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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 20223 | . . 3 ⊢ (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring)) |
6 | eqid 2728 | . . . . . 6 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾) | |
7 | eqid 2728 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
8 | 6, 7 | mgpbas 20079 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘(mulGrp‘𝐾)) |
9 | 1, 8 | eqtrdi 2784 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐾))) |
10 | eqid 2728 | . . . . . 6 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿) | |
11 | eqid 2728 | . . . . . 6 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
12 | 10, 11 | mgpbas 20079 | . . . . 5 ⊢ (Base‘𝐿) = (Base‘(mulGrp‘𝐿)) |
13 | 2, 12 | eqtrdi 2784 | . . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(mulGrp‘𝐿))) |
14 | eqid 2728 | . . . . . . 7 ⊢ (.r‘𝐾) = (.r‘𝐾) | |
15 | 6, 14 | mgpplusg 20077 | . . . . . 6 ⊢ (.r‘𝐾) = (+g‘(mulGrp‘𝐾)) |
16 | 15 | oveqi 7433 | . . . . 5 ⊢ (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘(mulGrp‘𝐾))𝑦) |
17 | eqid 2728 | . . . . . . 7 ⊢ (.r‘𝐿) = (.r‘𝐿) | |
18 | 10, 17 | mgpplusg 20077 | . . . . . 6 ⊢ (.r‘𝐿) = (+g‘(mulGrp‘𝐿)) |
19 | 18 | oveqi 7433 | . . . . 5 ⊢ (𝑥(.r‘𝐿)𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦) |
20 | 4, 16, 19 | 3eqtr3g 2791 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘(mulGrp‘𝐾))𝑦) = (𝑥(+g‘(mulGrp‘𝐿))𝑦)) |
21 | 9, 13, 20 | cmnpropd 19745 | . . 3 ⊢ (𝜑 → ((mulGrp‘𝐾) ∈ CMnd ↔ (mulGrp‘𝐿) ∈ CMnd)) |
22 | 5, 21 | anbi12d 631 | . 2 ⊢ (𝜑 → ((𝐾 ∈ Ring ∧ (mulGrp‘𝐾) ∈ CMnd) ↔ (𝐿 ∈ Ring ∧ (mulGrp‘𝐿) ∈ CMnd))) |
23 | 6 | iscrng 20179 | . 2 ⊢ (𝐾 ∈ CRing ↔ (𝐾 ∈ Ring ∧ (mulGrp‘𝐾) ∈ CMnd)) |
24 | 10 | iscrng 20179 | . 2 ⊢ (𝐿 ∈ CRing ↔ (𝐿 ∈ Ring ∧ (mulGrp‘𝐿) ∈ CMnd)) |
25 | 22, 23, 24 | 3bitr4g 314 | 1 ⊢ (𝜑 → (𝐾 ∈ CRing ↔ 𝐿 ∈ CRing)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ‘cfv 6548 (class class class)co 7420 Basecbs 17179 +gcplusg 17232 .rcmulr 17233 CMndccmn 19734 mulGrpcmgp 20073 Ringcrg 20172 CRingccrg 20173 |
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 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-plusg 17245 df-0g 17422 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18892 df-cmn 19736 df-mgp 20074 df-ring 20174 df-cring 20175 |
This theorem is referenced by: fldpropd 20661 zncrng 21477 opsrcrng 22002 |
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