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Mirrors > Home > MPE Home > Th. List > oppr1 | Structured version Visualization version GIF version |
Description: Multiplicative identity of an opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.) |
Ref | Expression |
---|---|
opprbas.1 | ⊢ 𝑂 = (oppr‘𝑅) |
oppr1.2 | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
oppr1 | ⊢ 1 = (1r‘𝑂) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2818 | . . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
3 | opprbas.1 | . . . . . . . . 9 ⊢ 𝑂 = (oppr‘𝑅) | |
4 | eqid 2818 | . . . . . . . . 9 ⊢ (.r‘𝑂) = (.r‘𝑂) | |
5 | 1, 2, 3, 4 | opprmul 19305 | . . . . . . . 8 ⊢ (𝑥(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑥) |
6 | 5 | eqeq1i 2823 | . . . . . . 7 ⊢ ((𝑥(.r‘𝑂)𝑦) = 𝑦 ↔ (𝑦(.r‘𝑅)𝑥) = 𝑦) |
7 | 1, 2, 3, 4 | opprmul 19305 | . . . . . . . 8 ⊢ (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦) |
8 | 7 | eqeq1i 2823 | . . . . . . 7 ⊢ ((𝑦(.r‘𝑂)𝑥) = 𝑦 ↔ (𝑥(.r‘𝑅)𝑦) = 𝑦) |
9 | 6, 8 | anbi12ci 627 | . . . . . 6 ⊢ (((𝑥(.r‘𝑂)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑂)𝑥) = 𝑦) ↔ ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦)) |
10 | 9 | ralbii 3162 | . . . . 5 ⊢ (∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑂)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑂)𝑥) = 𝑦) ↔ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦)) |
11 | 10 | anbi2i 622 | . . . 4 ⊢ ((𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑂)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑂)𝑥) = 𝑦)) ↔ (𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦))) |
12 | 11 | iotabii 6333 | . . 3 ⊢ (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑂)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑂)𝑥) = 𝑦))) = (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦))) |
13 | eqid 2818 | . . . . 5 ⊢ (mulGrp‘𝑂) = (mulGrp‘𝑂) | |
14 | 3, 1 | opprbas 19308 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑂) |
15 | 13, 14 | mgpbas 19174 | . . . 4 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑂)) |
16 | 13, 4 | mgpplusg 19172 | . . . 4 ⊢ (.r‘𝑂) = (+g‘(mulGrp‘𝑂)) |
17 | eqid 2818 | . . . 4 ⊢ (0g‘(mulGrp‘𝑂)) = (0g‘(mulGrp‘𝑂)) | |
18 | 15, 16, 17 | grpidval 17859 | . . 3 ⊢ (0g‘(mulGrp‘𝑂)) = (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑂)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑂)𝑥) = 𝑦))) |
19 | eqid 2818 | . . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
20 | 19, 1 | mgpbas 19174 | . . . 4 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅)) |
21 | 19, 2 | mgpplusg 19172 | . . . 4 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅)) |
22 | eqid 2818 | . . . 4 ⊢ (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅)) | |
23 | 20, 21, 22 | grpidval 17859 | . . 3 ⊢ (0g‘(mulGrp‘𝑅)) = (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦))) |
24 | 12, 18, 23 | 3eqtr4i 2851 | . 2 ⊢ (0g‘(mulGrp‘𝑂)) = (0g‘(mulGrp‘𝑅)) |
25 | eqid 2818 | . . 3 ⊢ (1r‘𝑂) = (1r‘𝑂) | |
26 | 13, 25 | ringidval 19182 | . 2 ⊢ (1r‘𝑂) = (0g‘(mulGrp‘𝑂)) |
27 | oppr1.2 | . . 3 ⊢ 1 = (1r‘𝑅) | |
28 | 19, 27 | ringidval 19182 | . 2 ⊢ 1 = (0g‘(mulGrp‘𝑅)) |
29 | 24, 26, 28 | 3eqtr4ri 2852 | 1 ⊢ 1 = (1r‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ℩cio 6305 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 .rcmulr 16554 0gc0g 16701 mulGrpcmgp 19168 1rcur 19180 opprcoppr 19301 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-nn 11627 df-2 11688 df-3 11689 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-plusg 16566 df-mulr 16567 df-0g 16703 df-mgp 19169 df-ur 19181 df-oppr 19302 |
This theorem is referenced by: opprunit 19340 isdrngrd 19457 opprsubrg 19485 srng1 19559 issrngd 19561 fidomndrng 20008 rhmopp 30819 ldual1 36164 lduallmodlem 36168 ldualvsub 36171 lcd1 38625 lcdvsub 38633 |
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