Theorem addsrid 28355

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 7427	. . 3 ⊢ (𝑎 = 𝑏 → (𝑎 +s 0s ) = (𝑏 +s 0s ))
2		id 23	. . 3 ⊢ (𝑎 = 𝑏 → 𝑎 = 𝑏)
3	1, 2	eqeq12d 2776	. 2 ⊢ (𝑎 = 𝑏 → ((𝑎 +s 0s ) = 𝑎 ↔ (𝑏 +s 0s ) = 𝑏))
4		oveq1 7427	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 +s 0s ) = (𝐴 +s 0s ))
5		id 23	. . 3 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
6	4, 5	eqeq12d 2776	. 2 ⊢ (𝑎 = 𝐴 → ((𝑎 +s 0s ) = 𝑎 ↔ (𝐴 +s 0s ) = 𝐴))
7		0no 28200	. . . . . 6 ⊢ 0s ∈ No
8		addsval 28353	. . . . . 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 704	. . . . 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 486	. . . 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 4180	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (L‘𝑎) → 𝑦 ∈ ((L‘𝑎) ∪ (R‘𝑎)))
12		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))(𝑏 +s 0s ) = 𝑏) → ∀𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))(𝑏 +s 0s ) = 𝑏)
13		oveq1 7427	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑦 → (𝑏 +s 0s ) = (𝑦 +s 0s ))
14		id 23	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑦 → 𝑏 = 𝑦)
15	13, 14	eqeq12d 2776	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑦 → ((𝑏 +s 0s ) = 𝑏 ↔ (𝑦 +s 0s ) = 𝑦))
16	15	rspcva 3574	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ((L‘𝑎) ∪ (R‘𝑎)) ∧ ∀𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))(𝑏 +s 0s ) = 𝑏) → (𝑦 +s 0s ) = 𝑦)
17	11, 12, 16	syl2anr 609	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ No ∧ ∀𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑦 ∈ (L‘𝑎)) → (𝑦 +s 0s ) = 𝑦)
18	17	eqeq2d 2771	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ No ∧ ∀𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑦 ∈ (L‘𝑎)) → (𝑥 = (𝑦 +s 0s ) ↔ 𝑥 = 𝑦))
19		equcom 2050	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 ↔ 𝑦 = 𝑥)
20	18, 19	bitrdi 290	. . . . . . . . . 10 ⊢ (((𝑎 ∈ No ∧ ∀𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑦 ∈ (L‘𝑎)) → (𝑥 = (𝑦 +s 0s ) ↔ 𝑦 = 𝑥))
21	20	rexbidva 3184	. . . . . . . . 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 3237	. . . . . . . . 9 ⊢ (𝑥 ∈ (L‘𝑎) ↔ ∃𝑦 ∈ (L‘𝑎)𝑦 = 𝑥)
23	21, 22	bitr4di 292	. . . . . . . 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 2894	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))(𝑏 +s 0s ) = 𝑏) → {𝑥 ∣ ∃𝑦 ∈ (L‘𝑎)𝑥 = (𝑦 +s 0s )} = (L‘𝑎))
25		rex0 4307	. . . . . . . . . 10 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑧 = (𝑎 +s 𝑦)
26		left0s 28284	. . . . . . . . . . 11 ⊢ (L‘ 0s ) = ∅
27	26	rexeqi 3318	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ (L‘ 0s )𝑧 = (𝑎 +s 𝑦) ↔ ∃𝑦 ∈ ∅ 𝑧 = (𝑎 +s 𝑦))
28	25, 27	mtbir 326	. . . . . . . . 9 ⊢ ¬ ∃𝑦 ∈ (L‘ 0s )𝑧 = (𝑎 +s 𝑦)
29	28	abf 4363	. . . . . . . 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 4168	. . . . . 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 4343	. . . . . 6 ⊢ ((L‘𝑎) ∪ ∅) = (L‘𝑎)
33	31, 32	eqtrdi 2811	. . . . 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 4181	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (R‘𝑎) → 𝑤 ∈ ((L‘𝑎) ∪ (R‘𝑎)))
35		oveq1 7427	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑤 → (𝑏 +s 0s ) = (𝑤 +s 0s ))
36		id 23	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑤 → 𝑏 = 𝑤)
37	35, 36	eqeq12d 2776	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑤 → ((𝑏 +s 0s ) = 𝑏 ↔ (𝑤 +s 0s ) = 𝑤))
38	37	rspcva 3574	. . . . . . . . . . . . 13 ⊢ ((𝑤 ∈ ((L‘𝑎) ∪ (R‘𝑎)) ∧ ∀𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))(𝑏 +s 0s ) = 𝑏) → (𝑤 +s 0s ) = 𝑤)
39	34, 12, 38	syl2anr 609	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ No ∧ ∀𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑤 ∈ (R‘𝑎)) → (𝑤 +s 0s ) = 𝑤)
40	39	eqeq2d 2771	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ No ∧ ∀𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑤 ∈ (R‘𝑎)) → (𝑥 = (𝑤 +s 0s ) ↔ 𝑥 = 𝑤))
41		equcom 2050	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 ↔ 𝑤 = 𝑥)
42	40, 41	bitrdi 290	. . . . . . . . . 10 ⊢ (((𝑎 ∈ No ∧ ∀𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑤 ∈ (R‘𝑎)) → (𝑥 = (𝑤 +s 0s ) ↔ 𝑤 = 𝑥))
43	42	rexbidva 3184	. . . . . . . . 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 3237	. . . . . . . . 9 ⊢ (𝑥 ∈ (R‘𝑎) ↔ ∃𝑤 ∈ (R‘𝑎)𝑤 = 𝑥)
45	43, 44	bitr4di 292	. . . . . . . 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 2894	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))(𝑏 +s 0s ) = 𝑏) → {𝑥 ∣ ∃𝑤 ∈ (R‘𝑎)𝑥 = (𝑤 +s 0s )} = (R‘𝑎))
47		rex0 4307	. . . . . . . . . 10 ⊢ ¬ ∃𝑤 ∈ ∅ 𝑧 = (𝑎 +s 𝑤)
48		right0s 28285	. . . . . . . . . . 11 ⊢ (R‘ 0s ) = ∅
49	48	rexeqi 3318	. . . . . . . . . 10 ⊢ (∃𝑤 ∈ (R‘ 0s )𝑧 = (𝑎 +s 𝑤) ↔ ∃𝑤 ∈ ∅ 𝑧 = (𝑎 +s 𝑤))
50	47, 49	mtbir 326	. . . . . . . . 9 ⊢ ¬ ∃𝑤 ∈ (R‘ 0s )𝑧 = (𝑎 +s 𝑤)
51	50	abf 4363	. . . . . . . 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 4168	. . . . . 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 4343	. . . . . 6 ⊢ ((R‘𝑎) ∪ ∅) = (R‘𝑎)
55	53, 54	eqtrdi 2811	. . . . 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 7438	. . . 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 28295	. . . . 5 ⊢ (𝑎 ∈ No → ((L‘𝑎) |s (R‘𝑎)) = 𝑎)
58	57	adantr 486	. . . 4 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))(𝑏 +s 0s ) = 𝑏) → ((L‘𝑎) |s (R‘𝑎)) = 𝑎)
59	10, 56, 58	3eqtrd 2799	. . 3 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))(𝑏 +s 0s ) = 𝑏) → (𝑎 +s 0s ) = 𝑎)
60	59	ex 418	. 2 ⊢ (𝑎 ∈ No → (∀𝑏 ∈ ((L‘𝑎) ∪ (R‘𝑎))(𝑏 +s 0s ) = 𝑏 → (𝑎 +s 0s ) = 𝑎))
61	3, 6, 60	noinds 28336	1 ⊢ (𝐴 ∈ No → (𝐴 +s 0s ) = 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086   ∪ cun 4149  ∅c0 4278  ‘cfv 6538  (class class class)co 7420  Nocsur 28002   |s ccuts 28150   0_s c0s 28196  Lcleft 28216  Rcright 28217   +_s cadds 28350

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 4585  df-pr 4587  df-tp 4589  df-op 4835  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 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-slts 28149  df-cuts 28151  df-0s 28198  df-made 28218  df-old 28219  df-left 28221  df-right 28222  df-norec2 28340  df-adds 28351

This theorem is used by:  addsridd  28356  addslid  28359  addsfo  28374  addsgt0d  28405  subsfo  28456  subsid1  28459  precsexlem11  28608  twocut  28814