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Theorem matecl 21574

Description: Each entry (according to Wikipedia "Matrix (mathematics)", 30-Dec-2018, https://en.wikipedia.org/wiki/Matrix_(mathematics)#Definition (or element or component or coefficient or cell) of a matrix is an element of the underlying ring. (Contributed by AV, 16-Dec-2018.)

**Hypotheses**
Ref	Expression
matecl.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
matecl.k	⊢ 𝐾 = (Base‘𝑅)

**Assertion**
Ref	Expression
matecl	⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝐼𝑀𝐽) ∈ 𝐾)

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

Step	Hyp	Ref	Expression
1		matecl.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		eqid 2738	. . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴)
3	1, 2	matrcl 21559	. . 3 ⊢ (𝑀 ∈ (Base‘𝐴) → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
4	3	3ad2ant3 1134	. 2 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
5		matecl.k	. . . . . . . . 9 ⊢ 𝐾 = (Base‘𝑅)
6	1, 5	matbas2 21570	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝐾 ↑_m (𝑁 × 𝑁)) = (Base‘𝐴))
7	6	eqcomd 2744	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘𝐴) = (𝐾 ↑_m (𝑁 × 𝑁)))
8	7	eleq2d 2824	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑀 ∈ (Base‘𝐴) ↔ 𝑀 ∈ (𝐾 ↑_m (𝑁 × 𝑁))))
9	5	fvexi 6788	. . . . . . . . 9 ⊢ 𝐾 ∈ V
10	9	a1i 11	. . . . . . . 8 ⊢ (𝑅 ∈ V → 𝐾 ∈ V)
11		sqxpexg 7605	. . . . . . . 8 ⊢ (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ V)
12		elmapg 8628	. . . . . . . 8 ⊢ ((𝐾 ∈ V ∧ (𝑁 × 𝑁) ∈ V) → (𝑀 ∈ (𝐾 ↑_m (𝑁 × 𝑁)) ↔ 𝑀:(𝑁 × 𝑁)⟶𝐾))
13	10, 11, 12	syl2anr 597	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑀 ∈ (𝐾 ↑_m (𝑁 × 𝑁)) ↔ 𝑀:(𝑁 × 𝑁)⟶𝐾))
14		ffnov 7401	. . . . . . . 8 ⊢ (𝑀:(𝑁 × 𝑁)⟶𝐾 ↔ (𝑀 Fn (𝑁 × 𝑁) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) ∈ 𝐾))
15		oveq1 7282	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐼 → (𝑖𝑀𝑗) = (𝐼𝑀𝑗))
16	15	eleq1d 2823	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → ((𝑖𝑀𝑗) ∈ 𝐾 ↔ (𝐼𝑀𝑗) ∈ 𝐾))
17		oveq2 7283	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → (𝐼𝑀𝑗) = (𝐼𝑀𝐽))
18	17	eleq1d 2823	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → ((𝐼𝑀𝑗) ∈ 𝐾 ↔ (𝐼𝑀𝐽) ∈ 𝐾))
19	16, 18	rspc2v 3570	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) ∈ 𝐾 → (𝐼𝑀𝐽) ∈ 𝐾))
20	19	com12 32	. . . . . . . . . 10 ⊢ (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) ∈ 𝐾 → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝐼𝑀𝐽) ∈ 𝐾))
21	20	adantl 482	. . . . . . . . 9 ⊢ ((𝑀 Fn (𝑁 × 𝑁) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) ∈ 𝐾) → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝐼𝑀𝐽) ∈ 𝐾))
22	21	a1i 11	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → ((𝑀 Fn (𝑁 × 𝑁) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) ∈ 𝐾) → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝐼𝑀𝐽) ∈ 𝐾)))
23	14, 22	syl5bi 241	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑀:(𝑁 × 𝑁)⟶𝐾 → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝐼𝑀𝐽) ∈ 𝐾)))
24	13, 23	sylbid 239	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑀 ∈ (𝐾 ↑_m (𝑁 × 𝑁)) → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝐼𝑀𝐽) ∈ 𝐾)))
25	8, 24	sylbid 239	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑀 ∈ (Base‘𝐴) → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝐼𝑀𝐽) ∈ 𝐾)))
26	25	com13 88	. . . 4 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝑀 ∈ (Base‘𝐴) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝐼𝑀𝐽) ∈ 𝐾)))
27	26	ex 413	. . 3 ⊢ (𝐼 ∈ 𝑁 → (𝐽 ∈ 𝑁 → (𝑀 ∈ (Base‘𝐴) → ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝐼𝑀𝐽) ∈ 𝐾))))
28	27	3imp1 1346	. 2 ⊢ (((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) ∧ (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) → (𝐼𝑀𝐽) ∈ 𝐾)
29	4, 28	mpdan 684	1 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝐼𝑀𝐽) ∈ 𝐾)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 Vcvv 3432 × cxp 5587 Fn wfn 6428 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 Fincfn 8733 Basecbs 16912 Mat cmat 21554

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-hom 16986 df-cco 16987 df-0g 17152 df-prds 17158 df-pws 17160 df-sra 20434 df-rgmod 20435 df-dsmm 20939 df-frlm 20954 df-mat 21555

This theorem is referenced by: matecld 21575 matinvgcell 21584 matepmcl 21611 matepm2cl 21612 dmatmul 21646 marrepcl 21713 marepvcl 21718 mulmarep1el 21721 mulmarep1gsum1 21722 submabas 21727 m1detdiag 21746 mdetdiag 21748 m2detleib 21780 marep01ma 21809 smadiadetlem4 21818 mat2pmatbas 21875 decpmatmul 21921 pm2mpghm 21965 chpscmat 21991 chpscmatgsumbin 21993 chpscmatgsummon 21994 mdetlap1 31776

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