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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 2733 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2733 | . . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
3 | opprbas.1 | . . . . . . . . 9 ⊢ 𝑂 = (oppr‘𝑅) | |
4 | eqid 2733 | . . . . . . . . 9 ⊢ (.r‘𝑂) = (.r‘𝑂) | |
5 | 1, 2, 3, 4 | opprmul 20060 | . . . . . . . 8 ⊢ (𝑥(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑥) |
6 | 5 | eqeq1i 2738 | . . . . . . 7 ⊢ ((𝑥(.r‘𝑂)𝑦) = 𝑦 ↔ (𝑦(.r‘𝑅)𝑥) = 𝑦) |
7 | 1, 2, 3, 4 | opprmul 20060 | . . . . . . . 8 ⊢ (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦) |
8 | 7 | eqeq1i 2738 | . . . . . . 7 ⊢ ((𝑦(.r‘𝑂)𝑥) = 𝑦 ↔ (𝑥(.r‘𝑅)𝑦) = 𝑦) |
9 | 6, 8 | anbi12ci 629 | . . . . . 6 ⊢ (((𝑥(.r‘𝑂)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑂)𝑥) = 𝑦) ↔ ((𝑥(.r‘𝑅)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑅)𝑥) = 𝑦)) |
10 | 9 | ralbii 3093 | . . . . 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 624 | . . . 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 6485 | . . 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 2733 | . . . . 5 ⊢ (mulGrp‘𝑂) = (mulGrp‘𝑂) | |
14 | 3, 1 | opprbas 20064 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑂) |
15 | 13, 14 | mgpbas 19910 | . . . 4 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑂)) |
16 | 13, 4 | mgpplusg 19908 | . . . 4 ⊢ (.r‘𝑂) = (+g‘(mulGrp‘𝑂)) |
17 | eqid 2733 | . . . 4 ⊢ (0g‘(mulGrp‘𝑂)) = (0g‘(mulGrp‘𝑂)) | |
18 | 15, 16, 17 | grpidval 18524 | . . 3 ⊢ (0g‘(mulGrp‘𝑂)) = (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑂)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑂)𝑥) = 𝑦))) |
19 | eqid 2733 | . . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
20 | 19, 1 | mgpbas 19910 | . . . 4 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅)) |
21 | 19, 2 | mgpplusg 19908 | . . . 4 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅)) |
22 | eqid 2733 | . . . 4 ⊢ (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅)) | |
23 | 20, 21, 22 | grpidval 18524 | . . 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 2771 | . 2 ⊢ (0g‘(mulGrp‘𝑂)) = (0g‘(mulGrp‘𝑅)) |
25 | eqid 2733 | . . 3 ⊢ (1r‘𝑂) = (1r‘𝑂) | |
26 | 13, 25 | ringidval 19923 | . 2 ⊢ (1r‘𝑂) = (0g‘(mulGrp‘𝑂)) |
27 | oppr1.2 | . . 3 ⊢ 1 = (1r‘𝑅) | |
28 | 19, 27 | ringidval 19923 | . 2 ⊢ 1 = (0g‘(mulGrp‘𝑅)) |
29 | 24, 26, 28 | 3eqtr4ri 2772 | 1 ⊢ 1 = (1r‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3061 ℩cio 6450 ‘cfv 6500 (class class class)co 7361 Basecbs 17091 .rcmulr 17142 0gc0g 17329 mulGrpcmgp 19904 1rcur 19921 opprcoppr 20056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-3 12225 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-plusg 17154 df-mulr 17155 df-0g 17331 df-mgp 19905 df-ur 19922 df-oppr 20057 |
This theorem is referenced by: opprunit 20098 rhmopp 20192 isdrngrd 20250 isdrngrdOLD 20252 opprsubrg 20285 srng1 20361 issrngd 20363 fidomndrng 20801 ldual1 37660 lduallmodlem 37664 ldualvsub 37667 lcd1 40122 lcdvsub 40130 |
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