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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 19425 | . . . . . 6 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿)) |
5 | 4 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (Unit‘𝐾) = (Unit‘𝐿)) |
6 | 1, 2 | eqtr3d 2857 | . . . . . . 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 17850 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → (0g‘𝐾) = (0g‘𝐿)) |
13 | 12 | sneqd 4560 | . . . . . 6 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → {(0g‘𝐾)} = {(0g‘𝐿)}) |
14 | 7, 13 | difeq12d 4083 | . . . . 5 ⊢ ((𝜑 ∧ 𝐾 ∈ Ring) → ((Base‘𝐾) ∖ {(0g‘𝐾)}) = ((Base‘𝐿) ∖ {(0g‘𝐿)})) |
15 | 5, 14 | eqeq12d 2836 | . . . 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 19310 | . . . 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 2820 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
21 | eqid 2820 | . . 3 ⊢ (Unit‘𝐾) = (Unit‘𝐾) | |
22 | eqid 2820 | . . 3 ⊢ (0g‘𝐾) = (0g‘𝐾) | |
23 | 20, 21, 22 | isdrng 19484 | . 2 ⊢ (𝐾 ∈ DivRing ↔ (𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)}))) |
24 | eqid 2820 | . . 3 ⊢ (Base‘𝐿) = (Base‘𝐿) | |
25 | eqid 2820 | . . 3 ⊢ (Unit‘𝐿) = (Unit‘𝐿) | |
26 | eqid 2820 | . . 3 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
27 | 24, 25, 26 | isdrng 19484 | . 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 1537 ∈ wcel 2114 ∖ cdif 3916 {csn 4548 ‘cfv 6336 (class class class)co 7137 Basecbs 16461 +gcplusg 16543 .rcmulr 16544 0gc0g 16691 Ringcrg 19275 Unitcui 19367 DivRingcdr 19480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-mulcom 10582 ax-addass 10583 ax-mulass 10584 ax-distr 10585 ax-i2m1 10586 ax-1ne0 10587 ax-1rid 10588 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 ax-pre-ltadd 10594 ax-pre-mulgt0 10595 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7095 df-ov 7140 df-oprab 7141 df-mpo 7142 df-om 7562 df-tpos 7873 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-er 8270 df-en 8491 df-dom 8492 df-sdom 8493 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-sub 10853 df-neg 10854 df-nn 11620 df-2 11682 df-3 11683 df-ndx 16464 df-slot 16465 df-base 16467 df-sets 16468 df-plusg 16556 df-mulr 16557 df-0g 16693 df-mgm 17830 df-sgrp 17879 df-mnd 17890 df-grp 18084 df-mgp 19218 df-ur 19230 df-ring 19277 df-oppr 19351 df-dvdsr 19369 df-unit 19370 df-drng 19482 |
This theorem is referenced by: fldpropd 19508 lvecprop2d 19916 hlhildrng 39115 |
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