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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 2821 | . . . . . . . . 9 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2821 | . . . . . . . . 9 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
3 | opprbas.1 | . . . . . . . . 9 ⊢ 𝑂 = (oppr‘𝑅) | |
4 | eqid 2821 | . . . . . . . . 9 ⊢ (.r‘𝑂) = (.r‘𝑂) | |
5 | 1, 2, 3, 4 | opprmul 19376 | . . . . . . . 8 ⊢ (𝑥(.r‘𝑂)𝑦) = (𝑦(.r‘𝑅)𝑥) |
6 | 5 | eqeq1i 2826 | . . . . . . 7 ⊢ ((𝑥(.r‘𝑂)𝑦) = 𝑦 ↔ (𝑦(.r‘𝑅)𝑥) = 𝑦) |
7 | 1, 2, 3, 4 | opprmul 19376 | . . . . . . . 8 ⊢ (𝑦(.r‘𝑂)𝑥) = (𝑥(.r‘𝑅)𝑦) |
8 | 7 | eqeq1i 2826 | . . . . . . 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 3165 | . . . . 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 6340 | . . 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 2821 | . . . . 5 ⊢ (mulGrp‘𝑂) = (mulGrp‘𝑂) | |
14 | 3, 1 | opprbas 19379 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑂) |
15 | 13, 14 | mgpbas 19245 | . . . 4 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑂)) |
16 | 13, 4 | mgpplusg 19243 | . . . 4 ⊢ (.r‘𝑂) = (+g‘(mulGrp‘𝑂)) |
17 | eqid 2821 | . . . 4 ⊢ (0g‘(mulGrp‘𝑂)) = (0g‘(mulGrp‘𝑂)) | |
18 | 15, 16, 17 | grpidval 17871 | . . 3 ⊢ (0g‘(mulGrp‘𝑂)) = (℩𝑥(𝑥 ∈ (Base‘𝑅) ∧ ∀𝑦 ∈ (Base‘𝑅)((𝑥(.r‘𝑂)𝑦) = 𝑦 ∧ (𝑦(.r‘𝑂)𝑥) = 𝑦))) |
19 | eqid 2821 | . . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
20 | 19, 1 | mgpbas 19245 | . . . 4 ⊢ (Base‘𝑅) = (Base‘(mulGrp‘𝑅)) |
21 | 19, 2 | mgpplusg 19243 | . . . 4 ⊢ (.r‘𝑅) = (+g‘(mulGrp‘𝑅)) |
22 | eqid 2821 | . . . 4 ⊢ (0g‘(mulGrp‘𝑅)) = (0g‘(mulGrp‘𝑅)) | |
23 | 20, 21, 22 | grpidval 17871 | . . 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 2854 | . 2 ⊢ (0g‘(mulGrp‘𝑂)) = (0g‘(mulGrp‘𝑅)) |
25 | eqid 2821 | . . 3 ⊢ (1r‘𝑂) = (1r‘𝑂) | |
26 | 13, 25 | ringidval 19253 | . 2 ⊢ (1r‘𝑂) = (0g‘(mulGrp‘𝑂)) |
27 | oppr1.2 | . . 3 ⊢ 1 = (1r‘𝑅) | |
28 | 19, 27 | ringidval 19253 | . 2 ⊢ 1 = (0g‘(mulGrp‘𝑅)) |
29 | 24, 26, 28 | 3eqtr4ri 2855 | 1 ⊢ 1 = (1r‘𝑂) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ℩cio 6312 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 .rcmulr 16566 0gc0g 16713 mulGrpcmgp 19239 1rcur 19251 opprcoppr 19372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-tpos 7892 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-plusg 16578 df-mulr 16579 df-0g 16715 df-mgp 19240 df-ur 19252 df-oppr 19373 |
This theorem is referenced by: opprunit 19411 isdrngrd 19528 opprsubrg 19556 srng1 19630 issrngd 19632 fidomndrng 20080 rhmopp 30892 ldual1 36299 lduallmodlem 36303 ldualvsub 36306 lcd1 38760 lcdvsub 38768 |
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