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Theorem cramerlem2 22631

Description: Lemma 2 for cramer 22634. (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 1214	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ 𝑧 ∈ 𝑉) ∧ (𝑋 · 𝑧) = 𝑌) → 𝑅 ∈ CRing)
2		simpll2 1215	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ 𝑧 ∈ 𝑉) ∧ (𝑋 · 𝑧) = 𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉))
3		simpll3 1216	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ 𝑧 ∈ 𝑉) ∧ (𝑋 · 𝑧) = 𝑌) → (𝐷‘𝑋) ∈ (Unit‘𝑅))
4		simplr 769	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ 𝑧 ∈ 𝑉) ∧ (𝑋 · 𝑧) = 𝑌) → 𝑧 ∈ 𝑉)
5		simpr 484	. . . 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 22630	. . . 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 1385	. . 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 412	. 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 3130	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 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ⟨cop 4574 ↦ cmpt 5167 ‘cfv 6490 (class class class)co 7358 ↑_m cmap 8764 Basecbs 17137 CRingccrg 20173 Unitcui 20293 /_rcdvr 20338 Mat cmat 22350 maVecMul cmvmul 22483 matRepV cmatrepV 22500 maDet cmdat 22527

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-addf 11106 ax-mulf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-xor 1514 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-ot 4577 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-tpos 8167 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-sup 9346 df-oi 9416 df-card 9852 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-xnn0 12476 df-z 12490 df-dec 12609 df-uz 12753 df-rp 12907 df-fz 13425 df-fzo 13572 df-seq 13926 df-exp 13986 df-hash 14255 df-word 14438 df-lsw 14487 df-concat 14495 df-s1 14521 df-substr 14566 df-pfx 14596 df-splice 14674 df-reverse 14683 df-s2 14772 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17138 df-ress 17159 df-plusg 17191 df-mulr 17192 df-starv 17193 df-sca 17194 df-vsca 17195 df-ip 17196 df-tset 17197 df-ple 17198 df-ds 17200 df-unif 17201 df-hom 17202 df-cco 17203 df-0g 17362 df-gsum 17363 df-prds 17368 df-pws 17370 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18709 df-submnd 18710 df-efmnd 18795 df-grp 18870 df-minusg 18871 df-sbg 18872 df-mulg 19002 df-subg 19057 df-ghm 19146 df-gim 19192 df-cntz 19250 df-oppg 19279 df-symg 19303 df-pmtr 19375 df-psgn 19424 df-evpm 19425 df-cmn 19715 df-abl 19716 df-mgp 20080 df-rng 20092 df-ur 20121 df-srg 20126 df-ring 20174 df-cring 20175 df-oppr 20275 df-dvdsr 20295 df-unit 20296 df-invr 20326 df-dvr 20339 df-rhm 20410 df-subrng 20481 df-subrg 20505 df-drng 20666 df-lmod 20815 df-lss 20885 df-sra 21127 df-rgmod 21128 df-cnfld 21312 df-zring 21404 df-zrh 21460 df-dsmm 21689 df-frlm 21704 df-mamu 22334 df-mat 22351 df-mvmul 22484 df-marrep 22501 df-marepv 22502 df-subma 22520 df-mdet 22528 df-minmar1 22578

This theorem is referenced by: cramerlem3 22632

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