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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 20267 | . . 3 ⊢ (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring)) |
6 | eqid 2726 | . . . . . 6 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾) | |
7 | eqid 2726 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
8 | 6, 7 | mgpbas 20123 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾)) |
9 | 1, 8 | eqtrdi 2782 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐾))) |
10 | eqid 2726 | . . . . . 6 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿) | |
11 | eqid 2726 | . . . . . 6 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
12 | 10, 11 | mgpbas 20123 | . . . . 5 ⊢ (Base‘𝐿) = (Base‘(mulGrp‘𝐿)) |
13 | 2, 12 | eqtrdi 2782 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐿))) |
14 | eqid 2726 | . . . . . . 7 ⊢ (.r‘𝐾) = (.r‘𝐾) | |
15 | 6, 14 | mgpplusg 20121 | . . . . . 6 ⊢ (.r‘𝐾) = (+g‘(mulGrp‘𝐾)) |
16 | 15 | oveqi 7437 | . . . . 5 ⊢ (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘(mulGrp‘𝐾))𝑦) |
17 | eqid 2726 | . . . . . . 7 ⊢ (.r‘𝐿) = (.r‘𝐿) | |
18 | 10, 17 | mgpplusg 20121 | . . . . . 6 ⊢ (.r‘𝐿) = (+g‘(mulGrp‘𝐿)) |
19 | 18 | oveqi 7437 | . . . . 5 ⊢ (𝑥(.r‘𝐿)𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦) |
20 | 4, 16, 19 | 3eqtr3g 2789 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘(mulGrp‘𝐾))𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦)) |
21 | 9, 13, 20 | cmnpropd 19789 | . . 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 20223 | . 2 ⊢ (𝐾 ∈ CRing ↔ (𝐾 ∈ Ring ∧ (mulGrp‘𝐾) ∈ CMnd)) |
24 | 10 | iscrng 20223 | . 2 ⊢ (𝐿 ∈ CRing ↔ (𝐿 ∈ Ring ∧ (mulGrp‘𝐿) ∈ CMnd)) |
25 | 22, 23, 24 | 3bitr4g 313 | 1 ⊢ (𝜑 → (𝐾 ∈ CRing ↔ 𝐿 ∈ CRing)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 +gcplusg 17266 .rcmulr 17267 CMndccmn 19778 mulGrpcmgp 20117 Ringcrg 20216 CRingccrg 20217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-plusg 17279 df-0g 17456 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-grp 18931 df-cmn 19780 df-mgp 20118 df-ring 20218 df-cring 20219 |
This theorem is referenced by: fldpropd 20748 zncrng 21542 opsrcrng 22072 |
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