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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 2737 | . . . . 5 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
| 2 | 1 | ringmgp 20236 | . . . 4 ⊢ (𝑅 ∈ Ring → (mulGrp‘𝑅) ∈ Mnd) | 
| 3 | 2 | 3ad2ant1 1134 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (mulGrp‘𝑅) ∈ Mnd) | 
| 4 | ringgrp 20235 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
| 5 | invginvrid.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
| 6 | invginvrid.u | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑅) | |
| 7 | 5, 6 | unitcl 20375 | . . . . 5 ⊢ (𝑌 ∈ 𝑈 → 𝑌 ∈ 𝐵) | 
| 8 | invginvrid.n | . . . . . 6 ⊢ 𝑁 = (invg‘𝑅) | |
| 9 | 5, 8 | grpinvcl 19005 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝑌 ∈ 𝐵) → (𝑁‘𝑌) ∈ 𝐵) | 
| 10 | 4, 7, 9 | syl2an 596 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑈) → (𝑁‘𝑌) ∈ 𝐵) | 
| 11 | 10 | 3adant2 1132 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑁‘𝑌) ∈ 𝐵) | 
| 12 | 6, 8 | unitnegcl 20397 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑈) → (𝑁‘𝑌) ∈ 𝑈) | 
| 13 | invginvrid.i | . . . . . 6 ⊢ 𝐼 = (invr‘𝑅) | |
| 14 | 6, 13, 5 | ringinvcl 20392 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑁‘𝑌) ∈ 𝑈) → (𝐼‘(𝑁‘𝑌)) ∈ 𝐵) | 
| 15 | 12, 14 | syldan 591 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑈) → (𝐼‘(𝑁‘𝑌)) ∈ 𝐵) | 
| 16 | 15 | 3adant2 1132 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝐼‘(𝑁‘𝑌)) ∈ 𝐵) | 
| 17 | simp2 1138 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → 𝑋 ∈ 𝐵) | |
| 18 | 1, 5 | mgpbas 20142 | . . . . 5 ⊢ 𝐵 = (Base‘(mulGrp‘𝑅)) | 
| 19 | invginvrid.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
| 20 | 1, 19 | mgpplusg 20141 | . . . . 5 ⊢ · = (+g‘(mulGrp‘𝑅)) | 
| 21 | 18, 20 | mndass 18756 | . . . 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 2743 | . . 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 1374 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑁‘𝑌) · ((𝐼‘(𝑁‘𝑌)) · 𝑋)) = (((𝑁‘𝑌) · (𝐼‘(𝑁‘𝑌))) · 𝑋)) | 
| 24 | simp1 1137 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → 𝑅 ∈ Ring) | |
| 25 | 12 | 3adant2 1132 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (𝑁‘𝑌) ∈ 𝑈) | 
| 26 | eqid 2737 | . . . . 5 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
| 27 | 6, 13, 19, 26 | unitrinv 20394 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑁‘𝑌) ∈ 𝑈) → ((𝑁‘𝑌) · (𝐼‘(𝑁‘𝑌))) = (1r‘𝑅)) | 
| 28 | 24, 25, 27 | syl2anc 584 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((𝑁‘𝑌) · (𝐼‘(𝑁‘𝑌))) = (1r‘𝑅)) | 
| 29 | 28 | oveq1d 7446 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → (((𝑁‘𝑌) · (𝐼‘(𝑁‘𝑌))) · 𝑋) = ((1r‘𝑅) · 𝑋)) | 
| 30 | 5, 19, 26 | ringlidm 20266 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((1r‘𝑅) · 𝑋) = 𝑋) | 
| 31 | 30 | 3adant3 1133 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈) → ((1r‘𝑅) · 𝑋) = 𝑋) | 
| 32 | 23, 29, 31 | 3eqtrd 2781 | 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 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 .rcmulr 17298 Mndcmnd 18747 Grpcgrp 18951 invgcminusg 18952 mulGrpcmgp 20137 1rcur 20178 Ringcrg 20230 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-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| 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-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 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-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 df-dvdsr 20357 df-unit 20358 df-invr 20388 | 
| This theorem is referenced by: lincresunit3lem1 48396 | 
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