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Theorem addsid1 33765
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 7177 . . 3 (𝑎 = 𝑏 → (𝑎 +s 0s ) = (𝑏 +s 0s ))
2 id 22 . . 3 (𝑎 = 𝑏𝑎 = 𝑏)
31, 2eqeq12d 2754 . 2 (𝑎 = 𝑏 → ((𝑎 +s 0s ) = 𝑎 ↔ (𝑏 +s 0s ) = 𝑏))
4 oveq1 7177 . . 3 (𝑎 = 𝐴 → (𝑎 +s 0s ) = (𝐴 +s 0s ))
5 id 22 . . 3 (𝑎 = 𝐴𝑎 = 𝐴)
64, 5eqeq12d 2754 . 2 (𝑎 = 𝐴 → ((𝑎 +s 0s ) = 𝑎 ↔ (𝐴 +s 0s ) = 𝐴))
7 0sno 33661 . . . . . 6 0s ∈ No
8 addsov 33764 . . . . . 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 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)})))
109adantr 484 . . . 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 4066 . . . . . . . . . . . . 13 (𝑦 ∈ ( L ‘𝑎) → 𝑦 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎)))
12 simpr 488 . . . . . . . . . . . . 13 ((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏)
13 oveq1 7177 . . . . . . . . . . . . . . 15 (𝑏 = 𝑦 → (𝑏 +s 0s ) = (𝑦 +s 0s ))
14 id 22 . . . . . . . . . . . . . . 15 (𝑏 = 𝑦𝑏 = 𝑦)
1513, 14eqeq12d 2754 . . . . . . . . . . . . . 14 (𝑏 = 𝑦 → ((𝑏 +s 0s ) = 𝑏 ↔ (𝑦 +s 0s ) = 𝑦))
1615rspcva 3524 . . . . . . . . . . . . 13 ((𝑦 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎)) ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (𝑦 +s 0s ) = 𝑦)
1711, 12, 16syl2anr 600 . . . . . . . . . . . 12 (((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑦 ∈ ( L ‘𝑎)) → (𝑦 +s 0s ) = 𝑦)
1817eqeq2d 2749 . . . . . . . . . . 11 (((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑦 ∈ ( L ‘𝑎)) → (𝑥 = (𝑦 +s 0s ) ↔ 𝑥 = 𝑦))
19 equcom 2030 . . . . . . . . . . 11 (𝑥 = 𝑦𝑦 = 𝑥)
2018, 19bitrdi 290 . . . . . . . . . 10 (((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑦 ∈ ( L ‘𝑎)) → (𝑥 = (𝑦 +s 0s ) ↔ 𝑦 = 𝑥))
2120rexbidva 3206 . . . . . . . . 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 3177 . . . . . . . . 9 (𝑥 ∈ ( L ‘𝑎) ↔ ∃𝑦 ∈ ( L ‘𝑎)𝑦 = 𝑥)
2321, 22bitr4di 292 . . . . . . . 8 ((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (∃𝑦 ∈ ( L ‘𝑎)𝑥 = (𝑦 +s 0s ) ↔ 𝑥 ∈ ( L ‘𝑎)))
2423abbi1dv 2870 . . . . . . 7 ((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → {𝑥 ∣ ∃𝑦 ∈ ( L ‘𝑎)𝑥 = (𝑦 +s 0s )} = ( L ‘𝑎))
25 rex0 4246 . . . . . . . . . 10 ¬ ∃𝑦 ∈ ∅ 𝑧 = (𝑎 +s 𝑦)
26 left0s 33713 . . . . . . . . . . 11 ( L ‘ 0s ) = ∅
2726rexeqi 3315 . . . . . . . . . 10 (∃𝑦 ∈ ( L ‘ 0s )𝑧 = (𝑎 +s 𝑦) ↔ ∃𝑦 ∈ ∅ 𝑧 = (𝑎 +s 𝑦))
2825, 27mtbir 326 . . . . . . . . 9 ¬ ∃𝑦 ∈ ( L ‘ 0s )𝑧 = (𝑎 +s 𝑦)
2928abf 4291 . . . . . . . 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 4054 . . . . . 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 4279 . . . . . 6 (( L ‘𝑎) ∪ ∅) = ( L ‘𝑎)
3331, 32eqtrdi 2789 . . . . 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 4067 . . . . . . . . . . . . 13 (𝑤 ∈ ( R ‘𝑎) → 𝑤 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎)))
35 oveq1 7177 . . . . . . . . . . . . . . 15 (𝑏 = 𝑤 → (𝑏 +s 0s ) = (𝑤 +s 0s ))
36 id 22 . . . . . . . . . . . . . . 15 (𝑏 = 𝑤𝑏 = 𝑤)
3735, 36eqeq12d 2754 . . . . . . . . . . . . . 14 (𝑏 = 𝑤 → ((𝑏 +s 0s ) = 𝑏 ↔ (𝑤 +s 0s ) = 𝑤))
3837rspcva 3524 . . . . . . . . . . . . 13 ((𝑤 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎)) ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (𝑤 +s 0s ) = 𝑤)
3934, 12, 38syl2anr 600 . . . . . . . . . . . 12 (((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑤 ∈ ( R ‘𝑎)) → (𝑤 +s 0s ) = 𝑤)
4039eqeq2d 2749 . . . . . . . . . . 11 (((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑤 ∈ ( R ‘𝑎)) → (𝑥 = (𝑤 +s 0s ) ↔ 𝑥 = 𝑤))
41 equcom 2030 . . . . . . . . . . 11 (𝑥 = 𝑤𝑤 = 𝑥)
4240, 41bitrdi 290 . . . . . . . . . 10 (((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) ∧ 𝑤 ∈ ( R ‘𝑎)) → (𝑥 = (𝑤 +s 0s ) ↔ 𝑤 = 𝑥))
4342rexbidva 3206 . . . . . . . . 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 3177 . . . . . . . . 9 (𝑥 ∈ ( R ‘𝑎) ↔ ∃𝑤 ∈ ( R ‘𝑎)𝑤 = 𝑥)
4543, 44bitr4di 292 . . . . . . . 8 ((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (∃𝑤 ∈ ( R ‘𝑎)𝑥 = (𝑤 +s 0s ) ↔ 𝑥 ∈ ( R ‘𝑎)))
4645abbi1dv 2870 . . . . . . 7 ((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → {𝑥 ∣ ∃𝑤 ∈ ( R ‘𝑎)𝑥 = (𝑤 +s 0s )} = ( R ‘𝑎))
47 rex0 4246 . . . . . . . . . 10 ¬ ∃𝑤 ∈ ∅ 𝑧 = (𝑎 +s 𝑤)
48 right0s 33714 . . . . . . . . . . 11 ( R ‘ 0s ) = ∅
4948rexeqi 3315 . . . . . . . . . 10 (∃𝑤 ∈ ( R ‘ 0s )𝑧 = (𝑎 +s 𝑤) ↔ ∃𝑤 ∈ ∅ 𝑧 = (𝑎 +s 𝑤))
5047, 49mtbir 326 . . . . . . . . 9 ¬ ∃𝑤 ∈ ( R ‘ 0s )𝑧 = (𝑎 +s 𝑤)
5150abf 4291 . . . . . . . 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 4054 . . . . . 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 4279 . . . . . 6 (( R ‘𝑎) ∪ ∅) = ( R ‘𝑎)
5553, 54eqtrdi 2789 . . . . 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 7188 . . . 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 33721 . . . . 5 (𝑎 No → (( L ‘𝑎) |s ( R ‘𝑎)) = 𝑎)
5857adantr 484 . . . 4 ((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (( L ‘𝑎) |s ( R ‘𝑎)) = 𝑎)
5910, 56, 583eqtrd 2777 . . 3 ((𝑎 No ∧ ∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏) → (𝑎 +s 0s ) = 𝑎)
6059ex 416 . 2 (𝑎 No → (∀𝑏 ∈ (( L ‘𝑎) ∪ ( R ‘𝑎))(𝑏 +s 0s ) = 𝑏 → (𝑎 +s 0s ) = 𝑎))
613, 6, 60noinds 33739 1 (𝐴 No → (𝐴 +s 0s ) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  {cab 2716  wral 3053  wrex 3054  cun 3841  c0 4211  cfv 6339  (class class class)co 7170   No csur 33484   |s cscut 33618   0s c0s 33657   L cleft 33670   R cright 33671   +s cadds 33754
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-1o 8131  df-2o 8132  df-frecs 33436  df-no 33487  df-slt 33488  df-bday 33489  df-sslt 33617  df-scut 33619  df-0s 33659  df-made 33672  df-old 33673  df-left 33675  df-right 33676  df-norec2 33743  df-adds 33757
This theorem is referenced by:  addsid1d  33766
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