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Mirrors > Home > ILE Home > Th. List > crngpropd | 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 | . . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
2 | eqid 2177 | . . . . . . 7 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾) | |
3 | eqid 2177 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | 2, 3 | mgpbasg 13136 | . . . . . 6 ⊢ (𝐾 ∈ Ring → (Base‘𝐾) = (Base‘(mulGrp‘𝐾))) |
5 | 1, 4 | sylan9eq 2230 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → 𝐵 = (Base‘(mulGrp‘𝐾))) |
6 | ringpropd.2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
7 | 6 | adantr 276 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → 𝐵 = (Base‘𝐿)) |
8 | ringpropd.3 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) | |
9 | ringpropd.4 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) | |
10 | 1, 6, 8, 9 | ringpropd 13217 | . . . . . . . 8 ⊢ (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring)) |
11 | 10 | biimpa 296 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → 𝐿 ∈ Ring) |
12 | eqid 2177 | . . . . . . . 8 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿) | |
13 | eqid 2177 | . . . . . . . 8 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
14 | 12, 13 | mgpbasg 13136 | . . . . . . 7 ⊢ (𝐿 ∈ Ring → (Base‘𝐿) = (Base‘(mulGrp‘𝐿))) |
15 | 11, 14 | syl 14 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (Base‘𝐿) = (Base‘(mulGrp‘𝐿))) |
16 | 7, 15 | eqtrd 2210 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → 𝐵 = (Base‘(mulGrp‘𝐿))) |
17 | 9 | adantlr 477 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
18 | eqid 2177 | . . . . . . . . 9 ⊢ (.r‘𝐾) = (.r‘𝐾) | |
19 | 2, 18 | mgpplusgg 13134 | . . . . . . . 8 ⊢ (𝐾 ∈ Ring → (.r‘𝐾) = (+g‘(mulGrp‘𝐾))) |
20 | 19 | adantl 277 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (.r‘𝐾) = (+g‘(mulGrp‘𝐾))) |
21 | 20 | oveqdr 5903 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘(mulGrp‘𝐾))𝑦)) |
22 | eqid 2177 | . . . . . . . . 9 ⊢ (.r‘𝐿) = (.r‘𝐿) | |
23 | 12, 22 | mgpplusgg 13134 | . . . . . . . 8 ⊢ (𝐿 ∈ Ring → (.r‘𝐿) = (+g‘(mulGrp‘𝐿))) |
24 | 11, 23 | syl 14 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (.r‘𝐿) = (+g‘(mulGrp‘𝐿))) |
25 | 24 | oveqdr 5903 | . . . . . 6 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐿)𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦)) |
26 | 17, 21, 25 | 3eqtr3d 2218 | . . . . 5 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘(mulGrp‘𝐾))𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦)) |
27 | 5, 16, 26 | cmnpropd 13098 | . . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → ((mulGrp‘𝐾) ∈ CMnd ↔ (mulGrp‘𝐿) ∈ CMnd)) |
28 | 27 | pm5.32da 452 | . . 3 ⊢ (𝜑 → ((𝐾 ∈ Ring ∧ (mulGrp‘𝐾) ∈ CMnd) ↔ (𝐾 ∈ Ring ∧ (mulGrp‘𝐿) ∈ CMnd))) |
29 | 10 | anbi1d 465 | . . 3 ⊢ (𝜑 → ((𝐾 ∈ Ring ∧ (mulGrp‘𝐿) ∈ CMnd) ↔ (𝐿 ∈ Ring ∧ (mulGrp‘𝐿) ∈ CMnd))) |
30 | 28, 29 | bitrd 188 | . 2 ⊢ (𝜑 → ((𝐾 ∈ Ring ∧ (mulGrp‘𝐾) ∈ CMnd) ↔ (𝐿 ∈ Ring ∧ (mulGrp‘𝐿) ∈ CMnd))) |
31 | 2 | iscrng 13186 | . 2 ⊢ (𝐾 ∈ CRing ↔ (𝐾 ∈ Ring ∧ (mulGrp‘𝐾) ∈ CMnd)) |
32 | 12 | iscrng 13186 | . 2 ⊢ (𝐿 ∈ CRing ↔ (𝐿 ∈ Ring ∧ (mulGrp‘𝐿) ∈ CMnd)) |
33 | 30, 31, 32 | 3bitr4g 223 | 1 ⊢ (𝜑 → (𝐾 ∈ CRing ↔ 𝐿 ∈ CRing)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ‘cfv 5217 (class class class)co 5875 Basecbs 12462 +gcplusg 12536 .rcmulr 12537 CMndccmn 13088 mulGrpcmgp 13130 Ringcrg 13179 CRingccrg 13180 |
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 4122 ax-pow 4175 ax-pr 4210 ax-un 4434 ax-setind 4537 ax-cnex 7902 ax-resscn 7903 ax-1cn 7904 ax-1re 7905 ax-icn 7906 ax-addcl 7907 ax-addrcl 7908 ax-mulcl 7909 ax-addcom 7911 ax-addass 7913 ax-i2m1 7916 ax-0lt1 7917 ax-0id 7919 ax-rnegex 7920 ax-pre-ltirr 7923 ax-pre-ltadd 7927 |
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 2740 df-sbc 2964 df-csb 3059 df-dif 3132 df-un 3134 df-in 3136 df-ss 3143 df-nul 3424 df-pw 3578 df-sn 3599 df-pr 3600 df-op 3602 df-uni 3811 df-int 3846 df-br 4005 df-opab 4066 df-mpt 4067 df-id 4294 df-xp 4633 df-rel 4634 df-cnv 4635 df-co 4636 df-dm 4637 df-rn 4638 df-res 4639 df-iota 5179 df-fun 5219 df-fn 5220 df-fv 5225 df-riota 5831 df-ov 5878 df-oprab 5879 df-mpo 5880 df-pnf 7994 df-mnf 7995 df-ltxr 7997 df-inn 8920 df-2 8978 df-3 8979 df-ndx 12465 df-slot 12466 df-base 12468 df-sets 12469 df-plusg 12549 df-mulr 12550 df-0g 12707 df-mgm 12775 df-sgrp 12808 df-mnd 12818 df-grp 12880 df-cmn 13090 df-mgp 13131 df-ring 13181 df-cring 13182 |
This theorem is referenced by: (None) |
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