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Mirrors > Home > MPE Home > Th. List > drngprop | Structured version Visualization version GIF version |
Description: If two structures have the same ring components (properties), one is a division ring iff the other one is. (Contributed by Mario Carneiro, 11-Oct-2013.) (Revised by Mario Carneiro, 28-Dec-2014.) |
Ref | Expression |
---|---|
drngprop.b | ⊢ (Base‘𝐾) = (Base‘𝐿) |
drngprop.p | ⊢ (+g‘𝐾) = (+g‘𝐿) |
drngprop.m | ⊢ (.r‘𝐾) = (.r‘𝐿) |
Ref | Expression |
---|---|
drngprop | ⊢ (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2738 | . . . . . 6 ⊢ (𝐾 ∈ Ring → (Base‘𝐾) = (Base‘𝐾)) | |
2 | drngprop.b | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐿) | |
3 | 2 | a1i 11 | . . . . . 6 ⊢ (𝐾 ∈ Ring → (Base‘𝐾) = (Base‘𝐿)) |
4 | drngprop.m | . . . . . . . 8 ⊢ (.r‘𝐾) = (.r‘𝐿) | |
5 | 4 | oveqi 7226 | . . . . . . 7 ⊢ (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦) |
6 | 5 | a1i 11 | . . . . . 6 ⊢ ((𝐾 ∈ Ring ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
7 | 1, 3, 6 | unitpropd 19715 | . . . . 5 ⊢ (𝐾 ∈ Ring → (Unit‘𝐾) = (Unit‘𝐿)) |
8 | drngprop.p | . . . . . . . . . 10 ⊢ (+g‘𝐾) = (+g‘𝐿) | |
9 | 8 | oveqi 7226 | . . . . . . . . 9 ⊢ (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦) |
10 | 9 | a1i 11 | . . . . . . . 8 ⊢ ((𝐾 ∈ Ring ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) |
11 | 1, 3, 10 | grpidpropd 18134 | . . . . . . 7 ⊢ (𝐾 ∈ Ring → (0g‘𝐾) = (0g‘𝐿)) |
12 | 11 | sneqd 4553 | . . . . . 6 ⊢ (𝐾 ∈ Ring → {(0g‘𝐾)} = {(0g‘𝐿)}) |
13 | 12 | difeq2d 4037 | . . . . 5 ⊢ (𝐾 ∈ Ring → ((Base‘𝐾) ∖ {(0g‘𝐾)}) = ((Base‘𝐾) ∖ {(0g‘𝐿)})) |
14 | 7, 13 | eqeq12d 2753 | . . . 4 ⊢ (𝐾 ∈ Ring → ((Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)}) ↔ (Unit‘𝐿) = ((Base‘𝐾) ∖ {(0g‘𝐿)}))) |
15 | 14 | pm5.32i 578 | . . 3 ⊢ ((𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)})) ↔ (𝐾 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐾) ∖ {(0g‘𝐿)}))) |
16 | 2, 8, 4 | ringprop 19602 | . . . 4 ⊢ (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring) |
17 | 16 | anbi1i 627 | . . 3 ⊢ ((𝐾 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐾) ∖ {(0g‘𝐿)})) ↔ (𝐿 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐾) ∖ {(0g‘𝐿)}))) |
18 | 15, 17 | bitri 278 | . 2 ⊢ ((𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)})) ↔ (𝐿 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐾) ∖ {(0g‘𝐿)}))) |
19 | eqid 2737 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
20 | eqid 2737 | . . 3 ⊢ (Unit‘𝐾) = (Unit‘𝐾) | |
21 | eqid 2737 | . . 3 ⊢ (0g‘𝐾) = (0g‘𝐾) | |
22 | 19, 20, 21 | isdrng 19771 | . 2 ⊢ (𝐾 ∈ DivRing ↔ (𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)}))) |
23 | eqid 2737 | . . 3 ⊢ (Unit‘𝐿) = (Unit‘𝐿) | |
24 | eqid 2737 | . . 3 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
25 | 2, 23, 24 | isdrng 19771 | . 2 ⊢ (𝐿 ∈ DivRing ↔ (𝐿 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐾) ∖ {(0g‘𝐿)}))) |
26 | 18, 22, 25 | 3bitr4i 306 | 1 ⊢ (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∖ cdif 3863 {csn 4541 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 +gcplusg 16802 .rcmulr 16803 0gc0g 16944 Ringcrg 19562 Unitcui 19657 DivRingcdr 19767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-tpos 7968 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-er 8391 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-plusg 16815 df-mulr 16816 df-0g 16946 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-grp 18368 df-mgp 19505 df-ur 19517 df-ring 19564 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-drng 19769 |
This theorem is referenced by: (None) |
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