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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 6906 | . . . . . . 7 ⊢ (𝑟 = 𝑅 → (mulGrp‘𝑟) = (mulGrp‘𝑅)) | |
| 3 | fveq2 6906 | . . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅)) | |
| 4 | invrfval.u | . . . . . . . 8 ⊢ 𝑈 = (Unit‘𝑅) | |
| 5 | 3, 4 | eqtr4di 2795 | . . . . . . 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 2795 | . . . . 5 ⊢ (𝑟 = 𝑅 → ((mulGrp‘𝑟) ↾s (Unit‘𝑟)) = 𝐺) |
| 9 | 8 | fveq2d 6910 | . . . 4 ⊢ (𝑟 = 𝑅 → (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟))) = (invg‘𝐺)) |
| 10 | df-invr 20388 | . . . 4 ⊢ invr = (𝑟 ∈ V ↦ (invg‘((mulGrp‘𝑟) ↾s (Unit‘𝑟)))) | |
| 11 | fvex 6919 | . . . 4 ⊢ (invg‘𝐺) ∈ V | |
| 12 | 9, 10, 11 | fvmpt 7016 | . . 3 ⊢ (𝑅 ∈ V → (invr‘𝑅) = (invg‘𝐺)) |
| 13 | fvprc 6898 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → (invr‘𝑅) = ∅) | |
| 14 | base0 17252 | . . . . . . 7 ⊢ ∅ = (Base‘∅) | |
| 15 | eqid 2737 | . . . . . . 7 ⊢ (invg‘∅) = (invg‘∅) | |
| 16 | 14, 15 | grpinvfn 18999 | . . . . . 6 ⊢ (invg‘∅) Fn ∅ |
| 17 | fn0 6699 | . . . . . 6 ⊢ ((invg‘∅) Fn ∅ ↔ (invg‘∅) = ∅) | |
| 18 | 16, 17 | mpbi 230 | . . . . 5 ⊢ (invg‘∅) = ∅ |
| 19 | 13, 18 | eqtr4di 2795 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (invr‘𝑅) = (invg‘∅)) |
| 20 | fvprc 6898 | . . . . . . . 8 ⊢ (¬ 𝑅 ∈ V → (mulGrp‘𝑅) = ∅) | |
| 21 | 20 | oveq1d 7446 | . . . . . . 7 ⊢ (¬ 𝑅 ∈ V → ((mulGrp‘𝑅) ↾s 𝑈) = (∅ ↾s 𝑈)) |
| 22 | 7, 21 | eqtrid 2789 | . . . . . 6 ⊢ (¬ 𝑅 ∈ V → 𝐺 = (∅ ↾s 𝑈)) |
| 23 | ress0 17289 | . . . . . 6 ⊢ (∅ ↾s 𝑈) = ∅ | |
| 24 | 22, 23 | eqtrdi 2793 | . . . . 5 ⊢ (¬ 𝑅 ∈ V → 𝐺 = ∅) |
| 25 | 24 | fveq2d 6910 | . . . 4 ⊢ (¬ 𝑅 ∈ V → (invg‘𝐺) = (invg‘∅)) |
| 26 | 19, 25 | eqtr4d 2780 | . . 3 ⊢ (¬ 𝑅 ∈ V → (invr‘𝑅) = (invg‘𝐺)) |
| 27 | 12, 26 | pm2.61i 182 | . 2 ⊢ (invr‘𝑅) = (invg‘𝐺) |
| 28 | 1, 27 | eqtri 2765 | 1 ⊢ 𝐼 = (invg‘𝐺) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 ↾s cress 17274 invgcminusg 18952 mulGrpcmgp 20137 Unitcui 20355 invrcinvr 20387 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addcl 11215 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-minusg 18955 df-invr 20388 |
| This theorem is referenced by: unitinvcl 20390 unitinvinv 20391 unitlinv 20393 unitrinv 20394 rdivmuldivd 20413 invrpropd 20418 subrgugrp 20591 cntzsdrg 20803 cnmsubglem 21448 psgninv 21600 invrvald 22682 invrcn2 24188 nrginvrcn 24713 nrgtdrg 24714 sum2dchr 27318 ringinvval 33239 dvrcan5 33240 |
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