Theorem addsprop 27987

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 27763	. . . . 5 ⊢ ( bday ‘𝑋) ∈ On
2		bdayon 27763	. . . . 5 ⊢ ( bday ‘𝑌) ∈ On
3		naddcl 8604	. . . . 5 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑌) ∈ On) → (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On)
4	1, 2, 3	mp2an 693	. . . 4 ⊢ (( bday ‘𝑋) +no ( bday ‘𝑌)) ∈ On
5		bdayon 27763	. . . . 5 ⊢ ( bday ‘𝑍) ∈ On
6		naddcl 8604	. . . . 5 ⊢ ((( bday ‘𝑋) ∈ On ∧ ( bday ‘𝑍) ∈ On) → (( bday ‘𝑋) +no ( bday ‘𝑍)) ∈ On)
7	1, 5, 6	mp2an 693	. . . 4 ⊢ (( bday ‘𝑋) +no ( bday ‘𝑍)) ∈ On
8	4, 7	onun2i 6438	. . 3 ⊢ ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍))) ∈ On
9		risset 3213	. . 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 230	. 2 ⊢ ∃𝑎 ∈ On 𝑎 = ((( bday ‘𝑋) +no ( bday ‘𝑌)) ∪ (( bday ‘𝑋) +no ( bday ‘𝑍)))
11		eqeq1 2741	. . . . . . . . . 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 341	. . . . . . . . 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 3161	. . . . . . . 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 3202	. . . . . . 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 6832	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑝 → ( bday ‘𝑥) = ( bday ‘𝑝))
16	15	oveq1d 7373	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑝 → (( bday ‘𝑥) +no ( bday ‘𝑦)) = (( bday ‘𝑝) +no ( bday ‘𝑦)))
17	15	oveq1d 7373	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑝 → (( bday ‘𝑥) +no ( bday ‘𝑧)) = (( bday ‘𝑝) +no ( bday ‘𝑧)))
18	16, 17	uneq12d 4110	. . . . . . . . . 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 2748	. . . . . . . . 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 7365	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑝 → (𝑥 +s 𝑦) = (𝑝 +s 𝑦))
21	20	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑥 = 𝑝 → ((𝑥 +s 𝑦) ∈ No ↔ (𝑝 +s 𝑦) ∈ No ))
22		oveq2 7366	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑝 → (𝑦 +s 𝑥) = (𝑦 +s 𝑝))
23		oveq2 7366	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑝 → (𝑧 +s 𝑥) = (𝑧 +s 𝑝))
24	22, 23	breq12d 5099	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑝 → ((𝑦 +s 𝑥) <s (𝑧 +s 𝑥) ↔ (𝑦 +s 𝑝) <s (𝑧 +s 𝑝)))
25	24	imbi2d 340	. . . . . . . . . 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 633	. . . . . . . . 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 344	. . . . . . . 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 6832	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑞 → ( bday ‘𝑦) = ( bday ‘𝑞))
29	28	oveq2d 7374	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑞 → (( bday ‘𝑝) +no ( bday ‘𝑦)) = (( bday ‘𝑝) +no ( bday ‘𝑞)))
30	29	uneq1d 4108	. . . . . . . . . 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 2748	. . . . . . . . 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 7366	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑞 → (𝑝 +s 𝑦) = (𝑝 +s 𝑞))
33	32	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑦 = 𝑞 → ((𝑝 +s 𝑦) ∈ No ↔ (𝑝 +s 𝑞) ∈ No ))
34		breq1 5089	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑞 → (𝑦 <s 𝑧 ↔ 𝑞 <s 𝑧))
35		oveq1 7365	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑞 → (𝑦 +s 𝑝) = (𝑞 +s 𝑝))
36	35	breq1d 5096	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑞 → ((𝑦 +s 𝑝) <s (𝑧 +s 𝑝) ↔ (𝑞 +s 𝑝) <s (𝑧 +s 𝑝)))
37	34, 36	imbi12d 344	. . . . . . . . . 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 633	. . . . . . . . 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 344	. . . . . . . 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 6832	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑟 → ( bday ‘𝑧) = ( bday ‘𝑟))
41	40	oveq2d 7374	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑟 → (( bday ‘𝑝) +no ( bday ‘𝑧)) = (( bday ‘𝑝) +no ( bday ‘𝑟)))
42	41	uneq2d 4109	. . . . . . . . . 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 2748	. . . . . . . . 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 5090	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑟 → (𝑞 <s 𝑧 ↔ 𝑞 <s 𝑟))
45		oveq1 7365	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑟 → (𝑧 +s 𝑝) = (𝑟 +s 𝑝))
46	45	breq2d 5098	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑟 → ((𝑞 +s 𝑝) <s (𝑧 +s 𝑝) ↔ (𝑞 +s 𝑝) <s (𝑟 +s 𝑝)))
47	44, 46	imbi12d 344	. . . . . . . . . 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 631	. . . . . . . . 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 344	. . . . . . . 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 3222	. . . . . . 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 287	. . . . . 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 3269	. . . . . . . . 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 3266	. . . . . . . . . . 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 3165	. . . . . . . . . . . . 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 3213	. . . . . . . . . . . . . 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 349	. . . . . . . . . . . . 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 278	. . . . . . . . . . . 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 3084	. . . . . . . . . . 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 275	. . . . . . . . . 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 3113	. . . . . . . . 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 277	. . . . . . . 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 2826	. . . . . . . . . . . . . . . . 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 341	. . . . . . . . . . . . . . . 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 3161	. . . . . . . . . . . . . . 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 3202	. . . . . . . . . . . . . 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 632	. . . . . . . . . . . . 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 249	. . . . . . . . . . . 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 482	. . . . . . . . . . . . . . 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 779	. . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . 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 27982	. . . . . . . . . . . . . 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 1143	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . 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 27986	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 511	. . . . . . . . . . . 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 35	. . . . . . . . . . 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 467	. . . . . . . . . 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 3130	. . . . . . . . 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 3181	. . . . . . . 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 217	. . . . . . 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 7799	. . . . 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 6832	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → ( bday ‘𝑥) = ( bday ‘𝑋))
89	88	oveq1d 7373	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (( bday ‘𝑥) +no ( bday ‘𝑦)) = (( bday ‘𝑋) +no ( bday ‘𝑦)))
90	88	oveq1d 7373	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (( bday ‘𝑥) +no ( bday ‘𝑧)) = (( bday ‘𝑋) +no ( bday ‘𝑧)))
91	89, 90	uneq12d 4110	. . . . . . . 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 2748	. . . . . . 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 7365	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → (𝑥 +s 𝑦) = (𝑋 +s 𝑦))
94	93	eleq1d 2822	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → ((𝑥 +s 𝑦) ∈ No ↔ (𝑋 +s 𝑦) ∈ No ))
95		oveq2 7366	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑦 +s 𝑥) = (𝑦 +s 𝑋))
96		oveq2 7366	. . . . . . . . . 10 ⊢ (𝑥 = 𝑋 → (𝑧 +s 𝑥) = (𝑧 +s 𝑋))
97	95, 96	breq12d 5099	. . . . . . . . 9 ⊢ (𝑥 = 𝑋 → ((𝑦 +s 𝑥) <s (𝑧 +s 𝑥) ↔ (𝑦 +s 𝑋) <s (𝑧 +s 𝑋)))
98	97	imbi2d 340	. . . . . . . 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 633	. . . . . . 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 344	. . . . . 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 6832	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → ( bday ‘𝑦) = ( bday ‘𝑌))
102	101	oveq2d 7374	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (( bday ‘𝑋) +no ( bday ‘𝑦)) = (( bday ‘𝑋) +no ( bday ‘𝑌)))
103	102	uneq1d 4108	. . . . . . . 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 2748	. . . . . . 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 7366	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (𝑋 +s 𝑦) = (𝑋 +s 𝑌))
106	105	eleq1d 2822	. . . . . . . 8 ⊢ (𝑦 = 𝑌 → ((𝑋 +s 𝑦) ∈ No ↔ (𝑋 +s 𝑌) ∈ No ))
107		breq1 5089	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → (𝑦 <s 𝑧 ↔ 𝑌 <s 𝑧))
108		oveq1 7365	. . . . . . . . . 10 ⊢ (𝑦 = 𝑌 → (𝑦 +s 𝑋) = (𝑌 +s 𝑋))
109	108	breq1d 5096	. . . . . . . . 9 ⊢ (𝑦 = 𝑌 → ((𝑦 +s 𝑋) <s (𝑧 +s 𝑋) ↔ (𝑌 +s 𝑋) <s (𝑧 +s 𝑋)))
110	107, 109	imbi12d 344	. . . . . . . 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 633	. . . . . . 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 344	. . . . . 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 6832	. . . . . . . . . 10 ⊢ (𝑧 = 𝑍 → ( bday ‘𝑧) = ( bday ‘𝑍))
114	113	oveq2d 7374	. . . . . . . . 9 ⊢ (𝑧 = 𝑍 → (( bday ‘𝑋) +no ( bday ‘𝑧)) = (( bday ‘𝑋) +no ( bday ‘𝑍)))
115	114	uneq2d 4109	. . . . . . . 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 2748	. . . . . . 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 5090	. . . . . . . . 9 ⊢ (𝑧 = 𝑍 → (𝑌 <s 𝑧 ↔ 𝑌 <s 𝑍))
118		oveq1 7365	. . . . . . . . . 10 ⊢ (𝑧 = 𝑍 → (𝑧 +s 𝑋) = (𝑍 +s 𝑋))
119	118	breq2d 5098	. . . . . . . . 9 ⊢ (𝑧 = 𝑍 → ((𝑌 +s 𝑋) <s (𝑧 +s 𝑋) ↔ (𝑌 +s 𝑋) <s (𝑍 +s 𝑋)))
120	117, 119	imbi12d 344	. . . . . . . 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 631	. . . . . . 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 344	. . . . . 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 3581	. . . . 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 31	. . . 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 86	. . 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 3132	. 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))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062   ∪ cun 3888  {csn 4568   class class class wbr 5086  Oncon0 6315  ‘cfv 6490  (class class class)co 7358   +no cnadd 8592   No csur 27622   <s clts 27623   bday cbday 27624   <<s cslts 27768   L cleft 27836   R cright 27837   +_s cadds 27970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-1o 8396  df-2o 8397  df-nadd 8593  df-no 27625  df-lts 27626  df-bday 27627  df-slts 27769  df-cuts 27771  df-0s 27818  df-made 27838  df-old 27839  df-left 27841  df-right 27842  df-norec2 27960  df-adds 27971

This theorem is referenced by:  addcutslem  27988  ltadds1im  27996