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Theorem opprnegg 13206
Description: The negative function in an opposite ring. (Contributed by Mario Carneiro, 5-Dec-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
Hypotheses
Ref Expression
opprbas.1 𝑂 = (oppr𝑅)
opprneg.2 𝑁 = (invg𝑅)
Assertion
Ref Expression
opprnegg (𝑅𝑉𝑁 = (invg𝑂))

Proof of Theorem opprnegg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opprbas.1 . . . 4 𝑂 = (oppr𝑅)
2 eqid 2177 . . . 4 (Base‘𝑅) = (Base‘𝑅)
31, 2opprbasg 13200 . . 3 (𝑅𝑉 → (Base‘𝑅) = (Base‘𝑂))
4 eqid 2177 . . . . . . 7 (+g𝑅) = (+g𝑅)
51, 4oppraddg 13201 . . . . . 6 (𝑅𝑉 → (+g𝑅) = (+g𝑂))
65oveqd 5891 . . . . 5 (𝑅𝑉 → (𝑦(+g𝑅)𝑥) = (𝑦(+g𝑂)𝑥))
7 eqid 2177 . . . . . 6 (0g𝑅) = (0g𝑅)
81, 7oppr0g 13204 . . . . 5 (𝑅𝑉 → (0g𝑅) = (0g𝑂))
96, 8eqeq12d 2192 . . . 4 (𝑅𝑉 → ((𝑦(+g𝑅)𝑥) = (0g𝑅) ↔ (𝑦(+g𝑂)𝑥) = (0g𝑂)))
103, 9riotaeqbidv 5833 . . 3 (𝑅𝑉 → (𝑦 ∈ (Base‘𝑅)(𝑦(+g𝑅)𝑥) = (0g𝑅)) = (𝑦 ∈ (Base‘𝑂)(𝑦(+g𝑂)𝑥) = (0g𝑂)))
113, 10mpteq12dv 4085 . 2 (𝑅𝑉 → (𝑥 ∈ (Base‘𝑅) ↦ (𝑦 ∈ (Base‘𝑅)(𝑦(+g𝑅)𝑥) = (0g𝑅))) = (𝑥 ∈ (Base‘𝑂) ↦ (𝑦 ∈ (Base‘𝑂)(𝑦(+g𝑂)𝑥) = (0g𝑂))))
12 opprneg.2 . . 3 𝑁 = (invg𝑅)
132, 4, 7, 12grpinvfvalg 12869 . 2 (𝑅𝑉𝑁 = (𝑥 ∈ (Base‘𝑅) ↦ (𝑦 ∈ (Base‘𝑅)(𝑦(+g𝑅)𝑥) = (0g𝑅))))
141opprex 13198 . . 3 (𝑅𝑉𝑂 ∈ V)
15 eqid 2177 . . . 4 (Base‘𝑂) = (Base‘𝑂)
16 eqid 2177 . . . 4 (+g𝑂) = (+g𝑂)
17 eqid 2177 . . . 4 (0g𝑂) = (0g𝑂)
18 eqid 2177 . . . 4 (invg𝑂) = (invg𝑂)
1915, 16, 17, 18grpinvfvalg 12869 . . 3 (𝑂 ∈ V → (invg𝑂) = (𝑥 ∈ (Base‘𝑂) ↦ (𝑦 ∈ (Base‘𝑂)(𝑦(+g𝑂)𝑥) = (0g𝑂))))
2014, 19syl 14 . 2 (𝑅𝑉 → (invg𝑂) = (𝑥 ∈ (Base‘𝑂) ↦ (𝑦 ∈ (Base‘𝑂)(𝑦(+g𝑂)𝑥) = (0g𝑂))))
2111, 13, 203eqtr4d 2220 1 (𝑅𝑉𝑁 = (invg𝑂))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  Vcvv 2737  cmpt 4064  cfv 5216  crio 5829  (class class class)co 5874  Basecbs 12456  +gcplusg 12530  0gc0g 12695  invgcminusg 12832  opprcoppr 13192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-pre-ltirr 7922  ax-pre-lttrn 7924  ax-pre-ltadd 7926
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-tpos 6245  df-pnf 7992  df-mnf 7993  df-ltxr 7995  df-inn 8918  df-2 8976  df-3 8977  df-ndx 12459  df-slot 12460  df-base 12462  df-sets 12463  df-plusg 12543  df-mulr 12544  df-0g 12697  df-minusg 12835  df-oppr 13193
This theorem is referenced by:  unitnegcl  13252
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