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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ringinvval | Structured version Visualization version GIF version |
Description: The ring inverse expressed in terms of multiplication. (Contributed by Thierry Arnoux, 23-Oct-2017.) |
Ref | Expression |
---|---|
ringinvval.b | ⊢ 𝐵 = (Base‘𝑅) |
ringinvval.p | ⊢ ∗ = (.r‘𝑅) |
ringinvval.o | ⊢ 1 = (1r‘𝑅) |
ringinvval.n | ⊢ 𝑁 = (invr‘𝑅) |
ringinvval.u | ⊢ 𝑈 = (Unit‘𝑅) |
Ref | Expression |
---|---|
ringinvval | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋) = (℩𝑦 ∈ 𝑈 (𝑦 ∗ 𝑋) = 1 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringinvval.u | . . . . 5 ⊢ 𝑈 = (Unit‘𝑅) | |
2 | eqid 2727 | . . . . 5 ⊢ ((mulGrp‘𝑅) ↾s 𝑈) = ((mulGrp‘𝑅) ↾s 𝑈) | |
3 | 1, 2 | unitgrpbas 20310 | . . . 4 ⊢ 𝑈 = (Base‘((mulGrp‘𝑅) ↾s 𝑈)) |
4 | 1 | fvexi 6905 | . . . . 5 ⊢ 𝑈 ∈ V |
5 | eqid 2727 | . . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
6 | ringinvval.p | . . . . . . 7 ⊢ ∗ = (.r‘𝑅) | |
7 | 5, 6 | mgpplusg 20069 | . . . . . 6 ⊢ ∗ = (+g‘(mulGrp‘𝑅)) |
8 | 2, 7 | ressplusg 17262 | . . . . 5 ⊢ (𝑈 ∈ V → ∗ = (+g‘((mulGrp‘𝑅) ↾s 𝑈))) |
9 | 4, 8 | ax-mp 5 | . . . 4 ⊢ ∗ = (+g‘((mulGrp‘𝑅) ↾s 𝑈)) |
10 | eqid 2727 | . . . 4 ⊢ (0g‘((mulGrp‘𝑅) ↾s 𝑈)) = (0g‘((mulGrp‘𝑅) ↾s 𝑈)) | |
11 | ringinvval.n | . . . . 5 ⊢ 𝑁 = (invr‘𝑅) | |
12 | 1, 2, 11 | invrfval 20317 | . . . 4 ⊢ 𝑁 = (invg‘((mulGrp‘𝑅) ↾s 𝑈)) |
13 | 3, 9, 10, 12 | grpinvval 18928 | . . 3 ⊢ (𝑋 ∈ 𝑈 → (𝑁‘𝑋) = (℩𝑦 ∈ 𝑈 (𝑦 ∗ 𝑋) = (0g‘((mulGrp‘𝑅) ↾s 𝑈)))) |
14 | 13 | adantl 481 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋) = (℩𝑦 ∈ 𝑈 (𝑦 ∗ 𝑋) = (0g‘((mulGrp‘𝑅) ↾s 𝑈)))) |
15 | ringinvval.o | . . . . . . 7 ⊢ 1 = (1r‘𝑅) | |
16 | 1, 2, 15 | unitgrpid 20313 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 1 = (0g‘((mulGrp‘𝑅) ↾s 𝑈))) |
17 | 16 | adantr 480 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝑈) → 1 = (0g‘((mulGrp‘𝑅) ↾s 𝑈))) |
18 | 17 | eqeq2d 2738 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝑈) → ((𝑦 ∗ 𝑋) = 1 ↔ (𝑦 ∗ 𝑋) = (0g‘((mulGrp‘𝑅) ↾s 𝑈)))) |
19 | 18 | riotabidva 7390 | . . 3 ⊢ (𝑅 ∈ Ring → (℩𝑦 ∈ 𝑈 (𝑦 ∗ 𝑋) = 1 ) = (℩𝑦 ∈ 𝑈 (𝑦 ∗ 𝑋) = (0g‘((mulGrp‘𝑅) ↾s 𝑈)))) |
20 | 19 | adantr 480 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (℩𝑦 ∈ 𝑈 (𝑦 ∗ 𝑋) = 1 ) = (℩𝑦 ∈ 𝑈 (𝑦 ∗ 𝑋) = (0g‘((mulGrp‘𝑅) ↾s 𝑈)))) |
21 | 14, 20 | eqtr4d 2770 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋) = (℩𝑦 ∈ 𝑈 (𝑦 ∗ 𝑋) = 1 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3469 ‘cfv 6542 ℩crio 7369 (class class class)co 7414 Basecbs 17171 ↾s cress 17200 +gcplusg 17224 .rcmulr 17225 0gc0g 17412 mulGrpcmgp 20065 1rcur 20112 Ringcrg 20164 Unitcui 20283 invrcinvr 20315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-2 12297 df-3 12298 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-0g 17414 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-grp 18884 df-minusg 18885 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-oppr 20262 df-dvdsr 20285 df-unit 20286 df-invr 20316 |
This theorem is referenced by: (None) |
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