cramerimplem3 - Metamath Proof Explorer

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Theorem cramerimplem3 22634

Description: Lemma 3 for cramerimp 22635: 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 22361	. . . . . . . . 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 615	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing))
8	7	3adant3 1133	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing))
9		eqid 2737	. . . . . 6 ⊢ (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
10	2, 9	matmulr 22387	. . . . 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 7378	. . 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 6839	. 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 22633	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → (𝑋(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐸) = 𝐻)
19	18	fveq2d 6839	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → (𝐷‘(𝑋(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐸)) = (𝐷‘𝐻))
20		simp1l 1199	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → 𝑅 ∈ CRing)
21		simp2l 1201	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → 𝑋 ∈ 𝐵)
22		crngring 20185	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
23	22	adantr 480	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) → 𝑅 ∈ Ring)
24	23, 6	anim12i 614	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
25	24	3adant3 1133	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
26		ne0i 4294	. . . . . . . 8 ⊢ (𝐼 ∈ 𝑁 → 𝑁 ≠ ∅)
27	22, 26	anim12ci 615	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) → (𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring))
28	2, 3, 14, 17	slesolvec 22628	. . . . . . 7 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑍) = 𝑌 → 𝑍 ∈ 𝑉))
29	27, 28	sylan 581	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑍) = 𝑌 → 𝑍 ∈ 𝑉))
30	29	3impia 1118	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → 𝑍 ∈ 𝑉)
31		simp1r 1200	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → 𝐼 ∈ 𝑁)
32		eqid 2737	. . . . . 6 ⊢ (1_r‘𝐴) = (1_r‘𝐴)
33	2, 3, 14, 32	ma1repvcl 22519	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁)) → (((1_r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ 𝐵)
34	25, 30, 31, 33	syl12anc 837	. . . 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 2841	. . 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 2737	. . . 4 ⊢ (._r‘𝐴) = (._r‘𝐴)
39	2, 3, 36, 37, 38	mdetmul 22572	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝐸 ∈ 𝐵) → (𝐷‘(𝑋(._r‘𝐴)𝐸)) = ((𝐷‘𝑋) ⊗ (𝐷‘𝐸)))
40	20, 21, 35, 39	syl3anc 1374	. 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 2781	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 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2933 Vcvv 3441 ∅c0 4286 ⟨cop 4587 ⟨cotp 4589 ‘cfv 6493 (class class class)co 7361 ↑_m cmap 8768 Fincfn 8888 Basecbs 17141 ._rcmulr 17183 1_rcur 20121 Ringcrg 20173 CRingccrg 20174 maMul cmmul 22339 Mat cmat 22356 maVecMul cmvmul 22489 matRepV cmatrepV 22506 maDet cmdat 22533

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 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7683 ax-cnex 11087 ax-resscn 11088 ax-1cn 11089 ax-icn 11090 ax-addcl 11091 ax-addrcl 11092 ax-mulcl 11093 ax-mulrcl 11094 ax-mulcom 11095 ax-addass 11096 ax-mulass 11097 ax-distr 11098 ax-i2m1 11099 ax-1ne0 11100 ax-1rid 11101 ax-rnegex 11102 ax-rrecex 11103 ax-cnre 11104 ax-pre-lttri 11105 ax-pre-lttrn 11106 ax-pre-ltadd 11107 ax-pre-mulgt0 11108 ax-addf 11110 ax-mulf 11111

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 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-ot 4590 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-isom 6502 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-om 7812 df-1st 7936 df-2nd 7937 df-supp 8106 df-tpos 8171 df-frecs 8226 df-wrecs 8257 df-recs 8306 df-rdg 8344 df-1o 8400 df-2o 8401 df-er 8638 df-map 8770 df-pm 8771 df-ixp 8841 df-en 8889 df-dom 8890 df-sdom 8891 df-fin 8892 df-fsupp 9270 df-sup 9350 df-oi 9420 df-card 9856 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-div 11800 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-7 12218 df-8 12219 df-9 12220 df-n0 12407 df-xnn0 12480 df-z 12494 df-dec 12613 df-uz 12757 df-rp 12911 df-fz 13429 df-fzo 13576 df-seq 13930 df-exp 13990 df-hash 14259 df-word 14442 df-lsw 14491 df-concat 14499 df-s1 14525 df-substr 14570 df-pfx 14600 df-splice 14678 df-reverse 14687 df-s2 14776 df-struct 17079 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17142 df-ress 17163 df-plusg 17195 df-mulr 17196 df-starv 17197 df-sca 17198 df-vsca 17199 df-ip 17200 df-tset 17201 df-ple 17202 df-ds 17204 df-unif 17205 df-hom 17206 df-cco 17207 df-0g 17366 df-gsum 17367 df-prds 17372 df-pws 17374 df-mre 17510 df-mrc 17511 df-acs 17513 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-mhm 18713 df-submnd 18714 df-efmnd 18799 df-grp 18871 df-minusg 18872 df-sbg 18873 df-mulg 19003 df-subg 19058 df-ghm 19147 df-gim 19193 df-cntz 19251 df-oppg 19280 df-symg 19304 df-pmtr 19376 df-psgn 19425 df-evpm 19426 df-cmn 19716 df-abl 19717 df-mgp 20081 df-rng 20093 df-ur 20122 df-srg 20127 df-ring 20175 df-cring 20176 df-oppr 20278 df-dvdsr 20298 df-unit 20299 df-invr 20329 df-dvr 20342 df-rhm 20413 df-subrng 20484 df-subrg 20508 df-drng 20669 df-lmod 20818 df-lss 20888 df-sra 21130 df-rgmod 21131 df-cnfld 21315 df-zring 21407 df-zrh 21463 df-dsmm 21692 df-frlm 21707 df-mamu 22340 df-mat 22357 df-mvmul 22490 df-marepv 22508 df-mdet 22534

This theorem is referenced by: cramerimp 22635

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