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Mirrors > Home > MPE Home > Th. List > invrfval | Structured version Visualization version GIF version |
Description: Multiplicative inverse function for a division ring. (Contributed by NM, 21-Sep-2011.) (Revised by Mario Carneiro, 25-Dec-2014.) |
Ref | Expression |
---|---|
invrfval.u | ⊢ 𝑈 = (Unit‘𝑅) |
invrfval.g | ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) |
invrfval.i | ⊢ 𝐼 = (invr‘𝑅) |
Ref | Expression |
---|---|
invrfval | ⊢ 𝐼 = (invg‘𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | invrfval.i | . 2 ⊢ 𝐼 = (invr‘𝑅) | |
2 | fveq2 6907 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅)) | |
3 | fveq2 6907 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅)) | |
4 | invrfval.u | . . . . . . . 8 ⊢ 𝑈 = (Unit‘𝑅) | |
5 | 3, 4 | eqtr4di 2793 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈) |
6 | 2, 5 | oveq12d 7449 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = ((mulGrp‘𝑅) ↾s 𝑈)) |
7 | invrfval.g | . . . . . 6 ⊢ 𝐺 = ((mulGrp‘𝑅) ↾s 𝑈) | |
8 | 6, 7 | eqtr4di 2793 | . . . . 5 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = 𝐺) |
9 | 8 | fveq2d 6911 | . . . 4 ⊢ (𝑟 = 𝑅 → (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟))) = (invg‘𝐺)) |
10 | df-invr 20405 | . . . 4 ⊢ invr = (𝑟 ∈ V ↦ (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟)))) | |
11 | fvex 6920 | . . . 4 ⊢ (invg‘𝐺) ∈ V | |
12 | 9, 10, 11 | fvmpt 7016 | . . 3 ⊢ (𝑅 ∈ V → (invr‘𝑅) = (invg‘𝐺)) |
13 | fvprc 6899 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (invr‘𝑅) = ∅) | |
14 | base0 17250 | . . . . . . 7 ⊢ ∅ = (Base‘∅) | |
15 | eqid 2735 | . . . . . . 7 ⊢ (invg‘∅) = (invg‘∅) | |
16 | 14, 15 | grpinvfn 19012 | . . . . . 6 ⊢ (invg‘∅) Fn ∅ |
17 | fn0 6700 | . . . . . 6 ⊢ ((invg‘∅) Fn ∅ ↔ (invg‘∅) = ∅) | |
18 | 16, 17 | mpbi 230 | . . . . 5 ⊢ (invg‘∅) = ∅ |
19 | 13, 18 | eqtr4di 2793 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (invr‘𝑅) = (invg‘∅)) |
20 | fvprc 6899 | . . . . . . . 8 ⊢ (¬ 𝑅 ∈ V → (mulGrp‘𝑅) = ∅) | |
21 | 20 | oveq1d 7446 | . . . . . . 7 ⊢ (¬ 𝑅 ∈ V → ((mulGrp‘𝑅) ↾s 𝑈) = (∅ ↾s 𝑈)) |
22 | 7, 21 | eqtrid 2787 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐺 = (∅ ↾s 𝑈)) |
23 | ress0 17289 | . . . . . 6 ⊢ (∅ ↾s 𝑈) = ∅ | |
24 | 22, 23 | eqtrdi 2791 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐺 = ∅) |
25 | 24 | fveq2d 6911 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (invg‘𝐺) = (invg‘∅)) |
26 | 19, 25 | eqtr4d 2778 | . . 3 ⊢ (¬ 𝑅 ∈ V → (invr‘𝑅) = (invg‘𝐺)) |
27 | 12, 26 | pm2.61i 182 | . 2 ⊢ (invr‘𝑅) = (invg‘𝐺) |
28 | 1, 27 | eqtri 2763 | 1 ⊢ 𝐼 = (invg‘𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 ↾s cress 17274 invgcminusg 18965 mulGrpcmgp 20152 Unitcui 20372 invrcinvr 20404 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-1cn 11211 ax-addcl 11213 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-nn 12265 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-minusg 18968 df-invr 20405 |
This theorem is referenced by: unitinvcl 20407 unitinvinv 20408 unitlinv 20410 unitrinv 20411 rdivmuldivd 20430 invrpropd 20435 subrgugrp 20608 cntzsdrg 20820 cnmsubglem 21466 psgninv 21618 invrvald 22698 invrcn2 24204 nrginvrcn 24729 nrgtdrg 24730 sum2dchr 27333 ringinvval 33225 dvrcan5 33226 |
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