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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 19443 | . . . . . 6 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿)) |
5 | 4 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (Unit‘𝐾) = (Unit‘𝐿)) |
6 | 1, 2 | eqtr3d 2835 | . . . . . . 7 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) |
7 | 6 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (Base‘𝐾) = (Base‘𝐿)) |
8 | 1 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → 𝐵 = (Base‘𝐾)) |
9 | 2 | adantr 484 | . . . . . . . 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 714 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝐾 ∈ Ring) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
12 | 8, 9, 11 | grpidpropd 17864 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (0g‘𝐾) = (0g‘𝐿)) |
13 | 12 | sneqd 4537 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → {(0g‘𝐾)} = {(0g‘𝐿)}) |
14 | 7, 13 | difeq12d 4051 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → ((Base‘𝐾) ∖ {(0g‘𝐾)}) = ((Base‘𝐿) ∖ {(0g‘𝐿)})) |
15 | 5, 14 | eqeq12d 2814 | . . . 4 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → ((Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)}) ↔ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)}))) |
16 | 15 | pm5.32da 582 | . . 3 ⊢ (𝜑 → ((𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)})) ↔ (𝐾 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)})))) |
17 | 1, 2, 10, 3 | ringpropd 19328 | . . . 4 ⊢ (𝜑 → (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring)) |
18 | 17 | anbi1d 632 | . . 3 ⊢ (𝜑 → ((𝐾 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)})) ↔ (𝐿 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)})))) |
19 | 16, 18 | bitrd 282 | . 2 ⊢ (𝜑 → ((𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)})) ↔ (𝐿 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)})))) |
20 | eqid 2798 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
21 | eqid 2798 | . . 3 ⊢ (Unit‘𝐾) = (Unit‘𝐾) | |
22 | eqid 2798 | . . 3 ⊢ (0g‘𝐾) = (0g‘𝐾) | |
23 | 20, 21, 22 | isdrng 19499 | . 2 ⊢ (𝐾 ∈ DivRing ↔ (𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)}))) |
24 | eqid 2798 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
25 | eqid 2798 | . . 3 ⊢ (Unit‘𝐿) = (Unit‘𝐿) | |
26 | eqid 2798 | . . 3 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
27 | 24, 25, 26 | isdrng 19499 | . 2 ⊢ (𝐿 ∈ DivRing ↔ (𝐿 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐿) ∖ {(0g‘𝐿)}))) |
28 | 19, 23, 27 | 3bitr4g 317 | 1 ⊢ (𝜑 → (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∖ cdif 3878 {csn 4525 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 0gc0g 16705 Ringcrg 19290 Unitcui 19385 DivRingcdr 19495 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 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 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-mulr 16571 df-0g 16707 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-drng 19497 |
This theorem is referenced by: fldpropd 19523 lvecprop2d 19931 hlhildrng 39248 |
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