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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 20223 | . . . . . 6 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿)) |
5 | 4 | adantr 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (Unit‘𝐾) = (Unit‘𝐿)) |
6 | 1, 2 | eqtr3d 2774 | . . . . . . 7 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) |
7 | 6 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (Base‘𝐾) = (Base‘𝐿)) |
8 | 1 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → 𝐵 = (Base‘𝐾)) |
9 | 2 | adantr 481 | . . . . . . . 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 18577 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (0g‘𝐾) = (0g‘𝐿)) |
13 | 12 | sneqd 4639 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → {(0g‘𝐾)} = {(0g‘𝐿)}) |
14 | 7, 13 | difeq12d 4122 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → ((Base‘𝐾) ∖ {(0g‘𝐾)}) = ((Base‘𝐿) ∖ {(0g‘𝐿)})) |
15 | 5, 14 | eqeq12d 2748 | . . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → ((Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)}) ↔ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)}))) |
16 | 15 | pm5.32da 579 | . . 3 ⊢ (𝜑 → ((𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)})) ↔ (𝐾 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)})))) |
17 | 1, 2, 10, 3 | ringpropd 20095 | . . . 4 ⊢ (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring)) |
18 | 17 | anbi1d 630 | . . 3 ⊢ (𝜑 → ((𝐾 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)})) ↔ (𝐿 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)})))) |
19 | 16, 18 | bitrd 278 | . 2 ⊢ (𝜑 → ((𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)})) ↔ (𝐿 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)})))) |
20 | eqid 2732 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
21 | eqid 2732 | . . 3 ⊢ (Unit‘𝐾) = (Unit‘𝐾) | |
22 | eqid 2732 | . . 3 ⊢ (0g‘𝐾) = (0g‘𝐾) | |
23 | 20, 21, 22 | isdrng 20311 | . 2 ⊢ (𝐾 ∈ DivRing ↔ (𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)}))) |
24 | eqid 2732 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
25 | eqid 2732 | . . 3 ⊢ (Unit‘𝐿) = (Unit‘𝐿) | |
26 | eqid 2732 | . . 3 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
27 | 24, 25, 26 | isdrng 20311 | . 2 ⊢ (𝐿 ∈ DivRing ↔ (𝐿 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)}))) |
28 | 19, 23, 27 | 3bitr4g 313 | 1 ⊢ (𝜑 → (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∖ cdif 3944 {csn 4627 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 +gcplusg 17193 .rcmulr 17194 0gc0g 17381 Ringcrg 20049 Unitcui 20161 DivRingcdr 20307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-2nd 7972 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-mgp 19982 df-ur 19999 df-ring 20051 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-drng 20309 |
This theorem is referenced by: fldpropd 20345 lvecprop2d 20771 hlhildrng 40815 |
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