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Theorem marrepcl 22465

Description: Closure of the row replacement function for square matrices. (Contributed by AV, 13-Feb-2019.)

**Hypotheses**
Ref	Expression
marrepcl.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
marrepcl.b	⊢ 𝐵 = (Base‘𝐴)

**Assertion**
Ref	Expression
marrepcl	⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐿) ∈ 𝐵)

Proof of Theorem marrepcl Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		marrepcl.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		marrepcl.b	. . . 4 ⊢ 𝐵 = (Base‘𝐴)
3		eqid 2728	. . . 4 ⊢ (𝑁 matRRep 𝑅) = (𝑁 matRRep 𝑅)
4		eqid 2728	. . . 4 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
5	1, 2, 3, 4	marrepval 22463	. . 3 ⊢ (((𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐿) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, (0_g‘𝑅)), (𝑖𝑀𝑗))))
6	5	3adantl1 1164	. 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐿) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, (0_g‘𝑅)), (𝑖𝑀𝑗))))
7		eqid 2728	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
8	1, 2	matrcl 22311	. . . . . 6 ⊢ (𝑀 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
9	8	simpld 494	. . . . 5 ⊢ (𝑀 ∈ 𝐵 → 𝑁 ∈ Fin)
10	9	3ad2ant2 1132	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) → 𝑁 ∈ Fin)
11	10	adantr 480	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → 𝑁 ∈ Fin)
12		simpl1 1189	. . 3 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → 𝑅 ∈ Ring)
13		simp3 1136	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) → 𝑆 ∈ (Base‘𝑅))
14	7, 4	ring0cl 20202	. . . . . . . 8 ⊢ (𝑅 ∈ Ring → (0_g‘𝑅) ∈ (Base‘𝑅))
15	14	3ad2ant1 1131	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) → (0_g‘𝑅) ∈ (Base‘𝑅))
16	13, 15	ifcld 4575	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) → if(𝑗 = 𝐿, 𝑆, (0_g‘𝑅)) ∈ (Base‘𝑅))
17	16	adantr 480	. . . . 5 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → if(𝑗 = 𝐿, 𝑆, (0_g‘𝑅)) ∈ (Base‘𝑅))
18	17	3ad2ant1 1131	. . . 4 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑗 = 𝐿, 𝑆, (0_g‘𝑅)) ∈ (Base‘𝑅))
19		simp2 1135	. . . . 5 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
20		simp3 1136	. . . . 5 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
21	2	eleq2i 2821	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝐵 ↔ 𝑀 ∈ (Base‘𝐴))
22	21	biimpi 215	. . . . . . . 8 ⊢ (𝑀 ∈ 𝐵 → 𝑀 ∈ (Base‘𝐴))
23	22	3ad2ant2 1132	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) → 𝑀 ∈ (Base‘𝐴))
24	23	adantr 480	. . . . . 6 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → 𝑀 ∈ (Base‘𝐴))
25	24	3ad2ant1 1131	. . . . 5 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → 𝑀 ∈ (Base‘𝐴))
26	1, 7	matecl 22326	. . . . 5 ⊢ ((𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝑖𝑀𝑗) ∈ (Base‘𝑅))
27	19, 20, 25, 26	syl3anc 1369	. . . 4 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → (𝑖𝑀𝑗) ∈ (Base‘𝑅))
28	18, 27	ifcld 4575	. . 3 ⊢ ((((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑁) → if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, (0_g‘𝑅)), (𝑖𝑀𝑗)) ∈ (Base‘𝑅))
29	1, 7, 2, 11, 12, 28	matbas2d 22324	. 2 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, (0_g‘𝑅)), (𝑖𝑀𝑗))) ∈ 𝐵)
30	6, 29	eqeltrd 2829	1 ⊢ (((𝑅 ∈ Ring ∧ 𝑀 ∈ 𝐵 ∧ 𝑆 ∈ (Base‘𝑅)) ∧ (𝐾 ∈ 𝑁 ∧ 𝐿 ∈ 𝑁)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐿) ∈ 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 Vcvv 3471 ifcif 4529 ‘cfv 6548 (class class class)co 7420 ∈ cmpo 7422 Fincfn 8963 Basecbs 17179 0_gc0g 17420 Ringcrg 20172 Mat cmat 22306 matRRep cmarrep 22457

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-ot 4638 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-map 8846 df-ixp 8916 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-fsupp 9386 df-sup 9465 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-fz 13517 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-hom 17256 df-cco 17257 df-0g 17422 df-prds 17428 df-pws 17430 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-grp 18892 df-ring 20174 df-sra 21057 df-rgmod 21058 df-dsmm 21665 df-frlm 21680 df-mat 22307 df-marrep 22459

This theorem is referenced by: minmar1cl 22552 smadiadetg 22574 submatminr1 33411

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