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Theorem cramerlem1 22709

Description: Lemma 1 for cramer 22713. (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 1135	. . . . 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 1191	. . . 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 1130	. . . . . 6 ⊢ (((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌) → ((𝑋 · 𝑍) = 𝑌 ∧ (𝐷‘𝑋) ∈ (Unit‘𝑅)))
6	5	3ad2ant3 1134	. . . . 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 2735	. . . . 5 ⊢ (((1_r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝑎) = (((1_r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝑎)
12		eqid 2735	. . . . 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 22708	. . . 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 1370	. . 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 3144	. 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 8904	. . . . . 6 ⊢ (𝑍 ∈ ((Base‘𝑅) ↑_m 𝑁) → 𝑍 Fn 𝑁)
20	19, 10	eleq2s 2857	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → 𝑍 Fn 𝑁)
21	20	3ad2ant2 1133	. . . 4 ⊢ (((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌) → 𝑍 Fn 𝑁)
22	21	3ad2ant3 1134	. . 3 ⊢ ((𝑅 ∈ CRing ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ ((𝐷‘𝑋) ∈ (Unit‘𝑅) ∧ 𝑍 ∈ 𝑉 ∧ (𝑋 · 𝑍) = 𝑌)) → 𝑍 Fn 𝑁)
23		2fveq3 6912	. . . 4 ⊢ (𝑎 = 𝑖 → (𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎)) = (𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)))
24	23	oveq1d 7446	. . 3 ⊢ (𝑎 = 𝑖 → ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑎)) / (𝐷‘𝑋)) = ((𝐷‘((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝑖)) / (𝐷‘𝑋)))
25		ovexd 7466	. . 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 7466	. . 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 7061	. 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 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ⟨cop 4637 ↦ cmpt 5231 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 ↑_m cmap 8865 Basecbs 17245 1_rcur 20199 CRingccrg 20252 Unitcui 20372 /_rcdvr 20417 Mat cmat 22427 maVecMul cmvmul 22562 matRepV cmatrepV 22579 maDet cmdat 22606

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-addf 11232 ax-mulf 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-xor 1509 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-xnn0 12598 df-z 12612 df-dec 12732 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-seq 14040 df-exp 14100 df-hash 14367 df-word 14550 df-lsw 14598 df-concat 14606 df-s1 14631 df-substr 14676 df-pfx 14706 df-splice 14785 df-reverse 14794 df-s2 14884 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-0g 17488 df-gsum 17489 df-prds 17494 df-pws 17496 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-efmnd 18895 df-grp 18967 df-minusg 18968 df-sbg 18969 df-mulg 19099 df-subg 19154 df-ghm 19244 df-gim 19290 df-cntz 19348 df-oppg 19377 df-symg 19402 df-pmtr 19475 df-psgn 19524 df-evpm 19525 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-srg 20205 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-rhm 20489 df-subrng 20563 df-subrg 20587 df-drng 20748 df-lmod 20877 df-lss 20948 df-sra 21190 df-rgmod 21191 df-cnfld 21383 df-zring 21476 df-zrh 21532 df-dsmm 21770 df-frlm 21785 df-mamu 22411 df-mat 22428 df-mvmul 22563 df-marrep 22580 df-marepv 22581 df-subma 22599 df-mdet 22607 df-minmar1 22657

This theorem is referenced by: cramerlem2 22710 cramer 22713

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