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Theorem cramerlem2 20819

Description: Lemma 2 for cramer 20822. (Contributed by AV, 21-Feb-2019.) (Revised by AV, 1-Mar-2019.)

**Hypotheses**
Ref	Expression
cramer.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
cramer.b	⊢ 𝐵 = (Base‘𝐴)
cramer.v	⊢ 𝑉 = ((Base‘𝑅) ↑_𝑚 𝑁)
cramer.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
cramer.x	⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)
cramer.q	⊢ / = (/_r‘𝑅)

**Assertion**
Ref	Expression
cramerlem2	⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ∀𝑧 ∈ 𝑉 ((𝑋 · 𝑧) = 𝑌 → 𝑧 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))))

Distinct variable groups: 𝐵,𝑖 𝐷,𝑖 𝑖,𝑁 𝑅,𝑖 𝑖,𝑉 𝑖,𝑋 𝑖,𝑌 · ,𝑖 / ,𝑖 𝑧,𝐵 𝑧,𝐷 𝑧,𝑁,𝑖 𝑧,𝑅 𝑧,𝑉 𝑧,𝑋 𝑧,𝑌 Allowed substitution hints: 𝐴(𝑧,𝑖) / (𝑧) · (𝑧)

**Proof of Theorem cramerlem2**
Step	Hyp	Ref	Expression
1		simpll1 1270	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ 𝑧 ∈ 𝑉) ∧ (𝑋 · 𝑧) = 𝑌) → 𝑅 ∈ CRing)
2		simpll2 1272	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ 𝑧 ∈ 𝑉) ∧ (𝑋 · 𝑧) = 𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉))
3		simpll3 1274	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ 𝑧 ∈ 𝑉) ∧ (𝑋 · 𝑧) = 𝑌) → (𝐷‘𝑋) ∈ (Unit‘𝑅))
4		simplr 786	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ 𝑧 ∈ 𝑉) ∧ (𝑋 · 𝑧) = 𝑌) → 𝑧 ∈ 𝑉)
5		simpr 478	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ 𝑧 ∈ 𝑉) ∧ (𝑋 · 𝑧) = 𝑌) → (𝑋 · 𝑧) = 𝑌)
6		cramer.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
7		cramer.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
8		cramer.v	. . . . 5 ⊢ 𝑉 = ((Base‘𝑅) ↑_𝑚 𝑁)
9		cramer.d	. . . . 5 ⊢ 𝐷 = (𝑁 maDet 𝑅)
10		cramer.x	. . . . 5 ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)
11		cramer.q	. . . . 5 ⊢ / = (/_r‘𝑅)
12	6, 7, 8, 9, 10, 11	cramerlem1 20818	. . . 4 ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑧 ∈ 𝑉 ∧ (𝑋 · 𝑧) = 𝑌)) → 𝑧 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))))
13	1, 2, 3, 4, 5, 12	syl113anc 1502	. . 3 ⊢ ((((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ 𝑧 ∈ 𝑉) ∧ (𝑋 · 𝑧) = 𝑌) → 𝑧 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋))))
14	13	ex 402	. 2 ⊢ (((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ 𝑧 ∈ 𝑉) → ((𝑋 · 𝑧) = 𝑌 → 𝑧 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))))
15	14	ralrimiva 3145	1 ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) → ∀𝑧 ∈ 𝑉 ((𝑋 · 𝑧) = 𝑌 → 𝑧 = (𝑖 ∈ 𝑁 ↦ ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ∀wral 3087 ⟨cop 4372 ↦ cmpt 4920 ‘cfv 6099 (class class class)co 6876 ↑_𝑚 cmap 8093 Basecbs 16181 CRingccrg 18861 Unitcui 18952 /_rcdvr 18995 Mat cmat 20535 maVecMul cmvmul 20669 matRepV cmatrepV 20686 maDet cmdat 20713

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-rep 4962 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-inf2 8786 ax-cnex 10278 ax-resscn 10279 ax-1cn 10280 ax-icn 10281 ax-addcl 10282 ax-addrcl 10283 ax-mulcl 10284 ax-mulrcl 10285 ax-mulcom 10286 ax-addass 10287 ax-mulass 10288 ax-distr 10289 ax-i2m1 10290 ax-1ne0 10291 ax-1rid 10292 ax-rnegex 10293 ax-rrecex 10294 ax-cnre 10295 ax-pre-lttri 10296 ax-pre-lttrn 10297 ax-pre-ltadd 10298 ax-pre-mulgt0 10299 ax-addf 10301 ax-mulf 10302

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-xor 1635 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-reu 3094 df-rmo 3095 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-pss 3783 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-tp 4371 df-op 4373 df-ot 4375 df-uni 4627 df-int 4666 df-iun 4710 df-iin 4711 df-br 4842 df-opab 4904 df-mpt 4921 df-tr 4944 df-id 5218 df-eprel 5223 df-po 5231 df-so 5232 df-fr 5269 df-se 5270 df-we 5271 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-pred 5896 df-ord 5942 df-on 5943 df-lim 5944 df-suc 5945 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-isom 6108 df-riota 6837 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-of 7129 df-om 7298 df-1st 7399 df-2nd 7400 df-supp 7531 df-tpos 7588 df-wrecs 7643 df-recs 7705 df-rdg 7743 df-1o 7797 df-2o 7798 df-oadd 7801 df-er 7980 df-map 8095 df-pm 8096 df-ixp 8147 df-en 8194 df-dom 8195 df-sdom 8196 df-fin 8197 df-fsupp 8516 df-sup 8588 df-oi 8655 df-card 9049 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-sub 10556 df-neg 10557 df-div 10975 df-nn 11311 df-2 11372 df-3 11373 df-4 11374 df-5 11375 df-6 11376 df-7 11377 df-8 11378 df-9 11379 df-n0 11577 df-xnn0 11649 df-z 11663 df-dec 11780 df-uz 11927 df-rp 12071 df-fz 12577 df-fzo 12717 df-seq 13052 df-exp 13111 df-hash 13367 df-word 13531 df-lsw 13579 df-concat 13587 df-s1 13612 df-substr 13662 df-pfx 13711 df-splice 13818 df-reverse 13836 df-s2 13930 df-struct 16183 df-ndx 16184 df-slot 16185 df-base 16187 df-sets 16188 df-ress 16189 df-plusg 16277 df-mulr 16278 df-starv 16279 df-sca 16280 df-vsca 16281 df-ip 16282 df-tset 16283 df-ple 16284 df-ds 16286 df-unif 16287 df-hom 16288 df-cco 16289 df-0g 16414 df-gsum 16415 df-prds 16420 df-pws 16422 df-mre 16558 df-mrc 16559 df-acs 16561 df-mgm 17554 df-sgrp 17596 df-mnd 17607 df-mhm 17647 df-submnd 17648 df-grp 17738 df-minusg 17739 df-sbg 17740 df-mulg 17854 df-subg 17901 df-ghm 17968 df-gim 18011 df-cntz 18059 df-oppg 18085 df-symg 18107 df-pmtr 18171 df-psgn 18220 df-evpm 18221 df-cmn 18507 df-abl 18508 df-mgp 18803 df-ur 18815 df-srg 18819 df-ring 18862 df-cring 18863 df-oppr 18936 df-dvdsr 18954 df-unit 18955 df-invr 18985 df-dvr 18996 df-rnghom 19030 df-drng 19064 df-subrg 19093 df-lmod 19180 df-lss 19248 df-sra 19492 df-rgmod 19493 df-cnfld 20066 df-zring 20138 df-zrh 20171 df-dsmm 20398 df-frlm 20413 df-mamu 20512 df-mat 20536 df-mvmul 20670 df-marrep 20687 df-marepv 20688 df-subma 20706 df-mdet 20714 df-minmar1 20764

This theorem is referenced by: cramerlem3 20820

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