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Theorem cramerlem2 22674

Description: Lemma 2 for cramer 22677. (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 1220	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ 𝑧 ∈ 𝑉) ∧ (𝑋 · 𝑧) = 𝑌) → 𝑅 ∈ CRing)
2		simpll2 1221	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ 𝑧 ∈ 𝑉) ∧ (𝑋 · 𝑧) = 𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉))
3		simpll3 1222	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ 𝑧 ∈ 𝑉) ∧ (𝑋 · 𝑧) = 𝑌) → (𝐷‘𝑋) ∈ (Unit‘𝑅))
4		simplr 775	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ 𝑧 ∈ 𝑉) ∧ (𝑋 · 𝑧) = 𝑌) → 𝑧 ∈ 𝑉)
5		simpr 486	. . . 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 22673	. . . 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 1391	. . 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 414	. 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 3133	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 397 ∧ w3a 1093 = wceq 1548 ∈ wcel 2121 ∀wral 3055 ⟨cop 4563 ↦ cmpt 5155 ‘cfv 6488 (class class class)co 7359 ↑_m cmap 8767 Basecbs 17174 CRingccrg 20209 Unitcui 20329 /_rcdvr 20374 Mat cmat 22393 maVecMul cmvmul 22526 matRepV cmatrepV 22543 maDet cmdat 22570

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7681 ax-cnex 11090 ax-resscn 11091 ax-1cn 11092 ax-icn 11093 ax-addcl 11094 ax-addrcl 11095 ax-mulcl 11096 ax-mulrcl 11097 ax-mulcom 11098 ax-addass 11099 ax-mulass 11100 ax-distr 11101 ax-i2m1 11102 ax-1ne0 11103 ax-1rid 11104 ax-rnegex 11105 ax-rrecex 11106 ax-cnre 11107 ax-pre-lttri 11108 ax-pre-lttrn 11109 ax-pre-ltadd 11110 ax-pre-mulgt0 11111 ax-addf 11113 ax-mulf 11114

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-xor 1520 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3725 df-csb 3833 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-pss 3904 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-ot 4566 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-se 5574 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-isom 6497 df-riota 7316 df-ov 7362 df-oprab 7363 df-mpo 7364 df-of 7623 df-om 7810 df-1st 7933 df-2nd 7934 df-supp 8103 df-tpos 8168 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-pm 8770 df-ixp 8840 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fsupp 9269 df-sup 9349 df-oi 9419 df-card 9858 df-pnf 11177 df-mnf 11178 df-xr 11179 df-ltxr 11180 df-le 11181 df-sub 11375 df-neg 11376 df-div 11804 df-nn 12170 df-2 12239 df-3 12240 df-4 12241 df-5 12242 df-6 12243 df-7 12244 df-8 12245 df-9 12246 df-n0 12433 df-xnn0 12506 df-z 12520 df-dec 12640 df-uz 12784 df-rp 12938 df-fz 13457 df-fzo 13604 df-seq 13959 df-exp 14019 df-hash 14288 df-word 14471 df-lsw 14520 df-concat 14528 df-s1 14554 df-substr 14599 df-pfx 14629 df-splice 14707 df-reverse 14716 df-s2 14805 df-struct 17112 df-sets 17129 df-slot 17147 df-ndx 17159 df-base 17175 df-ress 17196 df-plusg 17228 df-mulr 17229 df-starv 17230 df-sca 17231 df-vsca 17232 df-ip 17233 df-tset 17234 df-ple 17235 df-ds 17237 df-unif 17238 df-hom 17239 df-cco 17240 df-0g 17399 df-gsum 17400 df-prds 17405 df-pws 17407 df-mre 17543 df-mrc 17544 df-acs 17546 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18746 df-submnd 18747 df-efmnd 18832 df-grp 18907 df-minusg 18908 df-sbg 18909 df-mulg 19039 df-subg 19094 df-ghm 19183 df-gim 19228 df-cntz 19286 df-oppg 19315 df-symg 19339 df-pmtr 19411 df-psgn 19460 df-evpm 19461 df-cmn 19751 df-abl 19752 df-mgp 20116 df-rng 20128 df-ur 20157 df-srg 20162 df-ring 20210 df-cring 20211 df-oppr 20311 df-dvdsr 20331 df-unit 20332 df-invr 20362 df-dvr 20375 df-rhm 20446 df-subrng 20521 df-subrg 20545 df-drng 20706 df-lmod 20855 df-lss 20925 df-sra 21166 df-rgmod 21167 df-cnfld 21351 df-zring 21425 df-zrh 21481 df-dsmm 21710 df-frlm 21725 df-mamu 22377 df-mat 22394 df-mvmul 22527 df-marrep 22544 df-marepv 22545 df-subma 22563 df-mdet 22571 df-minmar1 22621

This theorem is referenced by: cramerlem3 22675

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