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Mirrors > Home > MPE Home > Th. List > crngpropd | Structured version Visualization version GIF version |
Description: If two structures have the same group components (properties), one is a commutative ring iff the other one is. (Contributed by Mario Carneiro, 8-Feb-2015.) |
Ref | Expression |
---|---|
ringpropd.1 | ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) |
ringpropd.2 | ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) |
ringpropd.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
ringpropd.4 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
Ref | Expression |
---|---|
crngpropd | ⊢ (𝜑 → (𝐾 ∈ CRing ↔ 𝐿 ∈ CRing)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringpropd.1 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
2 | ringpropd.2 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
3 | ringpropd.3 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) | |
4 | ringpropd.4 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) | |
5 | 1, 2, 3, 4 | ringpropd 19068 | . . 3 ⊢ (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring)) |
6 | eqid 2773 | . . . . . 6 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾) | |
7 | eqid 2773 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
8 | 6, 7 | mgpbas 18981 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾)) |
9 | 1, 8 | syl6eq 2825 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐾))) |
10 | eqid 2773 | . . . . . 6 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿) | |
11 | eqid 2773 | . . . . . 6 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
12 | 10, 11 | mgpbas 18981 | . . . . 5 ⊢ (Base‘𝐿) = (Base‘(mulGrp‘𝐿)) |
13 | 2, 12 | syl6eq 2825 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐿))) |
14 | eqid 2773 | . . . . . . 7 ⊢ (.r‘𝐾) = (.r‘𝐾) | |
15 | 6, 14 | mgpplusg 18979 | . . . . . 6 ⊢ (.r‘𝐾) = (+g‘(mulGrp‘𝐾)) |
16 | 15 | oveqi 6988 | . . . . 5 ⊢ (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘(mulGrp‘𝐾))𝑦) |
17 | eqid 2773 | . . . . . . 7 ⊢ (.r‘𝐿) = (.r‘𝐿) | |
18 | 10, 17 | mgpplusg 18979 | . . . . . 6 ⊢ (.r‘𝐿) = (+g‘(mulGrp‘𝐿)) |
19 | 18 | oveqi 6988 | . . . . 5 ⊢ (𝑥(.r‘𝐿)𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦) |
20 | 4, 16, 19 | 3eqtr3g 2832 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘(mulGrp‘𝐾))𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦)) |
21 | 9, 13, 20 | cmnpropd 18688 | . . 3 ⊢ (𝜑 → ((mulGrp‘𝐾) ∈ CMnd ↔ (mulGrp‘𝐿) ∈ CMnd)) |
22 | 5, 21 | anbi12d 622 | . 2 ⊢ (𝜑 → ((𝐾 ∈ Ring ∧ (mulGrp‘𝐾) ∈ CMnd) ↔ (𝐿 ∈ Ring ∧ (mulGrp‘𝐿) ∈ CMnd))) |
23 | 6 | iscrng 19040 | . 2 ⊢ (𝐾 ∈ CRing ↔ (𝐾 ∈ Ring ∧ (mulGrp‘𝐾) ∈ CMnd)) |
24 | 10 | iscrng 19040 | . 2 ⊢ (𝐿 ∈ CRing ↔ (𝐿 ∈ Ring ∧ (mulGrp‘𝐿) ∈ CMnd)) |
25 | 22, 23, 24 | 3bitr4g 306 | 1 ⊢ (𝜑 → (𝐾 ∈ CRing ↔ 𝐿 ∈ CRing)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ‘cfv 6186 (class class class)co 6975 Basecbs 16338 +gcplusg 16420 .rcmulr 16421 CMndccmn 18679 mulGrpcmgp 18975 Ringcrg 19033 CRingccrg 19034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-pss 3840 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-tp 4441 df-op 4443 df-uni 4710 df-iun 4791 df-br 4927 df-opab 4989 df-mpt 5006 df-tr 5028 df-id 5309 df-eprel 5314 df-po 5323 df-so 5324 df-fr 5363 df-we 5365 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-ord 6030 df-on 6031 df-lim 6032 df-suc 6033 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-om 7396 df-wrecs 7749 df-recs 7811 df-rdg 7849 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-nn 11439 df-2 11502 df-ndx 16341 df-slot 16342 df-base 16344 df-sets 16345 df-plusg 16433 df-0g 16570 df-mgm 17723 df-sgrp 17765 df-mnd 17776 df-grp 17907 df-cmn 18681 df-mgp 18976 df-ring 19035 df-cring 19036 |
This theorem is referenced by: fldpropd 19266 opsrcrng 19994 zncrng 20409 |
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