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Mirrors > Home > MPE Home > Th. List > Mathboxes > invginvrid | Structured version Visualization version GIF version |
Description: Identity for a multiplication with additive and multiplicative inverses in a ring. (Contributed by AV, 18-May-2018.) |
Ref | Expression |
---|---|
invginvrid.b | ⊢ 𝐵 = (Base‘𝑅) |
invginvrid.u | ⊢ 𝑈 = (Unit‘𝑅) |
invginvrid.n | ⊢ 𝑁 = (invg‘𝑅) |
invginvrid.i | ⊢ 𝐼 = (invr‘𝑅) |
invginvrid.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
invginvrid | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑁‘𝑌) · ((𝐼‘(𝑁‘𝑌)) · 𝑋)) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2778 | . . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
2 | 1 | ringmgp 18944 | . . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) |
3 | 2 | 3ad2ant1 1124 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (mulGrp‘𝑅) ∈ Mnd) |
4 | ringgrp 18943 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
5 | invginvrid.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
6 | invginvrid.u | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑅) | |
7 | 5, 6 | unitcl 19050 | . . . . 5 ⊢ (𝑌 ∈ 𝑈 → 𝑌 ∈ 𝐵) |
8 | invginvrid.n | . . . . . 6 ⊢ 𝑁 = (invg‘𝑅) | |
9 | 5, 8 | grpinvcl 17858 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘𝑌) ∈ 𝐵) |
10 | 4, 7, 9 | syl2an 589 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑈) → (𝑁‘𝑌) ∈ 𝐵) |
11 | 10 | 3adant2 1122 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑁‘𝑌) ∈ 𝐵) |
12 | 6, 8 | unitnegcl 19072 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑈) → (𝑁‘𝑌) ∈ 𝑈) |
13 | invginvrid.i | . . . . . 6 ⊢ 𝐼 = (invr‘𝑅) | |
14 | 6, 13, 5 | ringinvcl 19067 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑁‘𝑌) ∈ 𝑈) → (𝐼‘(𝑁‘𝑌)) ∈ 𝐵) |
15 | 12, 14 | syldan 585 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑈) → (𝐼‘(𝑁‘𝑌)) ∈ 𝐵) |
16 | 15 | 3adant2 1122 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝐼‘(𝑁‘𝑌)) ∈ 𝐵) |
17 | simp2 1128 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ 𝐵) | |
18 | 1, 5 | mgpbas 18886 | . . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) |
19 | invginvrid.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
20 | 1, 19 | mgpplusg 18884 | . . . . 5 ⊢ · = (+g‘(mulGrp‘𝑅)) |
21 | 18, 20 | mndass 17692 | . . . 4 ⊢ (((mulGrp‘𝑅) ∈ Mnd ∧ ((𝑁‘𝑌) ∈ 𝐵 ∧ (𝐼‘(𝑁‘𝑌)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → (((𝑁‘𝑌) · (𝐼‘(𝑁‘𝑌))) · 𝑋) = ((𝑁‘𝑌) · ((𝐼‘(𝑁‘𝑌)) · 𝑋))) |
22 | 21 | eqcomd 2784 | . . 3 ⊢ (((mulGrp‘𝑅) ∈ Mnd ∧ ((𝑁‘𝑌) ∈ 𝐵 ∧ (𝐼‘(𝑁‘𝑌)) ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑁‘𝑌) · ((𝐼‘(𝑁‘𝑌)) · 𝑋)) = (((𝑁‘𝑌) · (𝐼‘(𝑁‘𝑌))) · 𝑋)) |
23 | 3, 11, 16, 17, 22 | syl13anc 1440 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑁‘𝑌) · ((𝐼‘(𝑁‘𝑌)) · 𝑋)) = (((𝑁‘𝑌) · (𝐼‘(𝑁‘𝑌))) · 𝑋)) |
24 | simp1 1127 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → 𝑅 ∈ Ring) | |
25 | 12 | 3adant2 1122 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑁‘𝑌) ∈ 𝑈) |
26 | eqid 2778 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
27 | 6, 13, 19, 26 | unitrinv 19069 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑁‘𝑌) ∈ 𝑈) → ((𝑁‘𝑌) · (𝐼‘(𝑁‘𝑌))) = (1r‘𝑅)) |
28 | 24, 25, 27 | syl2anc 579 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑁‘𝑌) · (𝐼‘(𝑁‘𝑌))) = (1r‘𝑅)) |
29 | 28 | oveq1d 6939 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (((𝑁‘𝑌) · (𝐼‘(𝑁‘𝑌))) · 𝑋) = ((1r‘𝑅) · 𝑋)) |
30 | 5, 19, 26 | ringlidm 18962 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((1r‘𝑅) · 𝑋) = 𝑋) |
31 | 30 | 3adant3 1123 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((1r‘𝑅) · 𝑋) = 𝑋) |
32 | 23, 29, 31 | 3eqtrd 2818 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑁‘𝑌) · ((𝐼‘(𝑁‘𝑌)) · 𝑋)) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ‘cfv 6137 (class class class)co 6924 Basecbs 16259 .rcmulr 16343 Mndcmnd 17684 Grpcgrp 17813 invgcminusg 17814 mulGrpcmgp 18880 1rcur 18892 Ringcrg 18938 Unitcui 19030 invrcinvr 19062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-om 7346 df-tpos 7636 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-nn 11379 df-2 11442 df-3 11443 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-ress 16267 df-plusg 16355 df-mulr 16356 df-0g 16492 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-grp 17816 df-minusg 17817 df-mgp 18881 df-ur 18893 df-ring 18940 df-oppr 19014 df-dvdsr 19032 df-unit 19033 df-invr 19063 |
This theorem is referenced by: lincresunit3lem1 43293 |
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