑝) <s (𝑧 +s 𝑝)))
37	34, 36	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑦 = 𝑞 → ((𝑦 <s 𝑧 → (𝑦 +s 𝑝) <s (𝑧 +s 𝑝)) ↔ (𝑞 <s 𝑧 → (𝑞 +s 𝑝) <s (𝑧 +s 𝑝))))
38	33, 37	anbi12d 644	. . . . . . . . 9 ⊢ (𝑦 = 𝑞 → (((𝑝 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑝) <s (𝑧 +s 𝑝))) ↔ ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑧 → (𝑞 +s 𝑝) <s (𝑧 +s 𝑝)))))
39	31, 38	imbi12d 347	. . . . . . . 8 ⊢ (𝑦 = 𝑞 → ((𝑏 = (((bday‘𝑝) +no (bday‘𝑦)) ∪ ((bday‘𝑝) +no (bday‘𝑧))) → ((𝑝 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑝) <s (𝑧 +s 𝑝)))) ↔ (𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑧 → (𝑞 +s 𝑝) <s (𝑧 +s 𝑝))))))
40		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑟 → (bday‘𝑧) = (bday‘𝑟))
41	40	oveq2d 7436	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑟 → ((bday‘𝑝) +no (bday‘𝑧)) = ((bday‘𝑝) +no (bday‘𝑟)))
42	41	uneq2d 4167	. . . . . . . . . 10 ⊢ (𝑧 = 𝑟 → (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑧))) = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))))
43	42	eqeq2d 2771	. . . . . . . . 9 ⊢ (𝑧 = 𝑟 → (𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑧))) ↔ 𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟)))))
44		breq2 5106	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑟 → (𝑞 <s 𝑧 ↔ 𝑞 <s 𝑟))
45		oveq1 7427	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑟 → (𝑧 +s 𝑝) = (𝑟 +s 𝑝))
46	45	breq2d 5114	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑟 → ((𝑞 +s 𝑝) <s (𝑧 +s 𝑝) ↔ (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))
47	44, 46	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑧 = 𝑟 → ((𝑞 <s 𝑧 → (𝑞 +s 𝑝) <s (𝑧 +s 𝑝)) ↔ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝))))
48	47	anbi2d 642	. . . . . . . . 9 ⊢ (𝑧 = 𝑟 → (((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑧 → (𝑞 +s 𝑝) <s (𝑧 +s 𝑝))) ↔ ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))))
49	43, 48	imbi12d 347	. . . . . . . 8 ⊢ (𝑧 = 𝑟 → ((𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑧 → (𝑞 +s 𝑝) <s (𝑧 +s 𝑝)))) ↔ (𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝))))))
50	27, 39, 49	cbvral3vw 3246	. . . . . . 7 ⊢ (∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (𝑏 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ ∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No (𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))))
51	14, 50	bitrdi 290	. . . . . 6 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (𝑎 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ ∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No (𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝))))))
52		ralrot3 3293	. . . . . . . . 9 ⊢ (∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑏 ∈ 𝑎 ∀𝑟 ∈ No (𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ↔ ∀𝑏 ∈ 𝑎 ∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No (𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))))
53		ralcom 3290	. . . . . . . . . . 11 ⊢ (∀𝑏 ∈ 𝑎 ∀𝑟 ∈ No (𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ↔ ∀𝑟 ∈ No ∀𝑏 ∈ 𝑎 (𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))))
54		r19.23v 3189	. . . . . . . . . . . . 13 ⊢ (∀𝑏 ∈ 𝑎 (𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ↔ (∃𝑏 ∈ 𝑎 𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))))
55		risset 3237	. . . . . . . . . . . . . 14 ⊢ ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ 𝑎 ↔ ∃𝑏 ∈ 𝑎 𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))))
56	55	imbi1i 352	. . . . . . . . . . . . 13 ⊢ (((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ 𝑎 → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ↔ (∃𝑏 ∈ 𝑎 𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))))
57	54, 56	bitr4i 281	. . . . . . . . . . . 12 ⊢ (∀𝑏 ∈ 𝑎 (𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ↔ ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ 𝑎 → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))))
58	57	ralbii 3108	. . . . . . . . . . 11 ⊢ (∀𝑟 ∈ No ∀𝑏 ∈ 𝑎 (𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ↔ ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ 𝑎 → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))))
59	53, 58	bitri 278	. . . . . . . . . 10 ⊢ (∀𝑏 ∈ 𝑎 ∀𝑟 ∈ No (𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ↔ ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ 𝑎 → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))))
60	59	2ralbii 3137	. . . . . . . . 9 ⊢ (∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑏 ∈ 𝑎 ∀𝑟 ∈ No (𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ↔ ∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ 𝑎 → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))))
61	52, 60	bitr3i 280	. . . . . . . 8 ⊢ (∀𝑏 ∈ 𝑎 ∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No (𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ↔ ∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ 𝑎 → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))))
62		eleq2 2849	. . . . . . . . . . . . . . . . 17 ⊢ (𝑎 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ 𝑎 ↔ (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧)))))
63	62	imbi1d 344	. . . . . . . . . . . . . . . 16 ⊢ (𝑎 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → (((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ 𝑎 → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ↔ ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝))))))
64	63	ralbidv 3185	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → (∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ 𝑎 → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ↔ ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝))))))
65	64	2ralbidv 3226	. . . . . . . . . . . . . 14 ⊢ (𝑎 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → (∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ 𝑎 → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ↔ ∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝))))))
66	65	anbi1d 643	. . . . . . . . . . . . 13 ⊢ (𝑎 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ 𝑎 → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ∧ ((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑧 ∈ No)) ↔ (∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ∧ ((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑧 ∈ No))))
67	66	biimpcd 252	. . . . . . . . . . . 12 ⊢ ((∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ 𝑎 → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ∧ ((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑧 ∈ No)) → (𝑎 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → (∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ∧ ((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑧 ∈ No))))
68		simpl 488	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ∧ ((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑧 ∈ No)) → ∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))))
69		simprll 791	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ∧ ((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑧 ∈ No)) → 𝑥 ∈ No)
70		simprlr 792	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ∧ ((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑧 ∈ No)) → 𝑦 ∈ No)
71	68, 69, 70	addsproplem3 28358	. . . . . . . . . . . . . 14 ⊢ ((∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ∧ ((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑧 ∈ No)) → ((𝑥 +s 𝑦) ∈ No ∧ ({𝑎 ∣ ∃𝑏 ∈ (L‘𝑥)𝑎 = (𝑏 +s 𝑦)} ∪ {𝑐 ∣ ∃𝑑 ∈ (L‘𝑦)𝑐 = (𝑥 +s 𝑑)}) <<s {(𝑥 +s 𝑦)} ∧ {(𝑥 +s 𝑦)} <<s ({𝑒 ∣ ∃𝑓 ∈ (R‘𝑥)𝑒 = (𝑓 +s 𝑦)} ∪ {𝑔 ∣ ∃ℎ ∈ (R‘𝑦)𝑔 = (𝑥 +s ℎ)})))
72	71	simp1d 1160	. . . . . . . . . . . . 13 ⊢ ((∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ∧ ((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑧 ∈ No)) → (𝑥 +s 𝑦) ∈ No)
73	68	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ∧ ((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑧 ∈ No)) ∧ 𝑦 <s 𝑧) → ∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))))
74	69	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ∧ ((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑧 ∈ No)) ∧ 𝑦 <s 𝑧) → 𝑥 ∈ No)
75	70	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ∧ ((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑧 ∈ No)) ∧ 𝑦 <s 𝑧) → 𝑦 ∈ No)
76		simplrr 790	. . . . . . . . . . . . . . 15 ⊢ (((∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ∧ ((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑧 ∈ No)) ∧ 𝑦 <s 𝑧) → 𝑧 ∈ No)
77		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ∧ ((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑧 ∈ No)) ∧ 𝑦 <s 𝑧) → 𝑦 <s 𝑧)
78	73, 74, 75, 76, 77	addsproplem7 28362	. . . . . . . . . . . . . 14 ⊢ (((∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ∧ ((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑧 ∈ No)) ∧ 𝑦 <s 𝑧) → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))
79	78	ex 418	. . . . . . . . . . . . 13 ⊢ ((∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ∧ ((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑧 ∈ No)) → (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))
80	72, 79	jca 521	. . . . . . . . . . . 12 ⊢ ((∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ∧ ((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑧 ∈ No)) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))
81	67, 80	syl6 36	. . . . . . . . . . 11 ⊢ ((∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ 𝑎 → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ∧ ((𝑥 ∈ No ∧ 𝑦 ∈ No) ∧ 𝑧 ∈ No)) → (𝑎 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
82	81	anassrs 473	. . . . . . . . . 10 ⊢ (((∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ 𝑎 → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ∧ (𝑥 ∈ No ∧ 𝑦 ∈ No)) ∧ 𝑧 ∈ No) → (𝑎 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
83	82	ralrimiva 3154	. . . . . . . . 9 ⊢ ((∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ 𝑎 → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) ∧ (𝑥 ∈ No ∧ 𝑦 ∈ No)) → ∀𝑧 ∈ No (𝑎 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
84	83	ralrimivva 3205	. . . . . . . 8 ⊢ (∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No ((((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) ∈ 𝑎 → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (𝑎 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
85	61, 84	sylbi 220	. . . . . . 7 ⊢ (∀𝑏 ∈ 𝑎 ∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No (𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (𝑎 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
86	85	a1i 11	. . . . . 6 ⊢ (𝑎 ∈ On → (∀𝑏 ∈ 𝑎 ∀𝑝 ∈ No ∀𝑞 ∈ No ∀𝑟 ∈ No (𝑏 = (((bday‘𝑝) +no (bday‘𝑞)) ∪ ((bday‘𝑝) +no (bday‘𝑟))) → ((𝑝 +s 𝑞) ∈ No ∧ (𝑞 <s 𝑟 → (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))) → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (𝑎 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))))))
87	51, 86	tfis2 7868	. . . . 5 ⊢ (𝑎 ∈ On → ∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (𝑎 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))))
88		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (bday‘𝑥) = (bday‘𝑋))
89	88	oveq1d 7435	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((bday‘𝑥) +no (bday‘𝑦)) = ((bday‘𝑋) +no (bday‘𝑦)))
90	88	oveq1d 7435	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((bday‘𝑥) +no (bday‘𝑧)) = ((bday‘𝑋) +no (bday‘𝑧)))
91	89, 90	uneq12d 4168	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) = (((bday‘𝑋) +no (bday‘𝑦)) ∪ ((bday‘𝑋) +no (bday‘𝑧))))
92	91	eqeq2d 2771	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑎 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) ↔ 𝑎 = (((bday‘𝑋) +no (bday‘𝑦)) ∪ ((bday‘𝑋) +no (bday‘𝑧)))))
93		oveq1 7427	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥 +s 𝑦) = (𝑋 +s 𝑦))
94	93	eleq1d 2845	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝑥 +s 𝑦) ∈ No ↔ (𝑋 +s 𝑦) ∈ No))
95		oveq2 7428	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑦 +s 𝑥) = (𝑦 +s 𝑋))
96		oveq2 7428	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑧 +s 𝑥) = (𝑧 +s 𝑋))
97	95, 96	breq12d 5115	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝑦 +s 𝑥) <s (𝑧 +s 𝑥) ↔ (𝑦 +s 𝑋) <s (𝑧 +s 𝑋)))
98	97	imbi2d 343	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)) ↔ (𝑦 <s 𝑧 → (𝑦 +s 𝑋) <s (𝑧 +s 𝑋))))
99	94, 98	anbi12d 644	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥))) ↔ ((𝑋 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑋) <s (𝑧 +s 𝑋)))))
100	92, 99	imbi12d 347	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑎 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) ↔ (𝑎 = (((bday‘𝑋) +no (bday‘𝑦)) ∪ ((bday‘𝑋) +no (bday‘𝑧))) → ((𝑋 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑋) <s (𝑧 +s 𝑋))))))
101		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → (bday‘𝑦) = (bday‘𝑌))
102	101	oveq2d 7436	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → ((bday‘𝑋) +no (bday‘𝑦)) = ((bday‘𝑋) +no (bday‘𝑌)))
103	102	uneq1d 4166	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → (((bday‘𝑋) +no (bday‘𝑦)) ∪ ((bday‘𝑋) +no (bday‘𝑧))) = (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑧))))
104	103	eqeq2d 2771	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑎 = (((bday‘𝑋) +no (bday‘𝑦)) ∪ ((bday‘𝑋) +no (bday‘𝑧))) ↔ 𝑎 = (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑧)))))
105		oveq2 7428	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (𝑋 +s 𝑦) = (𝑋 +s 𝑌))
106	105	eleq1d 2845	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ((𝑋 +s 𝑦) ∈ No ↔ (𝑋 +s 𝑌) ∈ No))
107		breq1 5105	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (𝑦 <s 𝑧 ↔ 𝑌 <s 𝑧))
108		oveq1 7427	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → (𝑦 +s 𝑋) = (𝑌 +s 𝑋))
109	108	breq1d 5112	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → ((𝑦 +s 𝑋) <s (𝑧 +s 𝑋) ↔ (𝑌 +s 𝑋) <s (𝑧 +s 𝑋)))
110	107, 109	imbi12d 347	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ((𝑦 <s 𝑧 → (𝑦 +s 𝑋) <s (𝑧 +s 𝑋)) ↔ (𝑌 <s 𝑧 → (𝑌 +s 𝑋) <s (𝑧 +s 𝑋))))
111	106, 110	anbi12d 644	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (((𝑋 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑋) <s (𝑧 +s 𝑋))) ↔ ((𝑋 +s 𝑌) ∈ No ∧ (𝑌 <s 𝑧 → (𝑌 +s 𝑋) <s (𝑧 +s 𝑋)))))
112	104, 111	imbi12d 347	. . . . . 6 ⊢ (𝑦 = 𝑌 → ((𝑎 = (((bday‘𝑋) +no (bday‘𝑦)) ∪ ((bday‘𝑋) +no (bday‘𝑧))) → ((𝑋 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑋) <s (𝑧 +s 𝑋)))) ↔ (𝑎 = (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑧))) → ((𝑋 +s 𝑌) ∈ No ∧ (𝑌 <s 𝑧 → (𝑌 +s 𝑋) <s (𝑧 +s 𝑋))))))
113		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑧 = 𝑍 → (bday‘𝑧) = (bday‘𝑍))
114	113	oveq2d 7436	. . . . . . . . 9 ⊢ (𝑧 = 𝑍 → ((bday‘𝑋) +no (bday‘𝑧)) = ((bday‘𝑋) +no (bday‘𝑍)))
115	114	uneq2d 4167	. . . . . . . 8 ⊢ (𝑧 = 𝑍 → (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑧))) = (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))))
116	115	eqeq2d 2771	. . . . . . 7 ⊢ (𝑧 = 𝑍 → (𝑎 = (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑧))) ↔ 𝑎 = (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍)))))
117		breq2 5106	. . . . . . . . 9 ⊢ (𝑧 = 𝑍 → (𝑌 <s 𝑧 ↔ 𝑌 <s 𝑍))
118		oveq1 7427	. . . . . . . . . 10 ⊢ (𝑧 = 𝑍 → (𝑧 +s 𝑋) = (𝑍 +s 𝑋))
119	118	breq2d 5114	. . . . . . . . 9 ⊢ (𝑧 = 𝑍 → ((𝑌 +s 𝑋) <s (𝑧 +s 𝑋) ↔ (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
120	117, 119	imbi12d 347	. . . . . . . 8 ⊢ (𝑧 = 𝑍 → ((𝑌 <s 𝑧 → (𝑌 +s 𝑋) <s (𝑧 +s 𝑋)) ↔ (𝑌 <s 𝑍 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))))
121	120	anbi2d 642	. . . . . . 7 ⊢ (𝑧 = 𝑍 → (((𝑋 +s 𝑌) ∈ No ∧ (𝑌 <s 𝑧 → (𝑌 +s 𝑋) <s (𝑧 +s 𝑋))) ↔ ((𝑋 +s 𝑌) ∈ No ∧ (𝑌 <s 𝑍 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))))
122	116, 121	imbi12d 347	. . . . . 6 ⊢ (𝑧 = 𝑍 → ((𝑎 = (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑧))) → ((𝑋 +s 𝑌) ∈ No ∧ (𝑌 <s 𝑧 → (𝑌 +s 𝑋) <s (𝑧 +s 𝑋)))) ↔ (𝑎 = (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) → ((𝑋 +s 𝑌) ∈ No ∧ (𝑌 <s 𝑍 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))))))
123	100, 112, 122	rspc3v 3591	. . . . 5 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑍 ∈ No) → (∀𝑥 ∈ No ∀𝑦 ∈ No ∀𝑧 ∈ No (𝑎 = (((bday‘𝑥) +no (bday‘𝑦)) ∪ ((bday‘𝑥) +no (bday‘𝑧))) → ((𝑥 +s 𝑦) ∈ No ∧ (𝑦 <s 𝑧 → (𝑦 +s 𝑥) <s (𝑧 +s 𝑥)))) → (𝑎 = (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) → ((𝑋 +s 𝑌) ∈ No ∧ (𝑌 <s 𝑍 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))))))
124	87, 123	syl5com 32	. . . 4 ⊢ (𝑎 ∈ On → ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑍 ∈ No) → (𝑎 = (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) → ((𝑋 +s 𝑌) ∈ No ∧ (𝑌 <s 𝑍 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))))))
125	124	com23 87	. . 3 ⊢ (𝑎 ∈ On → (𝑎 = (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) → ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑍 ∈ No) → ((𝑋 +s 𝑌) ∈ No ∧ (𝑌 <s 𝑍 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))))))
126	125	rexlimiv 3156	. 2 ⊢ (∃𝑎 ∈ On 𝑎 = (((bday‘𝑋) +no (bday‘𝑌)) ∪ ((bday‘𝑋) +no (bday‘𝑍))) → ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑍 ∈ No) → ((𝑋 +s 𝑌) ∈ No ∧ (𝑌 <s 𝑍 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))))
127	10, 126	ax-mp 5	1 ⊢ ((𝑋 ∈ No ∧ 𝑌 ∈ No ∧ 𝑍 ∈ No) → ((𝑋 +s 𝑌) ∈ No ∧ (𝑌 <s 𝑍 → (𝑌 +s 𝑋) <s (𝑍 +s 𝑋))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086   ∪ cun 4149  {csn 4583   class class class wbr 5102  Oncon0 6362  ‘cfv 6538  (class class class)co 7420   +no cnadd 8675  Nocsur 27998   <s clts 27999  bdaycbday 28000   <<s cslts 28144  Lcleft 28212  Rcright 28213   +_s cadds 28346

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-1o 8477  df-2o 8478  df-nadd 8676  df-no 28001  df-lts 28002  df-bday 28003  df-slts 28145  df-cuts 28147  df-0s 28194  df-made 28214  df-old 28215  df-left 28217  df-right 28218  df-norec2 28336  df-adds 28347

This theorem is used by:  addcutslem  28364  ltadds1im  28372