Theorem addsrid 28015

Description: Surreal addition to zero is identity. Part of Theorem 3 of [Conway] p. 17. (Contributed by Scott Fenton, 20-Aug-2024.)

Assertion

Ref	Expression
addsrid	⊢ (𝐴 ∈ No → (𝐴 +s 0s ) = 𝐴)

Proof of Theorem addsrid

Dummy variables 𝑎 𝑏 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7455	. . 3 ⊢ (𝑎 = 𝑏 → (𝑎 +s 0s ) = (𝑏 +s 0s ))
2		id 22	. . 3 ⊢ (𝑎 = 𝑏 → 𝑎 = 𝑏)
3	1, 2	eqeq12d 2756	. 2 ⊢ (𝑎 = 𝑏 → ((𝑎 +s 0s ) = 𝑎 ↔ (𝑏 +s 0s ) = 𝑏))
4		oveq1 7455	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 +s 0s ) = (𝐴 +s 0s ))
5		id 22	. . 3 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
6	4, 5	eqeq12d 2756	. 2 ⊢ (𝑎 = 𝐴 → ((𝑎 +s 0s ) = 𝑎 ↔ (𝐴 +s 0s ) = 𝐴))
7		0sno 27889	. . . . . 6 ⊢ 0s ∈ No
8		addsval 28013	. . . . . 6 ⊢ ((𝑎 ∈ No ∧ 0s ∈ No ) → (𝑎 +s 0s ) = (({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝑎)𝑥 = (𝑦 +s 0s )} ∪ {𝑧 ∣ ∃𝑦 ∈ ( L ‘ 0s )𝑧 = (𝑎 +s 𝑦)}) |s ({𝑥 ∣ ∃𝑤 ∈ ( R ‘𝑎)𝑥 = (𝑤 +s 0s )} ∪ {𝑧 ∣ ∃𝑤 ∈ ( R ‘ 0s )𝑧 = (𝑎 +s 𝑤)})))
9	7, 8	mpan2 690	. . . . 5 ⊢ (𝑎 ∈ No → (𝑎 +s 0s ) = (({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝑎)𝑥 = (𝑦 +s 0s )} ∪ {𝑧 ∣ ∃𝑦 ∈ ( L ‘ 0s )𝑧 = (𝑎 +s 𝑦)}) |s ({𝑥 ∣ ∃𝑤 ∈ ( R ‘𝑎)𝑥 = (𝑤 +s 0s )} ∪ {𝑧 ∣ ∃𝑤 ∈ ( R ‘ 0s )𝑧 = (𝑎 +s 𝑤)})))
10	9	adantr 480	. . . 4 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (𝑎 +s 0s ) = (({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝑎)𝑥 = (𝑦 +s 0s )} ∪ {𝑧 ∣ ∃𝑦 ∈ ( L ‘ 0s )𝑧 = (𝑎 +s 𝑦)}) |s ({𝑥 ∣ ∃𝑤 ∈ ( R ‘𝑎)𝑥 = (𝑤 +s 0s )} ∪ {𝑧 ∣ ∃𝑤 ∈ ( R ‘ 0s )𝑧 = (𝑎 +s 𝑤)})))
11		elun1 4205	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ( L ‘𝑎) → 𝑦 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎)))
12		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏)
13		oveq1 7455	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑦 → (𝑏 +s 0s ) = (𝑦 +s 0s ))
14		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑦 → 𝑏 = 𝑦)
15	13, 14	eqeq12d 2756	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑦 → ((𝑏 +s 0s ) = 𝑏 ↔ (𝑦 +s 0s ) = 𝑦))
16	15	rspcva 3633	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎)) ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (𝑦 +s 0s ) = 𝑦)
17	11, 12, 16	syl2anr 596	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑦 ∈ ( L ‘𝑎)) → (𝑦 +s 0s ) = 𝑦)
18	17	eqeq2d 2751	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑦 ∈ ( L ‘𝑎)) → (𝑥 = (𝑦 +s 0s ) ↔ 𝑥 = 𝑦))
19		equcom 2017	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
20	18, 19	bitrdi 287	. . . . . . . . . 10 ⊢ (((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑦 ∈ ( L ‘𝑎)) → (𝑥 = (𝑦 +s 0s ) ↔ 𝑦 = 𝑥))
21	20	rexbidva 3183	. . . . . . . . 9 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (∃𝑦 ∈ ( L ‘𝑎)𝑥 = (𝑦 +s 0s ) ↔ ∃𝑦 ∈ ( L ‘𝑎)𝑦 = 𝑥))
22		risset 3239	. . . . . . . . 9 ⊢ (𝑥 ∈ ( L ‘𝑎) ↔ ∃𝑦 ∈ ( L ‘𝑎)𝑦 = 𝑥)
23	21, 22	bitr4di 289	. . . . . . . 8 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (∃𝑦 ∈ ( L ‘𝑎)𝑥 = (𝑦 +s 0s ) ↔ 𝑥 ∈ ( L ‘𝑎)))
24	23	eqabcdv 2879	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝑎)𝑥 = (𝑦 +s 0s )} = ( L ‘𝑎))
25		rex0 4383	. . . . . . . . . 10 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑧 = (𝑎 +s 𝑦)
26		left0s 27949	. . . . . . . . . . 11 ⊢ ( L ‘ 0s ) = ∅
27	26	rexeqi 3333	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ ( L ‘ 0s )𝑧 = (𝑎 +s 𝑦) ↔ ∃𝑦 ∈ ∅ 𝑧 = (𝑎 +s 𝑦))
28	25, 27	mtbir 323	. . . . . . . . 9 ⊢ ¬ ∃𝑦 ∈ ( L ‘ 0s )𝑧 = (𝑎 +s 𝑦)
29	28	abf 4429	. . . . . . . 8 ⊢ {𝑧 ∣ ∃𝑦 ∈ ( L ‘ 0s )𝑧 = (𝑎 +s 𝑦)} = ∅
30	29	a1i 11	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → {𝑧 ∣ ∃𝑦 ∈ ( L ‘ 0s )𝑧 = (𝑎 +s 𝑦)} = ∅)
31	24, 30	uneq12d 4192	. . . . . 6 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → ({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝑎)𝑥 = (𝑦 +s 0s )} ∪ {𝑧 ∣ ∃𝑦 ∈ ( L ‘ 0s )𝑧 = (𝑎 +s 𝑦)}) = (( L ‘𝑎) ∪ ∅))
32		un0 4417	. . . . . 6 ⊢ (( L ‘𝑎) ∪ ∅) = ( L ‘𝑎)
33	31, 32	eqtrdi 2796	. . . . 5 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → ({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝑎)𝑥 = (𝑦 +s 0s )} ∪ {𝑧 ∣ ∃𝑦 ∈ ( L ‘ 0s )𝑧 = (𝑎 +s 𝑦)}) = ( L ‘𝑎))
34		elun2 4206	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ( R ‘𝑎) → 𝑤 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎)))
35		oveq1 7455	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑤 → (𝑏 +s 0s ) = (𝑤 +s 0s ))
36		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑤 → 𝑏 = 𝑤)
37	35, 36	eqeq12d 2756	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑤 → ((𝑏 +s 0s ) = 𝑏 ↔ (𝑤 +s 0s ) = 𝑤))
38	37	rspcva 3633	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎)) ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (𝑤 +s 0s ) = 𝑤)
39	34, 12, 38	syl2anr 596	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑤 ∈ ( R ‘𝑎)) → (𝑤 +s 0s ) = 𝑤)
40	39	eqeq2d 2751	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑤 ∈ ( R ‘𝑎)) → (𝑥 = (𝑤 +s 0s ) ↔ 𝑥 = 𝑤))
41		equcom 2017	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 ↔ 𝑤 = 𝑥)
42	40, 41	bitrdi 287	. . . . . . . . . 10 ⊢ (((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑤 ∈ ( R ‘𝑎)) → (𝑥 = (𝑤 +s 0s ) ↔ 𝑤 = 𝑥))
43	42	rexbidva 3183	. . . . . . . . 9 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (∃𝑤 ∈ ( R ‘𝑎)𝑥 = (𝑤 +s 0s ) ↔ ∃𝑤 ∈ ( R ‘𝑎)𝑤 = 𝑥))
44		risset 3239	. . . . . . . . 9 ⊢ (𝑥 ∈ ( R ‘𝑎) ↔ ∃𝑤 ∈ ( R ‘𝑎)𝑤 = 𝑥)
45	43, 44	bitr4di 289	. . . . . . . 8 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (∃𝑤 ∈ ( R ‘𝑎)𝑥 = (𝑤 +s 0s ) ↔ 𝑥 ∈ ( R ‘𝑎)))
46	45	eqabcdv 2879	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → {𝑥 ∣ ∃𝑤 ∈ ( R ‘𝑎)𝑥 = (𝑤 +s 0s )} = ( R ‘𝑎))
47		rex0 4383	. . . . . . . . . 10 ⊢ ¬ ∃𝑤 ∈ ∅ 𝑧 = (𝑎 +s 𝑤)
48		right0s 27950	. . . . . . . . . . 11 ⊢ ( R ‘ 0s ) = ∅
49	48	rexeqi 3333	. . . . . . . . . 10 ⊢ (∃𝑤 ∈ ( R ‘ 0s )𝑧 = (𝑎 +s 𝑤) ↔ ∃𝑤 ∈ ∅ 𝑧 = (𝑎 +s 𝑤))
50	47, 49	mtbir 323	. . . . . . . . 9 ⊢ ¬ ∃𝑤 ∈ ( R ‘ 0s )𝑧 = (𝑎 +s 𝑤)
51	50	abf 4429	. . . . . . . 8 ⊢ {𝑧 ∣ ∃𝑤 ∈ ( R ‘ 0s )𝑧 = (𝑎 +s 𝑤)} = ∅
52	51	a1i 11	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → {𝑧 ∣ ∃𝑤 ∈ ( R ‘ 0s )𝑧 = (𝑎 +s 𝑤)} = ∅)
53	46, 52	uneq12d 4192	. . . . . 6 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → ({𝑥 ∣ ∃𝑤 ∈ ( R ‘𝑎)𝑥 = (𝑤 +s 0s )} ∪ {𝑧 ∣ ∃𝑤 ∈ ( R ‘ 0s )𝑧 = (𝑎 +s 𝑤)}) = (( R ‘𝑎) ∪ ∅))
54		un0 4417	. . . . . 6 ⊢ (( R ‘𝑎) ∪ ∅) = ( R ‘𝑎)
55	53, 54	eqtrdi 2796	. . . . 5 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → ({𝑥 ∣ ∃𝑤 ∈ ( R ‘𝑎)𝑥 = (𝑤 +s 0s )} ∪ {𝑧 ∣ ∃𝑤 ∈ ( R ‘ 0s )𝑧 = (𝑎 +s 𝑤)}) = ( R ‘𝑎))
56	33, 55	oveq12d 7466	. . . 4 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (({𝑥 ∣ ∃𝑦 ∈ ( L ‘𝑎)𝑥 = (𝑦 +s 0s )} ∪ {𝑧 ∣ ∃𝑦 ∈ ( L ‘ 0s )𝑧 = (𝑎 +s 𝑦)}) |s ({𝑥 ∣ ∃𝑤 ∈ ( R ‘𝑎)𝑥 = (𝑤 +s 0s )} ∪ {𝑧 ∣ ∃𝑤 ∈ ( R ‘ 0s )𝑧 = (𝑎 +s 𝑤)})) = (( L ‘𝑎) |s ( R ‘𝑎)))
57		lrcut 27959	. . . . 5 ⊢ (𝑎 ∈ No → (( L ‘𝑎) |s ( R ‘𝑎)) = 𝑎)
58	57	adantr 480	. . . 4 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (( L ‘𝑎) |s ( R ‘𝑎)) = 𝑎)
59	10, 56, 58	3eqtrd 2784	. . 3 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (𝑎 +s 0s ) = 𝑎)
60	59	ex 412	. 2 ⊢ (𝑎 ∈ No → (∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏 → (𝑎 +s 0s ) = 𝑎))
61	3, 6, 60	noinds 27996	1 ⊢ (𝐴 ∈ No → (𝐴 +s 0s ) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  ∃wrex 3076   ∪ cun 3974  ∅c0 4352  ‘cfv 6573  (class class class)co 7448   No csur 27702   |s cscut 27845   0_s c0s 27885   L cleft 27902   R cright 27903   +_s cadds 28010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-1o 8522  df-2o 8523  df-no 27705  df-slt 27706  df-bday 27707  df-sslt 27844  df-scut 27846  df-0s 27887  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec2 28000  df-adds 28011

This theorem is referenced by:  addsridd  28016  addslid  28019  addsfo  28034  addsgt0d  28065  subsfo  28113  subsid1  28116  precsexlem11  28259  1p1e2s  28418  nohalf  28425