Theorem addsrid 28332

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 7419	. . 3 ⊢ (𝑎 = 𝑏 → (𝑎 +s 0s ) = (𝑏 +s 0s ))
2		id 23	. . 3 ⊢ (𝑎 = 𝑏 → 𝑎 = 𝑏)
3	1, 2	eqeq12d 2777	. 2 ⊢ (𝑎 = 𝑏 → ((𝑎 +s 0s ) = 𝑎 ↔ (𝑏 +s 0s ) = 𝑏))
4		oveq1 7419	. . 3 ⊢ (𝑎 = 𝐴 → (𝑎 +s 0s ) = (𝐴 +s 0s ))
5		id 23	. . 3 ⊢ (𝑎 = 𝐴 → 𝑎 = 𝐴)
6	4, 5	eqeq12d 2777	. 2 ⊢ (𝑎 = 𝐴 → ((𝑎 +s 0s ) = 𝑎 ↔ (𝐴 +s 0s ) = 𝐴))
7		0no 28177	. . . . . 6 ⊢ 0s ∈ No
8		addsval 28330	. . . . . 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 4128	. . . . . . . . . . . . 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 7419	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑦 → (𝑏 +s 0s ) = (𝑦 +s 0s ))
14		id 23	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑦 → 𝑏 = 𝑦)
15	13, 14	eqeq12d 2777	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑦 → ((𝑏 +s 0s ) = 𝑏 ↔ (𝑦 +s 0s ) = 𝑦))
16	15	rspcva 3575	. . . . . . . . . . . . 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 2772	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑦 ∈ ( L ‘𝑎)) → (𝑥 = (𝑦 +s 0s ) ↔ 𝑥 = 𝑦))
19		equcom 2051	. . . . . . . . . . 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 3185	. . . . . . . . 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 3238	. . . . . . . . 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 2895	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝑎)𝑥 = (𝑦 +s 0s )} = ( L ‘𝑎))
25		rex0 4308	. . . . . . . . . 10 ⊢ ¬ ∃𝑦 ∈ ∅ 𝑧 = (𝑎 +s 𝑦)
26		left0s 28261	. . . . . . . . . . 11 ⊢ ( L ‘ 0s ) = ∅
27	26	rexeqi 3319	. . . . . . . . . 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 4364	. . . . . . . 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 4116	. . . . . 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 4344	. . . . . 6 ⊢ (( L ‘𝑎) ∪ ∅) = ( L ‘𝑎)
33	31, 32	eqtrdi 2812	. . . . 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 4129	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ ( R ‘𝑎) → 𝑤 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎)))
35		oveq1 7419	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑤 → (𝑏 +s 0s ) = (𝑤 +s 0s ))
36		id 23	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑤 → 𝑏 = 𝑤)
37	35, 36	eqeq12d 2777	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑤 → ((𝑏 +s 0s ) = 𝑏 ↔ (𝑤 +s 0s ) = 𝑤))
38	37	rspcva 3575	. . . . . . . . . . . . 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 2772	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑤 ∈ ( R ‘𝑎)) → (𝑥 = (𝑤 +s 0s ) ↔ 𝑥 = 𝑤))
41		equcom 2051	. . . . . . . . . . 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 3185	. . . . . . . . 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 3238	. . . . . . . . 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 2895	. . . . . . 7 ⊢ ((𝑎 ∈ No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → {𝑥 ∣ ∃𝑤 ∈ ( R ‘𝑎)𝑥 = (𝑤 +s 0s )} = ( R ‘𝑎))
47		rex0 4308	. . . . . . . . . 10 ⊢ ¬ ∃𝑤 ∈ ∅ 𝑧 = (𝑎 +s 𝑤)
48		right0s 28262	. . . . . . . . . . 11 ⊢ ( R ‘ 0s ) = ∅
49	48	rexeqi 3319	. . . . . . . . . 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 4364	. . . . . . . 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 4116	. . . . . 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 4344	. . . . . 6 ⊢ (( R ‘𝑎) ∪ ∅) = ( R ‘𝑎)
55	53, 54	eqtrdi 2812	. . . . 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 7430	. . . 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 28272	. . . . 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 2800	. . 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 28313	1 ⊢ (𝐴 ∈ No → (𝐴 +s 0s ) = 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {cab 2739  ∀wral 3077  ∃wrex 3087   ∪ cun 3897  ∅c0 4279  ‘cfv 6531  (class class class)co 7412   No csur 27979   |s ccuts 28127   0_s c0s 28173   L cleft 28193   R cright 28194   +_s cadds 28327

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-1o 8460  df-2o 8461  df-no 27982  df-lts 27983  df-bday 27984  df-slts 28126  df-cuts 28128  df-0s 28175  df-made 28195  df-old 28196  df-left 28198  df-right 28199  df-norec2 28317  df-adds 28328

This theorem is used by:  addsridd  28333  addslid  28336  addsfo  28351  addsgt0d  28382  subsfo  28433  subsid1  28436  precsexlem11  28585  twocut  28791