ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  oppr0g GIF version

Theorem oppr0g 13173
Description: Additive identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
Hypotheses
Ref Expression
opprbas.1 𝑂 = (oppr𝑅)
oppr0.2 0 = (0g𝑅)
Assertion
Ref Expression
oppr0g (𝑅𝑉0 = (0g𝑂))

Proof of Theorem oppr0g
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 opprbas.1 . . . . . 6 𝑂 = (oppr𝑅)
2 eqid 2177 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
31, 2opprbasg 13169 . . . . 5 (𝑅𝑉 → (Base‘𝑅) = (Base‘𝑂))
43eleq2d 2247 . . . 4 (𝑅𝑉 → (𝑦 ∈ (Base‘𝑅) ↔ 𝑦 ∈ (Base‘𝑂)))
5 eqid 2177 . . . . . . . . 9 (+g𝑅) = (+g𝑅)
61, 5oppraddg 13170 . . . . . . . 8 (𝑅𝑉 → (+g𝑅) = (+g𝑂))
76oveqd 5888 . . . . . . 7 (𝑅𝑉 → (𝑦(+g𝑅)𝑥) = (𝑦(+g𝑂)𝑥))
87eqeq1d 2186 . . . . . 6 (𝑅𝑉 → ((𝑦(+g𝑅)𝑥) = 𝑥 ↔ (𝑦(+g𝑂)𝑥) = 𝑥))
96oveqd 5888 . . . . . . 7 (𝑅𝑉 → (𝑥(+g𝑅)𝑦) = (𝑥(+g𝑂)𝑦))
109eqeq1d 2186 . . . . . 6 (𝑅𝑉 → ((𝑥(+g𝑅)𝑦) = 𝑥 ↔ (𝑥(+g𝑂)𝑦) = 𝑥))
118, 10anbi12d 473 . . . . 5 (𝑅𝑉 → (((𝑦(+g𝑅)𝑥) = 𝑥 ∧ (𝑥(+g𝑅)𝑦) = 𝑥) ↔ ((𝑦(+g𝑂)𝑥) = 𝑥 ∧ (𝑥(+g𝑂)𝑦) = 𝑥)))
123, 11raleqbidv 2684 . . . 4 (𝑅𝑉 → (∀𝑥 ∈ (Base‘𝑅)((𝑦(+g𝑅)𝑥) = 𝑥 ∧ (𝑥(+g𝑅)𝑦) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝑂)((𝑦(+g𝑂)𝑥) = 𝑥 ∧ (𝑥(+g𝑂)𝑦) = 𝑥)))
134, 12anbi12d 473 . . 3 (𝑅𝑉 → ((𝑦 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑦(+g𝑅)𝑥) = 𝑥 ∧ (𝑥(+g𝑅)𝑦) = 𝑥)) ↔ (𝑦 ∈ (Base‘𝑂) ∧ ∀𝑥 ∈ (Base‘𝑂)((𝑦(+g𝑂)𝑥) = 𝑥 ∧ (𝑥(+g𝑂)𝑦) = 𝑥))))
1413iotabidv 5197 . 2 (𝑅𝑉 → (℩𝑦(𝑦 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑦(+g𝑅)𝑥) = 𝑥 ∧ (𝑥(+g𝑅)𝑦) = 𝑥))) = (℩𝑦(𝑦 ∈ (Base‘𝑂) ∧ ∀𝑥 ∈ (Base‘𝑂)((𝑦(+g𝑂)𝑥) = 𝑥 ∧ (𝑥(+g𝑂)𝑦) = 𝑥))))
15 oppr0.2 . . 3 0 = (0g𝑅)
162, 5, 15grpidvalg 12723 . 2 (𝑅𝑉0 = (℩𝑦(𝑦 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑦(+g𝑅)𝑥) = 𝑥 ∧ (𝑥(+g𝑅)𝑦) = 𝑥))))
171opprex 13167 . . 3 (𝑅𝑉𝑂 ∈ V)
18 eqid 2177 . . . 4 (Base‘𝑂) = (Base‘𝑂)
19 eqid 2177 . . . 4 (+g𝑂) = (+g𝑂)
20 eqid 2177 . . . 4 (0g𝑂) = (0g𝑂)
2118, 19, 20grpidvalg 12723 . . 3 (𝑂 ∈ V → (0g𝑂) = (℩𝑦(𝑦 ∈ (Base‘𝑂) ∧ ∀𝑥 ∈ (Base‘𝑂)((𝑦(+g𝑂)𝑥) = 𝑥 ∧ (𝑥(+g𝑂)𝑦) = 𝑥))))
2217, 21syl 14 . 2 (𝑅𝑉 → (0g𝑂) = (℩𝑦(𝑦 ∈ (Base‘𝑂) ∧ ∀𝑥 ∈ (Base‘𝑂)((𝑦(+g𝑂)𝑥) = 𝑥 ∧ (𝑥(+g𝑂)𝑦) = 𝑥))))
2314, 16, 223eqtr4d 2220 1 (𝑅𝑉0 = (0g𝑂))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wral 2455  Vcvv 2737  cio 5174  cfv 5214  (class class class)co 5871  Basecbs 12453  +gcplusg 12527  0gc0g 12692  opprcoppr 13161
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-sep 4120  ax-nul 4128  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535  ax-cnex 7898  ax-resscn 7899  ax-1cn 7900  ax-1re 7901  ax-icn 7902  ax-addcl 7903  ax-addrcl 7904  ax-mulcl 7905  ax-addcom 7907  ax-addass 7909  ax-i2m1 7912  ax-0lt1 7913  ax-0id 7915  ax-rnegex 7916  ax-pre-ltirr 7919  ax-pre-lttrn 7921  ax-pre-ltadd 7923
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-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-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5176  df-fun 5216  df-fn 5217  df-fv 5222  df-riota 5827  df-ov 5874  df-oprab 5875  df-mpo 5876  df-tpos 6242  df-pnf 7989  df-mnf 7990  df-ltxr 7992  df-inn 8915  df-2 8973  df-3 8974  df-ndx 12456  df-slot 12457  df-base 12459  df-sets 12460  df-plusg 12540  df-mulr 12541  df-0g 12694  df-oppr 13162
This theorem is referenced by:  opprnegg  13175
  Copyright terms: Public domain W3C validator