Theorem addsrid 27917

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 7362	. . 3 ⊢ (𝑎 = 𝑏 → (𝑎 +s 0s ) = (𝑏 +s 0s ))
2		id 22	. . 3 ⊢ (𝑎 = 𝑏 → 𝑎 = 𝑏)
3	1, 2	eqeq12d 2749	. 2 ⊢ (𝑎 = 𝑏 → ((𝑎 +s 0s ) = 𝑎 ↔ (𝑏 +s 0s ) = 𝑏))
4		oveq1 7362	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 +s 0s ) = (𝐴 +s 0s ))
5		id 22	. . 3 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
6	4, 5	eqeq12d 2749	. 2 ⊢ (𝑎 = 𝐴 → ((𝑎 +s 0s ) = 𝑎 ↔ (𝐴 +s 0s ) = 𝐴))
7		0sno 27780	. . . . . 6 ⊢ 0s ∈ No
8		addsval 27915	. . . . . 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 691	. . . . 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 4133	. . . . . . . . . . . . 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 7362	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑦 → (𝑏 +s 0s ) = (𝑦 +s 0s ))
14		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑦 → 𝑏 = 𝑦)
15	13, 14	eqeq12d 2749	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑦 → ((𝑏 +s 0s ) = 𝑏 ↔ (𝑦 +s 0s ) = 𝑦))
16	15	rspcva 3572	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎)) ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (𝑦 +s 0s ) = 𝑦)
17	11, 12, 16	syl2anr 597	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑦 ∈ ( L ‘𝑎)) → (𝑦 +s 0s ) = 𝑦)
18	17	eqeq2d 2744	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑦 ∈ ( L ‘𝑎)) → (𝑥 = (𝑦 +s 0s ) ↔ 𝑥 = 𝑦))
19		equcom 2019	. . . . . . . . . . 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 3156	. . . . . . . . 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 3209	. . . . . . . . 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 2867	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝑎)𝑥 = (𝑦 +s 0s )} = ( L ‘𝑎))
25		rex0 4311	. . . . . . . . . 10 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑧 = (𝑎 +s 𝑦)
26		left0s 27848	. . . . . . . . . . 11 ⊢ ( L ‘ 0s ) = ∅
27	26	rexeqi 3293	. . . . . . . . . 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 4357	. . . . . . . 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 4120	. . . . . 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 4345	. . . . . 6 ⊢ (( L ‘𝑎) ∪ ∅) = ( L ‘𝑎)
33	31, 32	eqtrdi 2784	. . . . 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 4134	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ( R ‘𝑎) → 𝑤 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎)))
35		oveq1 7362	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑤 → (𝑏 +s 0s ) = (𝑤 +s 0s ))
36		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑤 → 𝑏 = 𝑤)
37	35, 36	eqeq12d 2749	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑤 → ((𝑏 +s 0s ) = 𝑏 ↔ (𝑤 +s 0s ) = 𝑤))
38	37	rspcva 3572	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎)) ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (𝑤 +s 0s ) = 𝑤)
39	34, 12, 38	syl2anr 597	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑤 ∈ ( R ‘𝑎)) → (𝑤 +s 0s ) = 𝑤)
40	39	eqeq2d 2744	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑤 ∈ ( R ‘𝑎)) → (𝑥 = (𝑤 +s 0s ) ↔ 𝑥 = 𝑤))
41		equcom 2019	. . . . . . . . . . 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 3156	. . . . . . . . 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 3209	. . . . . . . . 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 2867	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → {𝑥 ∣ ∃𝑤 ∈ ( R ‘𝑎)𝑥 = (𝑤 +s 0s )} = ( R ‘𝑎))
47		rex0 4311	. . . . . . . . . 10 ⊢ ¬ ∃𝑤 ∈ ∅ 𝑧 = (𝑎 +s 𝑤)
48		right0s 27849	. . . . . . . . . . 11 ⊢ ( R ‘ 0s ) = ∅
49	48	rexeqi 3293	. . . . . . . . . 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 4357	. . . . . . . 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 4120	. . . . . 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 4345	. . . . . 6 ⊢ (( R ‘𝑎) ∪ ∅) = ( R ‘𝑎)
55	53, 54	eqtrdi 2784	. . . . 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 7373	. . . 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 27859	. . . . 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 2772	. . 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 27898	1 ⊢ (𝐴 ∈ No → (𝐴 +s 0s ) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {cab 2711  ∀wral 3049  ∃wrex 3058   ∪ cun 3897  ∅c0 4284  ‘cfv 6489  (class class class)co 7355   No csur 27588   |s cscut 27732   0_s c0s 27776   L cleft 27796   R cright 27797   +_s cadds 27912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-1o 8394  df-2o 8395  df-no 27591  df-slt 27592  df-bday 27593  df-sslt 27731  df-scut 27733  df-0s 27778  df-made 27798  df-old 27799  df-left 27801  df-right 27802  df-norec2 27902  df-adds 27913

This theorem is referenced by:  addsridd  27918  addslid  27921  addsfo  27936  addsgt0d  27967  subsfo  28015  subsid1  28018  precsexlem11  28165  1p1e2s  28349  twocut  28356