Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > drngpropd | Structured version Visualization version GIF version |
Description: If two structures have the same group components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 27-Jun-2015.) |
Ref | Expression |
---|---|
drngpropd.1 | ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) |
drngpropd.2 | ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) |
drngpropd.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
drngpropd.4 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
Ref | Expression |
---|---|
drngpropd | ⊢ (𝜑 → (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drngpropd.1 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
2 | drngpropd.2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
3 | drngpropd.4 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) | |
4 | 1, 2, 3 | unitpropd 19441 | . . . . . 6 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿)) |
5 | 4 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (Unit‘𝐾) = (Unit‘𝐿)) |
6 | 1, 2 | eqtr3d 2858 | . . . . . . 7 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) |
7 | 6 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (Base‘𝐾) = (Base‘𝐿)) |
8 | 1 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → 𝐵 = (Base‘𝐾)) |
9 | 2 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → 𝐵 = (Base‘𝐿)) |
10 | drngpropd.3 | . . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) | |
11 | 10 | adantlr 713 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
12 | 8, 9, 11 | grpidpropd 17866 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (0g‘𝐾) = (0g‘𝐿)) |
13 | 12 | sneqd 4572 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → {(0g‘𝐾)} = {(0g‘𝐿)}) |
14 | 7, 13 | difeq12d 4099 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → ((Base‘𝐾) ∖ {(0g‘𝐾)}) = ((Base‘𝐿) ∖ {(0g‘𝐿)})) |
15 | 5, 14 | eqeq12d 2837 | . . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → ((Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)}) ↔ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)}))) |
16 | 15 | pm5.32da 581 | . . 3 ⊢ (𝜑 → ((𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)})) ↔ (𝐾 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)})))) |
17 | 1, 2, 10, 3 | ringpropd 19326 | . . . 4 ⊢ (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring)) |
18 | 17 | anbi1d 631 | . . 3 ⊢ (𝜑 → ((𝐾 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)})) ↔ (𝐿 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)})))) |
19 | 16, 18 | bitrd 281 | . 2 ⊢ (𝜑 → ((𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)})) ↔ (𝐿 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)})))) |
20 | eqid 2821 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
21 | eqid 2821 | . . 3 ⊢ (Unit‘𝐾) = (Unit‘𝐾) | |
22 | eqid 2821 | . . 3 ⊢ (0g‘𝐾) = (0g‘𝐾) | |
23 | 20, 21, 22 | isdrng 19500 | . 2 ⊢ (𝐾 ∈ DivRing ↔ (𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)}))) |
24 | eqid 2821 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
25 | eqid 2821 | . . 3 ⊢ (Unit‘𝐿) = (Unit‘𝐿) | |
26 | eqid 2821 | . . 3 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
27 | 24, 25, 26 | isdrng 19500 | . 2 ⊢ (𝐿 ∈ DivRing ↔ (𝐿 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)}))) |
28 | 19, 23, 27 | 3bitr4g 316 | 1 ⊢ (𝜑 → (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∖ cdif 3932 {csn 4560 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 .rcmulr 16560 0gc0g 16707 Ringcrg 19291 Unitcui 19383 DivRingcdr 19496 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 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 3496 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 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 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 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-mulr 16573 df-0g 16709 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-drng 19498 |
This theorem is referenced by: fldpropd 19524 lvecprop2d 19932 hlhildrng 39082 |
Copyright terms: Public domain | W3C validator |