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Theorem addsid1 27279
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
addsid1 (𝐴 No → (𝐴 +s 0s ) = 𝐴)

Proof of Theorem addsid1
Dummy variables 𝑎 𝑏 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7365 . . 3 (𝑎 = 𝑏 → (𝑎 +s 0s ) = (𝑏 +s 0s ))
2 id 22 . . 3 (𝑎 = 𝑏𝑎 = 𝑏)
31, 2eqeq12d 2753 . 2 (𝑎 = 𝑏 → ((𝑎 +s 0s ) = 𝑎 ↔ (𝑏 +s 0s ) = 𝑏))
4 oveq1 7365 . . 3 (𝑎 = 𝐴 → (𝑎 +s 0s ) = (𝐴 +s 0s ))
5 id 22 . . 3 (𝑎 = 𝐴𝑎 = 𝐴)
64, 5eqeq12d 2753 . 2 (𝑎 = 𝐴 → ((𝑎 +s 0s ) = 𝑎 ↔ (𝐴 +s 0s ) = 𝐴))
7 0sno 27168 . . . . . 6 0s No
8 addsval 27277 . . . . . 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)})))
97, 8mpan2 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)})))
109adantr 482 . . . 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 4137 . . . . . . . . . . . . 13 (𝑦 ∈ ( L ‘𝑎) → 𝑦 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎)))
12 simpr 486 . . . . . . . . . . . . 13 ((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏)
13 oveq1 7365 . . . . . . . . . . . . . . 15 (𝑏 = 𝑦 → (𝑏 +s 0s ) = (𝑦 +s 0s ))
14 id 22 . . . . . . . . . . . . . . 15 (𝑏 = 𝑦𝑏 = 𝑦)
1513, 14eqeq12d 2753 . . . . . . . . . . . . . 14 (𝑏 = 𝑦 → ((𝑏 +s 0s ) = 𝑏 ↔ (𝑦 +s 0s ) = 𝑦))
1615rspcva 3580 . . . . . . . . . . . . 13 ((𝑦 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎)) ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (𝑦 +s 0s ) = 𝑦)
1711, 12, 16syl2anr 598 . . . . . . . . . . . 12 (((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑦 ∈ ( L ‘𝑎)) → (𝑦 +s 0s ) = 𝑦)
1817eqeq2d 2748 . . . . . . . . . . 11 (((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑦 ∈ ( L ‘𝑎)) → (𝑥 = (𝑦 +s 0s ) ↔ 𝑥 = 𝑦))
19 equcom 2022 . . . . . . . . . . 11 (𝑥 = 𝑦𝑦 = 𝑥)
2018, 19bitrdi 287 . . . . . . . . . 10 (((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑦 ∈ ( L ‘𝑎)) → (𝑥 = (𝑦 +s 0s ) ↔ 𝑦 = 𝑥))
2120rexbidva 3174 . . . . . . . . 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 3222 . . . . . . . . 9 (𝑥 ∈ ( L ‘𝑎) ↔ ∃𝑦 ∈ ( L ‘𝑎)𝑦 = 𝑥)
2321, 22bitr4di 289 . . . . . . . 8 ((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (∃𝑦 ∈ ( L ‘𝑎)𝑥 = (𝑦 +s 0s ) ↔ 𝑥 ∈ ( L ‘𝑎)))
2423abbi1dv 2873 . . . . . . 7 ((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝑎)𝑥 = (𝑦 +s 0s )} = ( L ‘𝑎))
25 rex0 4318 . . . . . . . . . 10 ¬ ∃𝑦 ∈ ∅ 𝑧 = (𝑎 +s 𝑦)
26 left0s 27225 . . . . . . . . . . 11 ( L ‘ 0s ) = ∅
2726rexeqi 3313 . . . . . . . . . 10 (∃𝑦 ∈ ( L ‘ 0s )𝑧 = (𝑎 +s 𝑦) ↔ ∃𝑦 ∈ ∅ 𝑧 = (𝑎 +s 𝑦))
2825, 27mtbir 323 . . . . . . . . 9 ¬ ∃𝑦 ∈ ( L ‘ 0s )𝑧 = (𝑎 +s 𝑦)
2928abf 4363 . . . . . . . 8 {𝑧 ∣ ∃𝑦 ∈ ( L ‘ 0s )𝑧 = (𝑎 +s 𝑦)} = ∅
3029a1i 11 . . . . . . 7 ((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → {𝑧 ∣ ∃𝑦 ∈ ( L ‘ 0s )𝑧 = (𝑎 +s 𝑦)} = ∅)
3124, 30uneq12d 4125 . . . . . 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 4351 . . . . . 6 (( L ‘𝑎) ∪ ∅) = ( L ‘𝑎)
3331, 32eqtrdi 2793 . . . . 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 4138 . . . . . . . . . . . . 13 (𝑤 ∈ ( R ‘𝑎) → 𝑤 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎)))
35 oveq1 7365 . . . . . . . . . . . . . . 15 (𝑏 = 𝑤 → (𝑏 +s 0s ) = (𝑤 +s 0s ))
36 id 22 . . . . . . . . . . . . . . 15 (𝑏 = 𝑤𝑏 = 𝑤)
3735, 36eqeq12d 2753 . . . . . . . . . . . . . 14 (𝑏 = 𝑤 → ((𝑏 +s 0s ) = 𝑏 ↔ (𝑤 +s 0s ) = 𝑤))
3837rspcva 3580 . . . . . . . . . . . . 13 ((𝑤 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎)) ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (𝑤 +s 0s ) = 𝑤)
3934, 12, 38syl2anr 598 . . . . . . . . . . . 12 (((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑤 ∈ ( R ‘𝑎)) → (𝑤 +s 0s ) = 𝑤)
4039eqeq2d 2748 . . . . . . . . . . 11 (((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑤 ∈ ( R ‘𝑎)) → (𝑥 = (𝑤 +s 0s ) ↔ 𝑥 = 𝑤))
41 equcom 2022 . . . . . . . . . . 11 (𝑥 = 𝑤𝑤 = 𝑥)
4240, 41bitrdi 287 . . . . . . . . . 10 (((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑤 ∈ ( R ‘𝑎)) → (𝑥 = (𝑤 +s 0s ) ↔ 𝑤 = 𝑥))
4342rexbidva 3174 . . . . . . . . 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 3222 . . . . . . . . 9 (𝑥 ∈ ( R ‘𝑎) ↔ ∃𝑤 ∈ ( R ‘𝑎)𝑤 = 𝑥)
4543, 44bitr4di 289 . . . . . . . 8 ((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (∃𝑤 ∈ ( R ‘𝑎)𝑥 = (𝑤 +s 0s ) ↔ 𝑥 ∈ ( R ‘𝑎)))
4645abbi1dv 2873 . . . . . . 7 ((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → {𝑥 ∣ ∃𝑤 ∈ ( R ‘𝑎)𝑥 = (𝑤 +s 0s )} = ( R ‘𝑎))
47 rex0 4318 . . . . . . . . . 10 ¬ ∃𝑤 ∈ ∅ 𝑧 = (𝑎 +s 𝑤)
48 right0s 27226 . . . . . . . . . . 11 ( R ‘ 0s ) = ∅
4948rexeqi 3313 . . . . . . . . . 10 (∃𝑤 ∈ ( R ‘ 0s )𝑧 = (𝑎 +s 𝑤) ↔ ∃𝑤 ∈ ∅ 𝑧 = (𝑎 +s 𝑤))
5047, 49mtbir 323 . . . . . . . . 9 ¬ ∃𝑤 ∈ ( R ‘ 0s )𝑧 = (𝑎 +s 𝑤)
5150abf 4363 . . . . . . . 8 {𝑧 ∣ ∃𝑤 ∈ ( R ‘ 0s )𝑧 = (𝑎 +s 𝑤)} = ∅
5251a1i 11 . . . . . . 7 ((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → {𝑧 ∣ ∃𝑤 ∈ ( R ‘ 0s )𝑧 = (𝑎 +s 𝑤)} = ∅)
5346, 52uneq12d 4125 . . . . . 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 4351 . . . . . 6 (( R ‘𝑎) ∪ ∅) = ( R ‘𝑎)
5553, 54eqtrdi 2793 . . . . 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))
5633, 55oveq12d 7376 . . . 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 27235 . . . . 5 (𝑎 No → (( L ‘𝑎) |s ( R ‘𝑎)) = 𝑎)
5857adantr 482 . . . 4 ((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (( L ‘𝑎) |s ( R ‘𝑎)) = 𝑎)
5910, 56, 583eqtrd 2781 . . 3 ((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (𝑎 +s 0s ) = 𝑎)
6059ex 414 . 2 (𝑎 No → (∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏 → (𝑎 +s 0s ) = 𝑎))
613, 6, 60noinds 27260 1 (𝐴 No → (𝐴 +s 0s ) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  {cab 2714  wral 3065  wrex 3074  cun 3909  c0 4283  cfv 6497  (class class class)co 7358   No csur 26991   |s cscut 27125   0s c0s 27164   L cleft 27178   R cright 27179   +s cadds 27274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-1o 8413  df-2o 8414  df-no 26994  df-slt 26995  df-bday 26996  df-sslt 27124  df-scut 27126  df-0s 27166  df-made 27180  df-old 27181  df-left 27183  df-right 27184  df-norec2 27264  df-adds 27275
This theorem is referenced by:  addsid1d  27280  addsid2  27283  addsfo  27296  subsid1  27358
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