cramerimplem3 - Metamath Proof Explorer

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Theorem cramerimplem3 22578

Description: Lemma 3 for cramerimp 22579: The determinant of the matrix of a system of linear equations multiplied with the determinant of the identity matrix with the ith column replaced by the solution vector of the system of linear equations equals the determinant of the matrix of the system of linear equations with the ith column replaced by the right-hand side vector of the system of linear equations. (Contributed by AV, 19-Feb-2019.) (Revised by AV, 1-Mar-2019.)

**Hypotheses**
Ref	Expression
cramerimp.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
cramerimp.b	⊢ 𝐵 = (Base‘𝐴)
cramerimp.v	⊢ 𝑉 = ((Base‘𝑅) ↑_m 𝑁)
cramerimp.e	⊢ 𝐸 = (((1_r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼)
cramerimp.h	⊢ 𝐻 = ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝐼)
cramerimp.x	⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)
cramerimp.d	⊢ 𝐷 = (𝑁 maDet 𝑅)
cramerimp.t	⊢ ⊗ = (._r‘𝑅)

**Assertion**
Ref	Expression
cramerimplem3	⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → ((𝐷‘𝑋) ⊗ (𝐷‘𝐸)) = (𝐷‘𝐻))

**Proof of Theorem cramerimplem3**
Step	Hyp	Ref	Expression
1		simpl 482	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) → 𝑅 ∈ CRing)
2		cramerimp.a	. . . . . . . . . 10 ⊢ 𝐴 = (𝑁 Mat 𝑅)
3		cramerimp.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐴)
4	2, 3	matrcl 22305	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
5	4	simpld 494	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → 𝑁 ∈ Fin)
6	5	adantr 480	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) → 𝑁 ∈ Fin)
7	1, 6	anim12ci 614	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing))
8	7	3adant3 1132	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing))
9		eqid 2730	. . . . . 6 ⊢ (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
10	2, 9	matmulr 22331	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (._r‘𝐴))
11	8, 10	syl 17	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (._r‘𝐴))
12	11	oveqd 7406	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → (𝑋(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐸) = (𝑋(._r‘𝐴)𝐸))
13	12	fveq2d 6864	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → (𝐷‘(𝑋(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐸)) = (𝐷‘(𝑋(._r‘𝐴)𝐸)))
14		cramerimp.v	. . . 4 ⊢ 𝑉 = ((Base‘𝑅) ↑_m 𝑁)
15		cramerimp.e	. . . 4 ⊢ 𝐸 = (((1_r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼)
16		cramerimp.h	. . . 4 ⊢ 𝐻 = ((𝑋(𝑁 matRepV 𝑅)𝑌)‘𝐼)
17		cramerimp.x	. . . 4 ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)
18	2, 3, 14, 15, 16, 17, 9	cramerimplem2 22577	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → (𝑋(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐸) = 𝐻)
19	18	fveq2d 6864	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → (𝐷‘(𝑋(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐸)) = (𝐷‘𝐻))
20		simp1l 1198	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → 𝑅 ∈ CRing)
21		simp2l 1200	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → 𝑋 ∈ 𝐵)
22		crngring 20160	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
23	22	adantr 480	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) → 𝑅 ∈ Ring)
24	23, 6	anim12i 613	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
25	24	3adant3 1132	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
26		ne0i 4306	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑁 → 𝑁 ≠ ∅)
27	22, 26	anim12ci 614	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) → (𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring))
28	2, 3, 14, 17	slesolvec 22572	. . . . . . 7 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑍) = 𝑌 → 𝑍 ∈ 𝑉))
29	27, 28	sylan 580	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑍) = 𝑌 → 𝑍 ∈ 𝑉))
30	29	3impia 1117	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → 𝑍 ∈ 𝑉)
31		simp1r 1199	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → 𝐼 ∈ 𝑁)
32		eqid 2730	. . . . . 6 ⊢ (1_r‘𝐴) = (1_r‘𝐴)
33	2, 3, 14, 32	ma1repvcl 22463	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁)) → (((1_r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ 𝐵)
34	25, 30, 31, 33	syl12anc 836	. . . 4 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → (((1_r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ 𝐵)
35	15, 34	eqeltrid 2833	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → 𝐸 ∈ 𝐵)
36		cramerimp.d	. . . 4 ⊢ 𝐷 = (𝑁 maDet 𝑅)
37		cramerimp.t	. . . 4 ⊢ ⊗ = (._r‘𝑅)
38		eqid 2730	. . . 4 ⊢ (._r‘𝐴) = (._r‘𝐴)
39	2, 3, 36, 37, 38	mdetmul 22516	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝐸 ∈ 𝐵) → (𝐷‘(𝑋(._r‘𝐴)𝐸)) = ((𝐷‘𝑋) ⊗ (𝐷‘𝐸)))
40	20, 21, 35, 39	syl3anc 1373	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → (𝐷‘(𝑋(._r‘𝐴)𝐸)) = ((𝐷‘𝑋) ⊗ (𝐷‘𝐸)))
41	13, 19, 40	3eqtr3rd 2774	1 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → ((𝐷‘𝑋) ⊗ (𝐷‘𝐸)) = (𝐷‘𝐻))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 Vcvv 3450 ∅c0 4298 ⟨cop 4597 ⟨cotp 4599 ‘cfv 6513 (class class class)co 7389 ↑_m cmap 8801 Fincfn 8920 Basecbs 17185 ._rcmulr 17227 1_rcur 20096 Ringcrg 20148 CRingccrg 20149 maMul cmmul 22283 Mat cmat 22300 maVecMul cmvmul 22433 matRepV cmatrepV 22450 maDet cmdat 22477

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 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-addf 11153 ax-mulf 11154

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 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-ot 4600 df-uni 4874 df-int 4913 df-iun 4959 df-iin 4960 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-se 5594 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-isom 6522 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-of 7655 df-om 7845 df-1st 7970 df-2nd 7971 df-supp 8142 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-2o 8437 df-er 8673 df-map 8803 df-pm 8804 df-ixp 8873 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-fsupp 9319 df-sup 9399 df-oi 9469 df-card 9898 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-7 12255 df-8 12256 df-9 12257 df-n0 12449 df-xnn0 12522 df-z 12536 df-dec 12656 df-uz 12800 df-rp 12958 df-fz 13475 df-fzo 13622 df-seq 13973 df-exp 14033 df-hash 14302 df-word 14485 df-lsw 14534 df-concat 14542 df-s1 14567 df-substr 14612 df-pfx 14642 df-splice 14721 df-reverse 14730 df-s2 14820 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17186 df-ress 17207 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-0g 17410 df-gsum 17411 df-prds 17416 df-pws 17418 df-mre 17553 df-mrc 17554 df-acs 17556 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18716 df-submnd 18717 df-efmnd 18802 df-grp 18874 df-minusg 18875 df-sbg 18876 df-mulg 19006 df-subg 19061 df-ghm 19151 df-gim 19197 df-cntz 19255 df-oppg 19284 df-symg 19306 df-pmtr 19378 df-psgn 19427 df-evpm 19428 df-cmn 19718 df-abl 19719 df-mgp 20056 df-rng 20068 df-ur 20097 df-srg 20102 df-ring 20150 df-cring 20151 df-oppr 20252 df-dvdsr 20272 df-unit 20273 df-invr 20303 df-dvr 20316 df-rhm 20387 df-subrng 20461 df-subrg 20485 df-drng 20646 df-lmod 20774 df-lss 20844 df-sra 21086 df-rgmod 21087 df-cnfld 21271 df-zring 21363 df-zrh 21419 df-dsmm 21647 df-frlm 21662 df-mamu 22284 df-mat 22301 df-mvmul 22434 df-marepv 22452 df-mdet 22478

This theorem is referenced by: cramerimp 22579

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