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Mirrors > Home > ILE Home > Th. List > oppr0g | GIF version |
Description: Additive identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) |
Ref | Expression |
---|---|
opprbas.1 | ⊢ 𝑂 = (oppr‘𝑅) |
oppr0.2 | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
oppr0g | ⊢ (𝑅 ∈ 𝑉 → 0 = (0g‘𝑂)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opprbas.1 | . . . . . 6 ⊢ 𝑂 = (oppr‘𝑅) | |
2 | eqid 2177 | . . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
3 | 1, 2 | opprbasg 13169 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → (Base‘𝑅) = (Base‘𝑂)) |
4 | 3 | eleq2d 2247 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → (𝑦 ∈ (Base‘𝑅) ↔ 𝑦 ∈ (Base‘𝑂))) |
5 | eqid 2177 | . . . . . . . . 9 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
6 | 1, 5 | oppraddg 13170 | . . . . . . . 8 ⊢ (𝑅 ∈ 𝑉 → (+g‘𝑅) = (+g‘𝑂)) |
7 | 6 | oveqd 5888 | . . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑦(+g‘𝑅)𝑥) = (𝑦(+g‘𝑂)𝑥)) |
8 | 7 | eqeq1d 2186 | . . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ((𝑦(+g‘𝑅)𝑥) = 𝑥 ↔ (𝑦(+g‘𝑂)𝑥) = 𝑥)) |
9 | 6 | oveqd 5888 | . . . . . . 7 ⊢ (𝑅 ∈ 𝑉 → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑂)𝑦)) |
10 | 9 | eqeq1d 2186 | . . . . . 6 ⊢ (𝑅 ∈ 𝑉 → ((𝑥(+g‘𝑅)𝑦) = 𝑥 ↔ (𝑥(+g‘𝑂)𝑦) = 𝑥)) |
11 | 8, 10 | anbi12d 473 | . . . . 5 ⊢ (𝑅 ∈ 𝑉 → (((𝑦(+g‘𝑅)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑅)𝑦) = 𝑥) ↔ ((𝑦(+g‘𝑂)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑂)𝑦) = 𝑥))) |
12 | 3, 11 | raleqbidv 2684 | . . . 4 ⊢ (𝑅 ∈ 𝑉 → (∀𝑥 ∈ (Base‘𝑅)((𝑦(+g‘𝑅)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑅)𝑦) = 𝑥) ↔ ∀𝑥 ∈ (Base‘𝑂)((𝑦(+g‘𝑂)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑂)𝑦) = 𝑥))) |
13 | 4, 12 | anbi12d 473 | . . 3 ⊢ (𝑅 ∈ 𝑉 → ((𝑦 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑦(+g‘𝑅)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑅)𝑦) = 𝑥)) ↔ (𝑦 ∈ (Base‘𝑂) ∧ ∀𝑥 ∈ (Base‘𝑂)((𝑦(+g‘𝑂)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑂)𝑦) = 𝑥)))) |
14 | 13 | iotabidv 5197 | . 2 ⊢ (𝑅 ∈ 𝑉 → (℩𝑦(𝑦 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑦(+g‘𝑅)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑅)𝑦) = 𝑥))) = (℩𝑦(𝑦 ∈ (Base‘𝑂) ∧ ∀𝑥 ∈ (Base‘𝑂)((𝑦(+g‘𝑂)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑂)𝑦) = 𝑥)))) |
15 | oppr0.2 | . . 3 ⊢ 0 = (0g‘𝑅) | |
16 | 2, 5, 15 | grpidvalg 12723 | . 2 ⊢ (𝑅 ∈ 𝑉 → 0 = (℩𝑦(𝑦 ∈ (Base‘𝑅) ∧ ∀𝑥 ∈ (Base‘𝑅)((𝑦(+g‘𝑅)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑅)𝑦) = 𝑥)))) |
17 | 1 | opprex 13167 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑂 ∈ V) |
18 | eqid 2177 | . . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂) | |
19 | eqid 2177 | . . . 4 ⊢ (+g‘𝑂) = (+g‘𝑂) | |
20 | eqid 2177 | . . . 4 ⊢ (0g‘𝑂) = (0g‘𝑂) | |
21 | 18, 19, 20 | grpidvalg 12723 | . . 3 ⊢ (𝑂 ∈ V → (0g‘𝑂) = (℩𝑦(𝑦 ∈ (Base‘𝑂) ∧ ∀𝑥 ∈ (Base‘𝑂)((𝑦(+g‘𝑂)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑂)𝑦) = 𝑥)))) |
22 | 17, 21 | syl 14 | . 2 ⊢ (𝑅 ∈ 𝑉 → (0g‘𝑂) = (℩𝑦(𝑦 ∈ (Base‘𝑂) ∧ ∀𝑥 ∈ (Base‘𝑂)((𝑦(+g‘𝑂)𝑥) = 𝑥 ∧ (𝑥(+g‘𝑂)𝑦) = 𝑥)))) |
23 | 14, 16, 22 | 3eqtr4d 2220 | 1 ⊢ (𝑅 ∈ 𝑉 → 0 = (0g‘𝑂)) |
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 |
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