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Mirrors > Home > MPE Home > Th. List > invrpropd | Structured version Visualization version GIF version |
Description: The ring inverse function depends only on the ring's base set and multiplication operation. (Contributed by Mario Carneiro, 26-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.) |
Ref | Expression |
---|---|
rngidpropd.1 | ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) |
rngidpropd.2 | ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) |
rngidpropd.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
Ref | Expression |
---|---|
invrpropd | ⊢ (𝜑 → (invr‘𝐾) = (invr‘𝐿)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . . . 5 ⊢ (Unit‘𝐾) = (Unit‘𝐾) | |
2 | eqid 2798 | . . . . 5 ⊢ ((mulGrp‘𝐾) ↾s (Unit‘𝐾)) = ((mulGrp‘𝐾) ↾s (Unit‘𝐾)) | |
3 | 1, 2 | unitgrpbas 19412 | . . . 4 ⊢ (Unit‘𝐾) = (Base‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))) |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → (Unit‘𝐾) = (Base‘((mulGrp‘𝐾) ↾s (Unit‘𝐾)))) |
5 | rngidpropd.1 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐾)) | |
6 | rngidpropd.2 | . . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) | |
7 | rngidpropd.3 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) | |
8 | 5, 6, 7 | unitpropd 19443 | . . . 4 ⊢ (𝜑 → (Unit‘𝐾) = (Unit‘𝐿)) |
9 | eqid 2798 | . . . . 5 ⊢ (Unit‘𝐿) = (Unit‘𝐿) | |
10 | eqid 2798 | . . . . 5 ⊢ ((mulGrp‘𝐿) ↾s (Unit‘𝐿)) = ((mulGrp‘𝐿) ↾s (Unit‘𝐿)) | |
11 | 9, 10 | unitgrpbas 19412 | . . . 4 ⊢ (Unit‘𝐿) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))) |
12 | 8, 11 | eqtrdi 2849 | . . 3 ⊢ (𝜑 → (Unit‘𝐾) = (Base‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))) |
13 | eqid 2798 | . . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
14 | 13, 1 | unitss 19406 | . . . . . . . 8 ⊢ (Unit‘𝐾) ⊆ (Base‘𝐾) |
15 | 14, 5 | sseqtrrid 3968 | . . . . . . 7 ⊢ (𝜑 → (Unit‘𝐾) ⊆ 𝐵) |
16 | 15 | sselda 3915 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (Unit‘𝐾)) → 𝑥 ∈ 𝐵) |
17 | 15 | sselda 3915 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (Unit‘𝐾)) → 𝑦 ∈ 𝐵) |
18 | 16, 17 | anim12dan 621 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) |
19 | 18, 7 | syldan 594 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥(.r‘𝐾)𝑦) = (𝑥(.r‘𝐿)𝑦)) |
20 | fvex 6658 | . . . . . 6 ⊢ (Unit‘𝐾) ∈ V | |
21 | eqid 2798 | . . . . . . . 8 ⊢ (mulGrp‘𝐾) = (mulGrp‘𝐾) | |
22 | eqid 2798 | . . . . . . . 8 ⊢ (.r‘𝐾) = (.r‘𝐾) | |
23 | 21, 22 | mgpplusg 19236 | . . . . . . 7 ⊢ (.r‘𝐾) = (+g‘(mulGrp‘𝐾)) |
24 | 2, 23 | ressplusg 16604 | . . . . . 6 ⊢ ((Unit‘𝐾) ∈ V → (.r‘𝐾) = (+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾)))) |
25 | 20, 24 | ax-mp 5 | . . . . 5 ⊢ (.r‘𝐾) = (+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))) |
26 | 25 | oveqi 7148 | . . . 4 ⊢ (𝑥(.r‘𝐾)𝑦) = (𝑥(+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾)))𝑦) |
27 | fvex 6658 | . . . . . 6 ⊢ (Unit‘𝐿) ∈ V | |
28 | eqid 2798 | . . . . . . . 8 ⊢ (mulGrp‘𝐿) = (mulGrp‘𝐿) | |
29 | eqid 2798 | . . . . . . . 8 ⊢ (.r‘𝐿) = (.r‘𝐿) | |
30 | 28, 29 | mgpplusg 19236 | . . . . . . 7 ⊢ (.r‘𝐿) = (+g‘(mulGrp‘𝐿)) |
31 | 10, 30 | ressplusg 16604 | . . . . . 6 ⊢ ((Unit‘𝐿) ∈ V → (.r‘𝐿) = (+g‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))) |
32 | 27, 31 | ax-mp 5 | . . . . 5 ⊢ (.r‘𝐿) = (+g‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))) |
33 | 32 | oveqi 7148 | . . . 4 ⊢ (𝑥(.r‘𝐿)𝑦) = (𝑥(+g‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))𝑦) |
34 | 19, 26, 33 | 3eqtr3g 2856 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Unit‘𝐾) ∧ 𝑦 ∈ (Unit‘𝐾))) → (𝑥(+g‘((mulGrp‘𝐾) ↾s (Unit‘𝐾)))𝑦) = (𝑥(+g‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))𝑦)) |
35 | 4, 12, 34 | grpinvpropd 18166 | . 2 ⊢ (𝜑 → (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿)))) |
36 | eqid 2798 | . . 3 ⊢ (invr‘𝐾) = (invr‘𝐾) | |
37 | 1, 2, 36 | invrfval 19419 | . 2 ⊢ (invr‘𝐾) = (invg‘((mulGrp‘𝐾) ↾s (Unit‘𝐾))) |
38 | eqid 2798 | . . 3 ⊢ (invr‘𝐿) = (invr‘𝐿) | |
39 | 9, 10, 38 | invrfval 19419 | . 2 ⊢ (invr‘𝐿) = (invg‘((mulGrp‘𝐿) ↾s (Unit‘𝐿))) |
40 | 35, 37, 39 | 3eqtr4g 2858 | 1 ⊢ (𝜑 → (invr‘𝐾) = (invr‘𝐿)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 ↾s cress 16476 +gcplusg 16557 .rcmulr 16558 invgcminusg 18096 mulGrpcmgp 19232 Unitcui 19385 invrcinvr 19417 |
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-ress 16483 df-plusg 16570 df-mulr 16571 df-0g 16707 df-minusg 18099 df-mgp 19233 df-ur 19245 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 |
This theorem is referenced by: (None) |
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