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Theorem cramerlem2 22750

Description: Lemma 2 for cramer 22753. (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 1227	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ 𝑧 ∈ 𝑉) ∧ (𝑋 · 𝑧) = 𝑌) → 𝑅 ∈ CRing)
2		simpll2 1228	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ 𝑧 ∈ 𝑉) ∧ (𝑋 · 𝑧) = 𝑌) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉))
3		simpll3 1229	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ 𝑧 ∈ 𝑉) ∧ (𝑋 · 𝑧) = 𝑌) → (𝐷‘𝑋) ∈ (Unit‘𝑅))
4		simplr 778	. . . 4 ⊢ ((((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)) ∧ 𝑧 ∈ 𝑉) ∧ (𝑋 · 𝑧) = 𝑌) → 𝑧 ∈ 𝑉)
5		simpr 488	. . . 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 22749	. . . 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 1403	. . 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 416	. 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 3156	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 399 ∧ w3a 1099 = wceq 1562 ∈ wcel 2144 ∀wral 3078 ⟨cop 4590 ↦ cmpt 5183 ‘cfv 6523 (class class class)co 7398 ↑_m cmap 8810 Basecbs 17247 CRingccrg 20286 Unitcui 20406 /_rcdvr 20451 Mat cmat 22469 maVecMul cmvmul 22602 matRepV cmatrepV 22619 maDet cmdat 22646

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-addf 11154 ax-mulf 11155

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-xor 1534 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-se 5603 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-isom 6532 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-of 7662 df-om 7849 df-1st 7972 df-2nd 7973 df-supp 8143 df-tpos 8208 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-2o 8440 df-er 8680 df-map 8812 df-pm 8813 df-ixp 8882 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-fsupp 9310 df-sup 9390 df-oi 9460 df-card 9899 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12484 df-xnn0 12557 df-z 12571 df-dec 12691 df-uz 12842 df-rp 12996 df-fz 13515 df-fzo 13662 df-seq 14017 df-exp 14077 df-hash 14346 df-word 14529 df-lsw 14578 df-concat 14586 df-s1 14612 df-substr 14657 df-pfx 14687 df-splice 14765 df-reverse 14774 df-s2 14863 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17248 df-ress 17269 df-plusg 17301 df-mulr 17302 df-starv 17303 df-sca 17304 df-vsca 17305 df-ip 17306 df-tset 17307 df-ple 17308 df-ds 17310 df-unif 17311 df-hom 17312 df-cco 17313 df-0g 17472 df-gsum 17473 df-prds 17478 df-pws 17480 df-mre 17616 df-mrc 17617 df-acs 17619 df-mgm 18676 df-sgrp 18755 df-mnd 18771 df-mhm 18819 df-submnd 18820 df-efmnd 18905 df-grp 18980 df-minusg 18981 df-sbg 18982 df-mulg 19112 df-subg 19167 df-ghm 19256 df-gim 19301 df-cntz 19359 df-oppg 19388 df-symg 19412 df-pmtr 19484 df-psgn 19533 df-evpm 19534 df-cmn 19824 df-abl 19825 df-mgp 20189 df-rng 20201 df-ur 20234 df-srg 20239 df-ring 20287 df-cring 20288 df-oppr 20388 df-dvdsr 20408 df-unit 20409 df-invr 20439 df-dvr 20452 df-rhm 20523 df-subrng 20598 df-subrg 20622 df-drng 20783 df-lmod 20931 df-lss 21001 df-sra 21242 df-rgmod 21243 df-cnfld 21427 df-zring 21501 df-zrh 21557 df-dsmm 21786 df-frlm 21801 df-mamu 22453 df-mat 22470 df-mvmul 22603 df-marrep 22620 df-marepv 22621 df-subma 22639 df-mdet 22647 df-minmar1 22697

This theorem is referenced by: cramerlem3 22751

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