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Theorem cramerlem1 22662

Description: Lemma 1 for cramer 22666. (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 1137	. . . . 5 ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) → 𝑅 ∈ CRing)
2	1	anim1i 616	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) ∧ 𝑎 ∈ 𝑁) → (𝑅 ∈ CRing ∧ 𝑎 ∈ 𝑁))
3		simpl2 1194	. . . 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 1132	. . . . . 6 ⊢ (((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌) → ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)))
6	5	3ad2ant3 1136	. . . . 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 2737	. . . . 5 ⊢ (((1_r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝑎) = (((1_r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝑎)
12		eqid 2737	. . . . 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 22661	. . . 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 1374	. . 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 3130	. 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 8805	. . . . . 6 ⊢ (𝑍 ∈ ((Base‘𝑅) ↑_m 𝑁) → 𝑍 Fn 𝑁)
20	19, 10	eleq2s 2855	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → 𝑍 Fn 𝑁)
21	20	3ad2ant2 1135	. . . 4 ⊢ (((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌) → 𝑍 Fn 𝑁)
22	21	3ad2ant3 1136	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) → 𝑍 Fn 𝑁)
23		2fveq3 6839	. . . 4 ⊢ (𝑎 = 𝑖 → (𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎)) = (𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)))
24	23	oveq1d 7375	. . 3 ⊢ (𝑎 = 𝑖 → ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎)) / (𝐷‘𝑋)) = ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))
25		ovexd 7395	. . 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 7395	. . 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 6987	. 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 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 Vcvv 3430 ⟨cop 4574 ↦ cmpt 5167 Fn wfn 6487 ‘cfv 6492 (class class class)co 7360 ↑_m cmap 8766 Basecbs 17170 1_rcur 20153 CRingccrg 20206 Unitcui 20326 /_rcdvr 20371 Mat cmat 22382 maVecMul cmvmul 22515 matRepV cmatrepV 22532 maDet cmdat 22559

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 5302 ax-pr 5370 ax-un 7682 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-addf 11108 ax-mulf 11109

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 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-supp 8104 df-tpos 8169 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8342 df-1o 8398 df-2o 8399 df-er 8636 df-map 8768 df-pm 8769 df-ixp 8839 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-fsupp 9268 df-sup 9348 df-oi 9418 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-xnn0 12502 df-z 12516 df-dec 12636 df-uz 12780 df-rp 12934 df-fz 13453 df-fzo 13600 df-seq 13955 df-exp 14015 df-hash 14284 df-word 14467 df-lsw 14516 df-concat 14524 df-s1 14550 df-substr 14595 df-pfx 14625 df-splice 14703 df-reverse 14712 df-s2 14801 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-0g 17395 df-gsum 17396 df-prds 17401 df-pws 17403 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-submnd 18743 df-efmnd 18828 df-grp 18903 df-minusg 18904 df-sbg 18905 df-mulg 19035 df-subg 19090 df-ghm 19179 df-gim 19225 df-cntz 19283 df-oppg 19312 df-symg 19336 df-pmtr 19408 df-psgn 19457 df-evpm 19458 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-srg 20159 df-ring 20207 df-cring 20208 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-dvr 20372 df-rhm 20443 df-subrng 20514 df-subrg 20538 df-drng 20699 df-lmod 20848 df-lss 20918 df-sra 21160 df-rgmod 21161 df-cnfld 21345 df-zring 21437 df-zrh 21493 df-dsmm 21722 df-frlm 21737 df-mamu 22366 df-mat 22383 df-mvmul 22516 df-marrep 22533 df-marepv 22534 df-subma 22552 df-mdet 22560 df-minmar1 22610

This theorem is referenced by: cramerlem2 22663 cramer 22666

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