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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 2772 | . . . . . 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 6806 | . . . . . . 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 18905 | . . . . 5 ⊢ (𝐾 ∈ Ring → (Unit‘𝐾) = (Unit‘𝐿)) |
8 | drngprop.p | . . . . . . . . . 10 ⊢ (+g‘𝐾) = (+g‘𝐿) | |
9 | 8 | oveqi 6806 | . . . . . . . . 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 17469 | . . . . . . 7 ⊢ (𝐾 ∈ Ring → (0g‘𝐾) = (0g‘𝐿)) |
12 | 11 | sneqd 4328 | . . . . . 6 ⊢ (𝐾 ∈ Ring → {(0g‘𝐾)} = {(0g‘𝐿)}) |
13 | 12 | difeq2d 3879 | . . . . 5 ⊢ (𝐾 ∈ Ring → ((Base‘𝐾) ∖ {(0g‘𝐾)}) = ((Base‘𝐾) ∖ {(0g‘𝐿)})) |
14 | 7, 13 | eqeq12d 2786 | . . . 4 ⊢ (𝐾 ∈ Ring → ((Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)}) ↔ (Unit‘𝐿) = ((Base‘𝐾) ∖ {(0g‘𝐿)}))) |
15 | 14 | pm5.32i 564 | . . 3 ⊢ ((𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)})) ↔ (𝐾 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐾) ∖ {(0g‘𝐿)}))) |
16 | 2, 8, 4 | ringprop 18792 | . . . 4 ⊢ (𝐾 ∈ Ring ↔ 𝐿 ∈ Ring) |
17 | 16 | anbi1i 610 | . . 3 ⊢ ((𝐾 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐾) ∖ {(0g‘𝐿)})) ↔ (𝐿 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐾) ∖ {(0g‘𝐿)}))) |
18 | 15, 17 | bitri 264 | . 2 ⊢ ((𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)})) ↔ (𝐿 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐾) ∖ {(0g‘𝐿)}))) |
19 | eqid 2771 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
20 | eqid 2771 | . . 3 ⊢ (Unit‘𝐾) = (Unit‘𝐾) | |
21 | eqid 2771 | . . 3 ⊢ (0g‘𝐾) = (0g‘𝐾) | |
22 | 19, 20, 21 | isdrng 18961 | . 2 ⊢ (𝐾 ∈ DivRing ↔ (𝐾 ∈ Ring ∧ (Unit‘𝐾) = ((Base‘𝐾) ∖ {(0g‘𝐾)}))) |
23 | eqid 2771 | . . 3 ⊢ (Unit‘𝐿) = (Unit‘𝐿) | |
24 | eqid 2771 | . . 3 ⊢ (0g‘𝐿) = (0g‘𝐿) | |
25 | 2, 23, 24 | isdrng 18961 | . 2 ⊢ (𝐿 ∈ DivRing ↔ (𝐿 ∈ Ring ∧ (Unit‘𝐿) = ((Base‘𝐾) ∖ {(0g‘𝐿)}))) |
26 | 18, 22, 25 | 3bitr4i 292 | 1 ⊢ (𝐾 ∈ DivRing ↔ 𝐿 ∈ DivRing) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∖ cdif 3720 {csn 4316 ‘cfv 6031 (class class class)co 6793 Basecbs 16064 +gcplusg 16149 .rcmulr 16150 0gc0g 16308 Ringcrg 18755 Unitcui 18847 DivRingcdr 18957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-tpos 7504 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-plusg 16162 df-mulr 16163 df-0g 16310 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-grp 17633 df-mgp 18698 df-ur 18710 df-ring 18757 df-oppr 18831 df-dvdsr 18849 df-unit 18850 df-drng 18959 |
This theorem is referenced by: (None) |
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