cramerimplem3 - Metamath Proof Explorer

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Theorem cramerimplem3 22752

Description: Lemma 3 for cramerimp 22753: 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 486	. . . . . . 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 22479	. . . . . . . . 9 ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
5	4	simpld 498	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → 𝑁 ∈ Fin)
6	5	adantr 484	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) → 𝑁 ∈ Fin)
7	1, 6	anim12ci 623	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing))
8	7	3adant3 1146	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → (𝑁 ∈ Fin ∧ 𝑅 ∈ CRing))
9		eqid 2763	. . . . . 6 ⊢ (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩) = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)
10	2, 9	matmulr 22505	. . . . 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 7413	. . 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 6871	. 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 22751	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → (𝑋(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐸) = 𝐻)
19	18	fveq2d 6871	. 2 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → (𝐷‘(𝑋(𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)𝐸)) = (𝐷‘𝐻))
20		simp1l 1212	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → 𝑅 ∈ CRing)
21		simp2l 1214	. . 3 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → 𝑋 ∈ 𝐵)
22		crngring 20305	. . . . . . . 8 ⊢ (𝑅 ∈ CRing → 𝑅 ∈ Ring)
23	22	adantr 484	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) → 𝑅 ∈ Ring)
24	23, 6	anim12i 622	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → (𝑅 ∈ Ring ∧ 𝑁 ∈ Fin))
25	24	3adant3 1146	. . . . 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 623	. . . . . . 7 ⊢ ((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) → (𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring))
28	2, 3, 14, 17	slesolvec 22746	. . . . . . 7 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑍) = 𝑌 → 𝑍 ∈ 𝑉))
29	27, 28	sylan 589	. . . . . 6 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑍) = 𝑌 → 𝑍 ∈ 𝑉))
30	29	3impia 1131	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → 𝑍 ∈ 𝑉)
31		simp1r 1213	. . . . 5 ⊢ (((𝑅 ∈ CRing ∧ 𝐼 ∈ 𝑁) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) ∧ (𝑋 · 𝑍) = 𝑌) → 𝐼 ∈ 𝑁)
32		eqid 2763	. . . . . 6 ⊢ (1_r‘𝐴) = (1_r‘𝐴)
33	2, 3, 14, 32	ma1repvcl 22637	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝑍 ∈ 𝑉 ∧ 𝐼 ∈ 𝑁)) → (((1_r‘𝐴)(𝑁 matRepV 𝑅)𝑍)‘𝐼) ∈ 𝐵)
34	25, 30, 31, 33	syl12anc 847	. . . 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 2867	. . 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 2763	. . . 4 ⊢ (._r‘𝐴) = (._r‘𝐴)
39	2, 3, 36, 37, 38	mdetmul 22690	. . 3 ⊢ ((𝑅 ∈ CRing ∧ 𝑋 ∈ 𝐵 ∧ 𝐸 ∈ 𝐵) → (𝐷‘(𝑋(._r‘𝐴)𝐸)) = ((𝐷‘𝑋) ⊗ (𝐷‘𝐸)))
40	20, 21, 35, 39	syl3anc 1392	. 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 2807	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 399 ∧ w3a 1099 = wceq 1561 ∈ wcel 2143 ≠ wne 2958 Vcvv 3455 ∅c0 4286 ⟨cop 4589 ⟨cotp 4591 ‘cfv 6521 (class class class)co 7396 ↑_m cmap 8808 Fincfn 8927 Basecbs 17255 ._rcmulr 17297 1_rcur 20241 Ringcrg 20293 CRingccrg 20294 maMul cmmul 22457 Mat cmat 22474 maVecMul cmvmul 22607 matRepV cmatrepV 22624 maDet cmdat 22651

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7718 ax-cnex 11140 ax-resscn 11141 ax-1cn 11142 ax-icn 11143 ax-addcl 11144 ax-addrcl 11145 ax-mulcl 11146 ax-mulrcl 11147 ax-mulcom 11148 ax-addass 11149 ax-mulass 11150 ax-distr 11151 ax-i2m1 11152 ax-1ne0 11153 ax-1rid 11154 ax-rnegex 11155 ax-rrecex 11156 ax-cnre 11157 ax-pre-lttri 11158 ax-pre-lttrn 11159 ax-pre-ltadd 11160 ax-pre-mulgt0 11161 ax-addf 11163 ax-mulf 11164

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-xor 1533 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-ot 4592 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-se 5602 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-om 7847 df-1st 7970 df-2nd 7971 df-supp 8141 df-tpos 8206 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9306 df-sup 9386 df-oi 9456 df-card 9909 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11427 df-neg 11428 df-div 11856 df-nn 12221 df-2 12290 df-3 12291 df-4 12292 df-5 12293 df-6 12294 df-7 12295 df-8 12296 df-9 12297 df-n0 12492 df-xnn0 12565 df-z 12579 df-dec 12699 df-uz 12850 df-rp 13004 df-fz 13523 df-fzo 13670 df-seq 14025 df-exp 14085 df-hash 14354 df-word 14537 df-lsw 14586 df-concat 14594 df-s1 14620 df-substr 14665 df-pfx 14695 df-splice 14773 df-reverse 14782 df-s2 14871 df-struct 17193 df-sets 17210 df-slot 17228 df-ndx 17240 df-base 17256 df-ress 17277 df-plusg 17309 df-mulr 17310 df-starv 17311 df-sca 17312 df-vsca 17313 df-ip 17314 df-tset 17315 df-ple 17316 df-ds 17318 df-unif 17319 df-hom 17320 df-cco 17321 df-0g 17480 df-gsum 17481 df-prds 17486 df-pws 17488 df-mre 17624 df-mrc 17625 df-acs 17627 df-mgm 18684 df-sgrp 18763 df-mnd 18779 df-mhm 18827 df-submnd 18828 df-efmnd 18913 df-grp 18988 df-minusg 18989 df-sbg 18990 df-mulg 19120 df-subg 19175 df-ghm 19264 df-gim 19309 df-cntz 19367 df-oppg 19396 df-symg 19420 df-pmtr 19492 df-psgn 19541 df-evpm 19542 df-cmn 19832 df-abl 19833 df-mgp 20197 df-rng 20209 df-ur 20242 df-srg 20247 df-ring 20295 df-cring 20296 df-oppr 20396 df-dvdsr 20416 df-unit 20417 df-invr 20447 df-dvr 20460 df-rhm 20531 df-subrng 20606 df-subrg 20630 df-drng 20790 df-lmod 20936 df-lss 21006 df-sra 21247 df-rgmod 21248 df-cnfld 21432 df-zring 21506 df-zrh 21562 df-dsmm 21791 df-frlm 21806 df-mamu 22458 df-mat 22475 df-mvmul 22608 df-marepv 22626 df-mdet 22652

This theorem is referenced by: cramerimp 22753

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