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Theorem matvscacell 22351

Description: Scalar multiplication in the matrix ring is cell-wise. (Contributed by AV, 7-Aug-2019.)

**Hypotheses**
Ref	Expression
matplusgcell.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
matplusgcell.b	⊢ 𝐵 = (Base‘𝐴)
matvscacell.k	⊢ 𝐾 = (Base‘𝑅)
matvscacell.v	⊢ · = ( ·_𝑠 ‘𝐴)
matvscacell.t	⊢ × = (._r‘𝑅)

**Assertion**
Ref	Expression
matvscacell	⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 · 𝑌)𝐽) = (𝑋 × (𝐼𝑌𝐽)))

**Proof of Theorem matvscacell**
Step	Hyp	Ref	Expression
1		matplusgcell.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		matplusgcell.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
3		matvscacell.k	. . . . 5 ⊢ 𝐾 = (Base‘𝑅)
4		matvscacell.v	. . . . 5 ⊢ · = ( ·_𝑠 ‘𝐴)
5		matvscacell.t	. . . . 5 ⊢ × = (._r‘𝑅)
6		eqid 2725	. . . . 5 ⊢ (𝑁 × 𝑁) = (𝑁 × 𝑁)
7	1, 2, 3, 4, 5, 6	matvsca2 22343	. . . 4 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) = (((𝑁 × 𝑁) × {𝑋}) ∘_f × 𝑌))
8	7	oveqd 7430	. . 3 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → (𝐼(𝑋 · 𝑌)𝐽) = (𝐼(((𝑁 × 𝑁) × {𝑋}) ∘_f × 𝑌)𝐽))
9	8	3ad2ant2 1131	. 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 · 𝑌)𝐽) = (𝐼(((𝑁 × 𝑁) × {𝑋}) ∘_f × 𝑌)𝐽))
10		df-ov 7416	. . 3 ⊢ (𝐼(((𝑁 × 𝑁) × {𝑋}) ∘_f × 𝑌)𝐽) = ((((𝑁 × 𝑁) × {𝑋}) ∘_f × 𝑌)‘⟨𝐼, 𝐽⟩)
11	10	a1i 11	. 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(((𝑁 × 𝑁) × {𝑋}) ∘_f × 𝑌)𝐽) = ((((𝑁 × 𝑁) × {𝑋}) ∘_f × 𝑌)‘⟨𝐼, 𝐽⟩))
12		opelxpi 5710	. . . 4 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → ⟨𝐼, 𝐽⟩ ∈ (𝑁 × 𝑁))
13	12	3ad2ant3 1132	. . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → ⟨𝐼, 𝐽⟩ ∈ (𝑁 × 𝑁))
14	1, 2	matrcl 22325	. . . . . . . 8 ⊢ (𝑌 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
15	14	simpld 493	. . . . . . 7 ⊢ (𝑌 ∈ 𝐵 → 𝑁 ∈ Fin)
16	15	adantl 480	. . . . . 6 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑁 ∈ Fin)
17	16	3ad2ant2 1131	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑁 ∈ Fin)
18		xpfi 9336	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑁 × 𝑁) ∈ Fin)
19	17, 17, 18	syl2anc 582	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝑁 × 𝑁) ∈ Fin)
20		simp2l 1196	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑋 ∈ 𝐾)
21	2	eleq2i 2817	. . . . . . . . 9 ⊢ (𝑌 ∈ 𝐵 ↔ 𝑌 ∈ (Base‘𝐴))
22	21	biimpi 215	. . . . . . . 8 ⊢ (𝑌 ∈ 𝐵 → 𝑌 ∈ (Base‘𝐴))
23	22	adantl 480	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) → 𝑌 ∈ (Base‘𝐴))
24	23	3ad2ant2 1131	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑌 ∈ (Base‘𝐴))
25		simp1 1133	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑅 ∈ Ring)
26		eqid 2725	. . . . . . . 8 ⊢ (Base‘𝑅) = (Base‘𝑅)
27	1, 26	matbas2 22336	. . . . . . 7 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((Base‘𝑅) ↑_m (𝑁 × 𝑁)) = (Base‘𝐴))
28	17, 25, 27	syl2anc 582	. . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → ((Base‘𝑅) ↑_m (𝑁 × 𝑁)) = (Base‘𝐴))
29	24, 28	eleqtrrd 2828	. . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑌 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)))
30		elmapfn 8877	. . . . 5 ⊢ (𝑌 ∈ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)) → 𝑌 Fn (𝑁 × 𝑁))
31	29, 30	syl 17	. . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → 𝑌 Fn (𝑁 × 𝑁))
32		df-ov 7416	. . . . . 6 ⊢ (𝐼𝑌𝐽) = (𝑌‘⟨𝐼, 𝐽⟩)
33	32	eqcomi 2734	. . . . 5 ⊢ (𝑌‘⟨𝐼, 𝐽⟩) = (𝐼𝑌𝐽)
34	33	a1i 11	. . . 4 ⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ ⟨𝐼, 𝐽⟩ ∈ (𝑁 × 𝑁)) → (𝑌‘⟨𝐼, 𝐽⟩) = (𝐼𝑌𝐽))
35	19, 20, 31, 34	ofc1 7706	. . 3 ⊢ (((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) ∧ ⟨𝐼, 𝐽⟩ ∈ (𝑁 × 𝑁)) → ((((𝑁 × 𝑁) × {𝑋}) ∘_f × 𝑌)‘⟨𝐼, 𝐽⟩) = (𝑋 × (𝐼𝑌𝐽)))
36	13, 35	mpdan 685	. 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → ((((𝑁 × 𝑁) × {𝑋}) ∘_f × 𝑌)‘⟨𝐼, 𝐽⟩) = (𝑋 × (𝐼𝑌𝐽)))
37	9, 11, 36	3eqtrd 2769	1 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐵) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁)) → (𝐼(𝑋 · 𝑌)𝐽) = (𝑋 × (𝐼𝑌𝐽)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 Vcvv 3463 {csn 4625 ⟨cop 4631 × cxp 5671 Fn wfn 6538 ‘cfv 6543 (class class class)co 7413 ∘_f cof 7677 ↑_m cmap 8838 Fincfn 8957 Basecbs 17174 ._rcmulr 17228 ·_𝑠 cvsca 17231 Ringcrg 20172 Mat cmat 22320

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-ot 4634 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-om 7866 df-1st 7987 df-2nd 7988 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9381 df-sup 9460 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-fz 13512 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-hom 17251 df-cco 17252 df-0g 17417 df-prds 17423 df-pws 17425 df-sra 21057 df-rgmod 21058 df-dsmm 21665 df-frlm 21680 df-mat 22321

This theorem is referenced by: dmatscmcl 22418 scmatscmide 22422 scmatscm 22428 mat2pmatlin 22650 monmatcollpw 22694 pmatcollpwlem 22695 chpmat1dlem 22750 chpdmatlem2 22754 chpdmatlem3 22755

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