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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 19263 | . . 3 ⊢ (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring)) |
6 | eqid 2821 | . . . . . 6 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾) | |
7 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
8 | 6, 7 | mgpbas 19176 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾)) |
9 | 1, 8 | syl6eq 2872 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐾))) |
10 | eqid 2821 | . . . . . 6 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿) | |
11 | eqid 2821 | . . . . . 6 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
12 | 10, 11 | mgpbas 19176 | . . . . 5 ⊢ (Base‘𝐿) = (Base‘(mulGrp‘𝐿)) |
13 | 2, 12 | syl6eq 2872 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐿))) |
14 | eqid 2821 | . . . . . . 7 ⊢ (.r‘𝐾) = (.r‘𝐾) | |
15 | 6, 14 | mgpplusg 19174 | . . . . . 6 ⊢ (.r‘𝐾) = (+g‘(mulGrp‘𝐾)) |
16 | 15 | oveqi 7158 | . . . . 5 ⊢ (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘(mulGrp‘𝐾))𝑦) |
17 | eqid 2821 | . . . . . . 7 ⊢ (.r‘𝐿) = (.r‘𝐿) | |
18 | 10, 17 | mgpplusg 19174 | . . . . . 6 ⊢ (.r‘𝐿) = (+g‘(mulGrp‘𝐿)) |
19 | 18 | oveqi 7158 | . . . . 5 ⊢ (𝑥(.r‘𝐿)𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦) |
20 | 4, 16, 19 | 3eqtr3g 2879 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘(mulGrp‘𝐾))𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦)) |
21 | 9, 13, 20 | cmnpropd 18847 | . . 3 ⊢ (𝜑 → ((mulGrp‘𝐾) ∈ CMnd ↔ (mulGrp‘𝐿) ∈ CMnd)) |
22 | 5, 21 | anbi12d 630 | . 2 ⊢ (𝜑 → ((𝐾 ∈ Ring ∧ (mulGrp‘𝐾) ∈ CMnd) ↔ (𝐿 ∈ Ring ∧ (mulGrp‘𝐿) ∈ CMnd))) |
23 | 6 | iscrng 19235 | . 2 ⊢ (𝐾 ∈ CRing ↔ (𝐾 ∈ Ring ∧ (mulGrp‘𝐾) ∈ CMnd)) |
24 | 10 | iscrng 19235 | . 2 ⊢ (𝐿 ∈ CRing ↔ (𝐿 ∈ Ring ∧ (mulGrp‘𝐿) ∈ CMnd)) |
25 | 22, 23, 24 | 3bitr4g 315 | 1 ⊢ (𝜑 → (𝐾 ∈ CRing ↔ 𝐿 ∈ CRing)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6349 (class class class)co 7145 Basecbs 16473 +gcplusg 16555 .rcmulr 16556 CMndccmn 18837 mulGrpcmgp 19170 Ringcrg 19228 CRingccrg 19229 |
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 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
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 2618 df-eu 2650 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 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-2 11689 df-ndx 16476 df-slot 16477 df-base 16479 df-sets 16480 df-plusg 16568 df-0g 16705 df-mgm 17842 df-sgrp 17891 df-mnd 17902 df-grp 18046 df-cmn 18839 df-mgp 19171 df-ring 19230 df-cring 19231 |
This theorem is referenced by: fldpropd 19461 opsrcrng 20198 zncrng 20621 |
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