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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 2797 | . . . . 5 ⊢ ((mulGrp‘𝑅) ↾s 𝑈) = ((mulGrp‘𝑅) ↾s 𝑈) | |
3 | 1, 2 | unitgrpbas 19110 | . . . 4 ⊢ 𝑈 = (Base‘((mulGrp‘𝑅) ↾s 𝑈)) |
4 | 1 | fvexi 6559 | . . . . 5 ⊢ 𝑈 ∈ V |
5 | eqid 2797 | . . . . . . 7 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
6 | ringinvval.p | . . . . . . 7 ⊢ ∗ = (.r‘𝑅) | |
7 | 5, 6 | mgpplusg 18937 | . . . . . 6 ⊢ ∗ = (+g‘(mulGrp‘𝑅)) |
8 | 2, 7 | ressplusg 16445 | . . . . 5 ⊢ (𝑈 ∈ V → ∗ = (+g‘((mulGrp‘𝑅) ↾s 𝑈))) |
9 | 4, 8 | ax-mp 5 | . . . 4 ⊢ ∗ = (+g‘((mulGrp‘𝑅) ↾s 𝑈)) |
10 | eqid 2797 | . . . 4 ⊢ (0g‘((mulGrp‘𝑅) ↾s 𝑈)) = (0g‘((mulGrp‘𝑅) ↾s 𝑈)) | |
11 | ringinvval.n | . . . . 5 ⊢ 𝑁 = (invr‘𝑅) | |
12 | 1, 2, 11 | invrfval 19117 | . . . 4 ⊢ 𝑁 = (invg‘((mulGrp‘𝑅) ↾s 𝑈)) |
13 | 3, 9, 10, 12 | grpinvval 17905 | . . 3 ⊢ (𝑋 ∈ 𝑈 → (𝑁‘𝑋) = (℩𝑦 ∈ 𝑈 (𝑦 ∗ 𝑋) = (0g‘((mulGrp‘𝑅) ↾s 𝑈)))) |
14 | 13 | adantl 482 | . 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 19113 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 1 = (0g‘((mulGrp‘𝑅) ↾s 𝑈))) |
17 | 16 | adantr 481 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝑈) → 1 = (0g‘((mulGrp‘𝑅) ↾s 𝑈))) |
18 | 17 | eqeq2d 2807 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑦 ∈ 𝑈) → ((𝑦 ∗ 𝑋) = 1 ↔ (𝑦 ∗ 𝑋) = (0g‘((mulGrp‘𝑅) ↾s 𝑈)))) |
19 | 18 | riotabidva 7000 | . . 3 ⊢ (𝑅 ∈ Ring → (℩𝑦 ∈ 𝑈 (𝑦 ∗ 𝑋) = 1 ) = (℩𝑦 ∈ 𝑈 (𝑦 ∗ 𝑋) = (0g‘((mulGrp‘𝑅) ↾s 𝑈)))) |
20 | 19 | adantr 481 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (℩𝑦 ∈ 𝑈 (𝑦 ∗ 𝑋) = 1 ) = (℩𝑦 ∈ 𝑈 (𝑦 ∗ 𝑋) = (0g‘((mulGrp‘𝑅) ↾s 𝑈)))) |
21 | 14, 20 | eqtr4d 2836 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋) = (℩𝑦 ∈ 𝑈 (𝑦 ∗ 𝑋) = 1 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1525 ∈ wcel 2083 Vcvv 3440 ‘cfv 6232 ℩crio 6983 (class class class)co 7023 Basecbs 16316 ↾s cress 16317 +gcplusg 16398 .rcmulr 16399 0gc0g 16546 mulGrpcmgp 18933 1rcur 18945 Ringcrg 18991 Unitcui 19083 invrcinvr 19115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-tpos 7750 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-nn 11493 df-2 11554 df-3 11555 df-ndx 16319 df-slot 16320 df-base 16322 df-sets 16323 df-ress 16324 df-plusg 16411 df-mulr 16412 df-0g 16548 df-mgm 17685 df-sgrp 17727 df-mnd 17738 df-grp 17868 df-minusg 17869 df-mgp 18934 df-ur 18946 df-ring 18993 df-oppr 19067 df-dvdsr 19085 df-unit 19086 df-invr 19116 |
This theorem is referenced by: (None) |
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