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Theorem cramerlem1 22581

Description: Lemma 1 for cramer 22585. (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
cramerlem1	⊢ ((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,𝑍 · ,𝑖 / ,𝑖 Allowed substitution hint: 𝐴(𝑖)

Proof of Theorem cramerlem1 Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1136	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) → 𝑅 ∈ CRing)
2	1	anim1i 615	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) ∧ 𝑎 ∈ 𝑁) → (𝑅 ∈ CRing ∧ 𝑎 ∈ 𝑁))
3		simpl2 1193	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) ∧ 𝑎 ∈ 𝑁) → (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉))
4		pm3.22 459	. . . . . . 7 ⊢ (((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ (𝑋 · 𝑍) = 𝑌) → ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)))
5	4	3adant2 1131	. . . . . 6 ⊢ (((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌) → ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)))
6	5	3ad2ant3 1135	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) → ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)))
7	6	adantr 480	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) ∧ 𝑎 ∈ 𝑁) → ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)))
8		cramer.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
9		cramer.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
10		cramer.v	. . . . 5 ⊢ 𝑉 = ((Base‘𝑅) ↑_m 𝑁)
11		eqid 2730	. . . . 5 ⊢ (((1_r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝑎) = (((1_r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝑎)
12		eqid 2730	. . . . 5 ⊢ ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎) = ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎)
13		cramer.x	. . . . 5 ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)
14		cramer.d	. . . . 5 ⊢ 𝐷 = (𝑁 maDet 𝑅)
15		cramer.q	. . . . 5 ⊢ / = (/_r‘𝑅)
16	8, 9, 10, 11, 12, 13, 14, 15	cramerimp 22580	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝑎 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅))) → (𝑍‘𝑎) = ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎)) / (𝐷‘𝑋)))
17	2, 3, 7, 16	syl3anc 1373	. . 3 ⊢ (((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) ∧ 𝑎 ∈ 𝑁) → (𝑍‘𝑎) = ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎)) / (𝐷‘𝑋)))
18	17	ralrimiva 3126	. 2 ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) → ∀𝑎 ∈ 𝑁 (𝑍‘𝑎) = ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎)) / (𝐷‘𝑋)))
19		elmapfn 8841	. . . . . 6 ⊢ (𝑍 ∈ ((Base‘𝑅) ↑_m 𝑁) → 𝑍 Fn 𝑁)
20	19, 10	eleq2s 2847	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → 𝑍 Fn 𝑁)
21	20	3ad2ant2 1134	. . . 4 ⊢ (((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌) → 𝑍 Fn 𝑁)
22	21	3ad2ant3 1135	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) → 𝑍 Fn 𝑁)
23		2fveq3 6866	. . . 4 ⊢ (𝑎 = 𝑖 → (𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎)) = (𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)))
24	23	oveq1d 7405	. . 3 ⊢ (𝑎 = 𝑖 → ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎)) / (𝐷‘𝑋)) = ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))
25		ovexd 7425	. . 3 ⊢ (((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) ∧ 𝑎 ∈ 𝑁) → ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎)) / (𝐷‘𝑋)) ∈ V)
26		ovexd 7425	. . 3 ⊢ (((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) ∧ 𝑖 ∈ 𝑁) → ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)) ∈ V)
27	22, 24, 25, 26	fnmptfvd 7016	. 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))) ↔ ∀𝑎 ∈ 𝑁 (𝑍‘𝑎) = ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎)) / (𝐷‘𝑋))))
28	18, 27	mpbird 257	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 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3045 Vcvv 3450 ⟨cop 4598 ↦ cmpt 5191 Fn wfn 6509 ‘cfv 6514 (class class class)co 7390 ↑_m cmap 8802 Basecbs 17186 1_rcur 20097 CRingccrg 20150 Unitcui 20271 /_rcdvr 20316 Mat cmat 22301 maVecMul cmvmul 22434 matRepV cmatrepV 22451 maDet cmdat 22478

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 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 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1512 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-ot 4601 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-sup 9400 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-xnn0 12523 df-z 12537 df-dec 12657 df-uz 12801 df-rp 12959 df-fz 13476 df-fzo 13623 df-seq 13974 df-exp 14034 df-hash 14303 df-word 14486 df-lsw 14535 df-concat 14543 df-s1 14568 df-substr 14613 df-pfx 14643 df-splice 14722 df-reverse 14731 df-s2 14821 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-0g 17411 df-gsum 17412 df-prds 17417 df-pws 17419 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-mhm 18717 df-submnd 18718 df-efmnd 18803 df-grp 18875 df-minusg 18876 df-sbg 18877 df-mulg 19007 df-subg 19062 df-ghm 19152 df-gim 19198 df-cntz 19256 df-oppg 19285 df-symg 19307 df-pmtr 19379 df-psgn 19428 df-evpm 19429 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-srg 20103 df-ring 20151 df-cring 20152 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-rhm 20388 df-subrng 20462 df-subrg 20486 df-drng 20647 df-lmod 20775 df-lss 20845 df-sra 21087 df-rgmod 21088 df-cnfld 21272 df-zring 21364 df-zrh 21420 df-dsmm 21648 df-frlm 21663 df-mamu 22285 df-mat 22302 df-mvmul 22435 df-marrep 22452 df-marepv 22453 df-subma 22471 df-mdet 22479 df-minmar1 22529

This theorem is referenced by: cramerlem2 22582 cramer 22585

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