ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  oppr1g GIF version

Theorem oppr1g 13077
Description: Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
Hypotheses
Ref Expression
opprbas.1 𝑂 = (oppr𝑅)
oppr1.2 1 = (1r𝑅)
Assertion
Ref Expression
oppr1g (𝑅𝑉1 = (1r𝑂))

Proof of Theorem oppr1g
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2177 . . . . . . . . . . 11 (Base‘𝑅) = (Base‘𝑅)
2 eqid 2177 . . . . . . . . . . 11 (.r𝑅) = (.r𝑅)
3 opprbas.1 . . . . . . . . . . 11 𝑂 = (oppr𝑅)
4 eqid 2177 . . . . . . . . . . 11 (.r𝑂) = (.r𝑂)
51, 2, 3, 4opprmulg 13068 . . . . . . . . . 10 ((𝑅𝑉𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r𝑂)𝑦) = (𝑦(.r𝑅)𝑥))
653expa 1203 . . . . . . . . 9 (((𝑅𝑉𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑥(.r𝑂)𝑦) = (𝑦(.r𝑅)𝑥))
76eqeq1d 2186 . . . . . . . 8 (((𝑅𝑉𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑥(.r𝑂)𝑦) = 𝑦 ↔ (𝑦(.r𝑅)𝑥) = 𝑦))
8 simpll 527 . . . . . . . . . 10 (((𝑅𝑉𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑅𝑉)
9 simpr 110 . . . . . . . . . 10 (((𝑅𝑉𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑦 ∈ (Base‘𝑅))
10 simplr 528 . . . . . . . . . 10 (((𝑅𝑉𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → 𝑥 ∈ (Base‘𝑅))
111, 2, 3, 4opprmulg 13068 . . . . . . . . . 10 ((𝑅𝑉𝑦 ∈ (Base‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑦(.r𝑂)𝑥) = (𝑥(.r𝑅)𝑦))
128, 9, 10, 11syl3anc 1238 . . . . . . . . 9 (((𝑅𝑉𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (𝑦(.r𝑂)𝑥) = (𝑥(.r𝑅)𝑦))
1312eqeq1d 2186 . . . . . . . 8 (((𝑅𝑉𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → ((𝑦(.r𝑂)𝑥) = 𝑦 ↔ (𝑥(.r𝑅)𝑦) = 𝑦))
147, 13anbi12d 473 . . . . . . 7 (((𝑅𝑉𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (((𝑥(.r𝑂)𝑦) = 𝑦 ∧ (𝑦(.r𝑂)𝑥) = 𝑦) ↔ ((𝑦(.r𝑅)𝑥) = 𝑦 ∧ (𝑥(.r𝑅)𝑦) = 𝑦)))
1514biancomd 271 . . . . . 6 (((𝑅𝑉𝑥 ∈ (Base‘𝑅)) ∧ 𝑦 ∈ (Base‘𝑅)) → (((𝑥(.r𝑂)𝑦) = 𝑦 ∧ (𝑦(.r𝑂)𝑥) = 𝑦) ↔ ((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦)))
1615ralbidva 2473 . . . . 5 ((𝑅𝑉𝑥 ∈ (Base‘𝑅)) → (∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑂)𝑦) = 𝑦 ∧ (𝑦(.r𝑂)𝑥) = 𝑦) ↔ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦)))
1716riotabidva 5841 . . . 4 (𝑅𝑉 → (𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑂)𝑦) = 𝑦 ∧ (𝑦(.r𝑂)𝑥) = 𝑦)) = (𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦)))
18 df-riota 5825 . . . 4 (𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑂)𝑦) = 𝑦 ∧ (𝑦(.r𝑂)𝑥) = 𝑦)) = (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑂)𝑦) = 𝑦 ∧ (𝑦(.r𝑂)𝑥) = 𝑦)))
19 df-riota 5825 . . . 4 (𝑥 ∈ (Base‘𝑅)∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦)) = (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦)))
2017, 18, 193eqtr3g 2233 . . 3 (𝑅𝑉 → (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑂)𝑦) = 𝑦 ∧ (𝑦(.r𝑂)𝑥) = 𝑦))) = (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦))))
213opprex 13070 . . . . 5 (𝑅𝑉𝑂 ∈ V)
22 eqid 2177 . . . . . 6 (mulGrp‘𝑂) = (mulGrp‘𝑂)
2322mgpex 12962 . . . . 5 (𝑂 ∈ V → (mulGrp‘𝑂) ∈ V)
24 eqid 2177 . . . . . 6 (Base‘(mulGrp‘𝑂)) = (Base‘(mulGrp‘𝑂))
25 eqid 2177 . . . . . 6 (+g‘(mulGrp‘𝑂)) = (+g‘(mulGrp‘𝑂))
26 eqid 2177 . . . . . 6 (0g‘(mulGrp‘𝑂)) = (0g‘(mulGrp‘𝑂))
2724, 25, 26grpidvalg 12684 . . . . 5 ((mulGrp‘𝑂) ∈ V → (0g‘(mulGrp‘𝑂)) = (℩𝑥(𝑥 ∈ (Base‘(mulGrp‘𝑂)) ∧ ∀𝑦 ∈ (Base‘(mulGrp‘𝑂))((𝑥(+g‘(mulGrp‘𝑂))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑂))𝑥) = 𝑦))))
2821, 23, 273syl 17 . . . 4 (𝑅𝑉 → (0g‘(mulGrp‘𝑂)) = (℩𝑥(𝑥 ∈ (Base‘(mulGrp‘𝑂)) ∧ ∀𝑦 ∈ (Base‘(mulGrp‘𝑂))((𝑥(+g‘(mulGrp‘𝑂))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑂))𝑥) = 𝑦))))
293, 1opprbasg 13072 . . . . . . . 8 (𝑅𝑉 → (Base‘𝑅) = (Base‘𝑂))
30 eqid 2177 . . . . . . . . . 10 (Base‘𝑂) = (Base‘𝑂)
3122, 30mgpbasg 12963 . . . . . . . . 9 (𝑂 ∈ V → (Base‘𝑂) = (Base‘(mulGrp‘𝑂)))
3221, 31syl 14 . . . . . . . 8 (𝑅𝑉 → (Base‘𝑂) = (Base‘(mulGrp‘𝑂)))
3329, 32eqtrd 2210 . . . . . . 7 (𝑅𝑉 → (Base‘𝑅) = (Base‘(mulGrp‘𝑂)))
3433eleq2d 2247 . . . . . 6 (𝑅𝑉 → (𝑥 ∈ (Base‘𝑅) ↔ 𝑥 ∈ (Base‘(mulGrp‘𝑂))))
3522, 4mgpplusgg 12961 . . . . . . . . . . 11 (𝑂 ∈ V → (.r𝑂) = (+g‘(mulGrp‘𝑂)))
3621, 35syl 14 . . . . . . . . . 10 (𝑅𝑉 → (.r𝑂) = (+g‘(mulGrp‘𝑂)))
3736oveqd 5886 . . . . . . . . 9 (𝑅𝑉 → (𝑥(.r𝑂)𝑦) = (𝑥(+g‘(mulGrp‘𝑂))𝑦))
3837eqeq1d 2186 . . . . . . . 8 (𝑅𝑉 → ((𝑥(.r𝑂)𝑦) = 𝑦 ↔ (𝑥(+g‘(mulGrp‘𝑂))𝑦) = 𝑦))
3936oveqd 5886 . . . . . . . . 9 (𝑅𝑉 → (𝑦(.r𝑂)𝑥) = (𝑦(+g‘(mulGrp‘𝑂))𝑥))
4039eqeq1d 2186 . . . . . . . 8 (𝑅𝑉 → ((𝑦(.r𝑂)𝑥) = 𝑦 ↔ (𝑦(+g‘(mulGrp‘𝑂))𝑥) = 𝑦))
4138, 40anbi12d 473 . . . . . . 7 (𝑅𝑉 → (((𝑥(.r𝑂)𝑦) = 𝑦 ∧ (𝑦(.r𝑂)𝑥) = 𝑦) ↔ ((𝑥(+g‘(mulGrp‘𝑂))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑂))𝑥) = 𝑦)))
4233, 41raleqbidv 2684 . . . . . 6 (𝑅𝑉 → (∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑂)𝑦) = 𝑦 ∧ (𝑦(.r𝑂)𝑥) = 𝑦) ↔ ∀𝑦 ∈ (Base‘(mulGrp‘𝑂))((𝑥(+g‘(mulGrp‘𝑂))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑂))𝑥) = 𝑦)))
4334, 42anbi12d 473 . . . . 5 (𝑅𝑉 → ((𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑂)𝑦) = 𝑦 ∧ (𝑦(.r𝑂)𝑥) = 𝑦)) ↔ (𝑥 ∈ (Base‘(mulGrp‘𝑂)) ∧ ∀𝑦 ∈ (Base‘(mulGrp‘𝑂))((𝑥(+g‘(mulGrp‘𝑂))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑂))𝑥) = 𝑦))))
4443iotabidv 5195 . . . 4 (𝑅𝑉 → (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑂)𝑦) = 𝑦 ∧ (𝑦(.r𝑂)𝑥) = 𝑦))) = (℩𝑥(𝑥 ∈ (Base‘(mulGrp‘𝑂)) ∧ ∀𝑦 ∈ (Base‘(mulGrp‘𝑂))((𝑥(+g‘(mulGrp‘𝑂))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑂))𝑥) = 𝑦))))
4528, 44eqtr4d 2213 . . 3 (𝑅𝑉 → (0g‘(mulGrp‘𝑂)) = (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑂)𝑦) = 𝑦 ∧ (𝑦(.r𝑂)𝑥) = 𝑦))))
46 eqid 2177 . . . . . 6 (mulGrp‘𝑅) = (mulGrp‘𝑅)
4746mgpex 12962 . . . . 5 (𝑅𝑉 → (mulGrp‘𝑅) ∈ V)
48 eqid 2177 . . . . . 6 (Base‘(mulGrp‘𝑅)) = (Base‘(mulGrp‘𝑅))
49 eqid 2177 . . . . . 6 (+g‘(mulGrp‘𝑅)) = (+g‘(mulGrp‘𝑅))
50 eqid 2177 . . . . . 6 (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅))
5148, 49, 50grpidvalg 12684 . . . . 5 ((mulGrp‘𝑅) ∈ V → (0g‘(mulGrp‘𝑅)) = (℩𝑥(𝑥 ∈ (Base‘(mulGrp‘𝑅)) ∧ ∀𝑦 ∈ (Base‘(mulGrp‘𝑅))((𝑥(+g‘(mulGrp‘𝑅))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑅))𝑥) = 𝑦))))
5247, 51syl 14 . . . 4 (𝑅𝑉 → (0g‘(mulGrp‘𝑅)) = (℩𝑥(𝑥 ∈ (Base‘(mulGrp‘𝑅)) ∧ ∀𝑦 ∈ (Base‘(mulGrp‘𝑅))((𝑥(+g‘(mulGrp‘𝑅))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑅))𝑥) = 𝑦))))
5346, 1mgpbasg 12963 . . . . . . 7 (𝑅𝑉 → (Base‘𝑅) = (Base‘(mulGrp‘𝑅)))
5453eleq2d 2247 . . . . . 6 (𝑅𝑉 → (𝑥 ∈ (Base‘𝑅) ↔ 𝑥 ∈ (Base‘(mulGrp‘𝑅))))
5546, 2mgpplusgg 12961 . . . . . . . . . 10 (𝑅𝑉 → (.r𝑅) = (+g‘(mulGrp‘𝑅)))
5655oveqd 5886 . . . . . . . . 9 (𝑅𝑉 → (𝑥(.r𝑅)𝑦) = (𝑥(+g‘(mulGrp‘𝑅))𝑦))
5756eqeq1d 2186 . . . . . . . 8 (𝑅𝑉 → ((𝑥(.r𝑅)𝑦) = 𝑦 ↔ (𝑥(+g‘(mulGrp‘𝑅))𝑦) = 𝑦))
5855oveqd 5886 . . . . . . . . 9 (𝑅𝑉 → (𝑦(.r𝑅)𝑥) = (𝑦(+g‘(mulGrp‘𝑅))𝑥))
5958eqeq1d 2186 . . . . . . . 8 (𝑅𝑉 → ((𝑦(.r𝑅)𝑥) = 𝑦 ↔ (𝑦(+g‘(mulGrp‘𝑅))𝑥) = 𝑦))
6057, 59anbi12d 473 . . . . . . 7 (𝑅𝑉 → (((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦) ↔ ((𝑥(+g‘(mulGrp‘𝑅))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑅))𝑥) = 𝑦)))
6153, 60raleqbidv 2684 . . . . . 6 (𝑅𝑉 → (∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦) ↔ ∀𝑦 ∈ (Base‘(mulGrp‘𝑅))((𝑥(+g‘(mulGrp‘𝑅))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑅))𝑥) = 𝑦)))
6254, 61anbi12d 473 . . . . 5 (𝑅𝑉 → ((𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦)) ↔ (𝑥 ∈ (Base‘(mulGrp‘𝑅)) ∧ ∀𝑦 ∈ (Base‘(mulGrp‘𝑅))((𝑥(+g‘(mulGrp‘𝑅))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑅))𝑥) = 𝑦))))
6362iotabidv 5195 . . . 4 (𝑅𝑉 → (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦))) = (℩𝑥(𝑥 ∈ (Base‘(mulGrp‘𝑅)) ∧ ∀𝑦 ∈ (Base‘(mulGrp‘𝑅))((𝑥(+g‘(mulGrp‘𝑅))𝑦) = 𝑦 ∧ (𝑦(+g‘(mulGrp‘𝑅))𝑥) = 𝑦))))
6452, 63eqtr4d 2213 . . 3 (𝑅𝑉 → (0g‘(mulGrp‘𝑅)) = (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r𝑅)𝑦) = 𝑦 ∧ (𝑦(.r𝑅)𝑥) = 𝑦))))
6520, 45, 643eqtr4d 2220 . 2 (𝑅𝑉 → (0g‘(mulGrp‘𝑂)) = (0g‘(mulGrp‘𝑅)))
66 eqid 2177 . . . 4 (1r𝑂) = (1r𝑂)
6722, 66ringidvalg 12970 . . 3 (𝑂 ∈ V → (1r𝑂) = (0g‘(mulGrp‘𝑂)))
6821, 67syl 14 . 2 (𝑅𝑉 → (1r𝑂) = (0g‘(mulGrp‘𝑂)))
69 oppr1.2 . . 3 1 = (1r𝑅)
7046, 69ringidvalg 12970 . 2 (𝑅𝑉1 = (0g‘(mulGrp‘𝑅)))
7165, 68, 703eqtr4rd 2221 1 (𝑅𝑉1 = (1r𝑂))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  wral 2455  Vcvv 2737  cio 5172  cfv 5212  crio 5824  (class class class)co 5869  Basecbs 12445  +gcplusg 12518  .rcmulr 12519  0gc0g 12653  mulGrpcmgp 12957  1rcur 12968  opprcoppr 13064
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 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-pre-ltirr 7914  ax-pre-lttrn 7916  ax-pre-ltadd 7918
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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-tpos 6240  df-pnf 7984  df-mnf 7985  df-ltxr 7987  df-inn 8909  df-2 8967  df-3 8968  df-ndx 12448  df-slot 12449  df-base 12451  df-sets 12452  df-plusg 12531  df-mulr 12532  df-0g 12655  df-mgp 12958  df-ur 12969  df-oppr 13065
This theorem is referenced by:  opprunitd  13104
  Copyright terms: Public domain W3C validator