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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 2796 | . . . . . 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 7029 | . . . . . . 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 19137 | . . . . 5 ⊢ (𝐾 ∈ Ring → (Unit‘𝐾) = (Unit‘𝐿)) |
8 | drngprop.p | . . . . . . . . . 10 ⊢ (+g‘𝐾) = (+g‘𝐿) | |
9 | 8 | oveqi 7029 | . . . . . . . . 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 17700 | . . . . . . 7 ⊢ (𝐾 ∈ Ring → (0g‘𝐾) = (0g‘𝐿)) |
12 | 11 | sneqd 4484 | . . . . . 6 ⊢ (𝐾 ∈ Ring → {(0g‘𝐾)} = {(0g‘𝐿)}) |
13 | 12 | difeq2d 4020 | . . . . 5 ⊢ (𝐾 ∈ Ring → ((Base‘𝐾) ∖ {(0g‘𝐾)}) = ((Base‘𝐾) ∖ {(0g‘𝐿)})) |
14 | 7, 13 | eqeq12d 2810 | . . . 4 ⊢ (𝐾 ∈ Ring → ((Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)}) ↔ (Unit‘𝐿) = ((Base‘𝐾) ∖ {(0g‘𝐿)}))) |
15 | 14 | pm5.32i 575 | . . 3 ⊢ ((𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)})) ↔ (𝐾 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐾) ∖ {(0g‘𝐿)}))) |
16 | 2, 8, 4 | ringprop 19024 | . . . 4 ⊢ (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring) |
17 | 16 | anbi1i 623 | . . 3 ⊢ ((𝐾 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐾) ∖ {(0g‘𝐿)})) ↔ (𝐿 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐾) ∖ {(0g‘𝐿)}))) |
18 | 15, 17 | bitri 276 | . 2 ⊢ ((𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)})) ↔ (𝐿 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐾) ∖ {(0g‘𝐿)}))) |
19 | eqid 2795 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
20 | eqid 2795 | . . 3 ⊢ (Unit‘𝐾) = (Unit‘𝐾) | |
21 | eqid 2795 | . . 3 ⊢ (0g‘𝐾) = (0g‘𝐾) | |
22 | 19, 20, 21 | isdrng 19196 | . 2 ⊢ (𝐾 ∈ DivRing ↔ (𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)}))) |
23 | eqid 2795 | . . 3 ⊢ (Unit‘𝐿) = (Unit‘𝐿) | |
24 | eqid 2795 | . . 3 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
25 | 2, 23, 24 | isdrng 19196 | . 2 ⊢ (𝐿 ∈ DivRing ↔ (𝐿 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐾) ∖ {(0g‘𝐿)}))) |
26 | 18, 22, 25 | 3bitr4i 304 | 1 ⊢ (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ∖ cdif 3856 {csn 4472 ‘cfv 6225 (class class class)co 7016 Basecbs 16312 +gcplusg 16394 .rcmulr 16395 0gc0g 16542 Ringcrg 18987 Unitcui 19079 DivRingcdr 19192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-tpos 7743 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-3 11549 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-plusg 16407 df-mulr 16408 df-0g 16544 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-grp 17864 df-mgp 18930 df-ur 18942 df-ring 18989 df-oppr 19063 df-dvdsr 19081 df-unit 19082 df-drng 19194 |
This theorem is referenced by: (None) |
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