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Theorem matecl 22447

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 2735	. . . 4 ⊢ (Base‘𝐴) = (Base‘𝐴)
3	1, 2	matrcl 22432	. . 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 22443	. . . . . . . 8 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝐾 ↑_m (𝑁 × 𝑁)) = (Base‘𝐴))
7	6	eqcomd 2741	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘𝐴) = (𝐾 ↑_m (𝑁 × 𝑁)))
8	7	eleq2d 2825	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑀 ∈ (Base‘𝐴) ↔ 𝑀 ∈ (𝐾 ↑_m (𝑁 × 𝑁))))
9	5	fvexi 6921	. . . . . . . . 9 ⊢ 𝐾 ∈ V
10	9	a1i 11	. . . . . . . 8 ⊢ (𝑅 ∈ V → 𝐾 ∈ V)
11		sqxpexg 7774	. . . . . . . 8 ⊢ (𝑁 ∈ Fin → (𝑁 × 𝑁) ∈ V)
12		elmapg 8878	. . . . . . . 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 7559	. . . . . . . 8 ⊢ (𝑀:(𝑁 × 𝑁)⟶𝐾 ↔ (𝑀 Fn (𝑁 × 𝑁) ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖𝑀𝑗) ∈ 𝐾))
15		oveq1 7438	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝐼 → (𝑖𝑀𝑗) = (𝐼𝑀𝑗))
16	15	eleq1d 2824	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → ((𝑖𝑀𝑗) ∈ 𝐾 ↔ (𝐼𝑀𝑗) ∈ 𝐾))
17		oveq2 7439	. . . . . . . . . . . . 13 ⊢ (𝑗 = 𝐽 → (𝐼𝑀𝑗) = (𝐼𝑀𝐽))
18	17	eleq1d 2824	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → ((𝐼𝑀𝑗) ∈ 𝐾 ↔ (𝐼𝑀𝐽) ∈ 𝐾))
19	16, 18	rspc2v 3633	. . . . . . . . . . 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 481	. . . . . . . . 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	biimtrid 242	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑀:(𝑁 × 𝑁)⟶𝐾 → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝐼𝑀𝐽) ∈ 𝐾)))
24	13, 23	sylbid 240	. . . . . 6 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑀 ∈ (𝐾 ↑_m (𝑁 × 𝑁)) → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝐼𝑀𝐽) ∈ 𝐾)))
25	8, 24	sylbid 240	. . . . 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 412	. . 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 687	1 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝑀 ∈ (Base‘𝐴)) → (𝐼𝑀𝐽) ∈ 𝐾)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 × cxp 5687 Fn wfn 6558 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑_m cmap 8865 Fincfn 8984 Basecbs 17245 Mat cmat 22427

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-sup 9480 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-hom 17322 df-cco 17323 df-0g 17488 df-prds 17494 df-pws 17496 df-sra 21190 df-rgmod 21191 df-dsmm 21770 df-frlm 21785 df-mat 22428

This theorem is referenced by: matecld 22448 matinvgcell 22457 matepmcl 22484 matepm2cl 22485 dmatmul 22519 marrepcl 22586 marepvcl 22591 mulmarep1el 22594 mulmarep1gsum1 22595 submabas 22600 m1detdiag 22619 mdetdiag 22621 m2detleib 22653 marep01ma 22682 smadiadetlem4 22691 mat2pmatbas 22748 decpmatmul 22794 pm2mpghm 22838 chpscmat 22864 chpscmatgsumbin 22866 chpscmatgsummon 22867 mdetlap1 33787

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