mavmulcl - Metamath Proof Explorer

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Theorem mavmulcl 22485

Description: Multiplication of an NxN matrix with an N-dimensional vector results in an N-dimensional vector. (Contributed by AV, 6-Dec-2018.) (Revised by AV, 23-Feb-2019.) (Proof shortened by AV, 23-Jul-2019.)

**Hypotheses**
Ref	Expression
mavmulcl.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
mavmulcl.m	⊢ × = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)
mavmulcl.b	⊢ 𝐵 = (Base‘𝑅)
mavmulcl.t	⊢ · = (._r‘𝑅)
mavmulcl.r	⊢ (𝜑 → 𝑅 ∈ Ring)
mavmulcl.n	⊢ (𝜑 → 𝑁 ∈ Fin)
mavmulcl.x	⊢ (𝜑 → 𝑋 ∈ (Base‘𝐴))
mavmulcl.y	⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑_m 𝑁))

**Assertion**
Ref	Expression
mavmulcl	⊢ (𝜑 → (𝑋 × 𝑌) ∈ (𝐵 ↑_m 𝑁))

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

Step	Hyp	Ref	Expression
1		mavmulcl.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		mavmulcl.m	. . 3 ⊢ × = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)
3		mavmulcl.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
4		mavmulcl.t	. . 3 ⊢ · = (._r‘𝑅)
5		mavmulcl.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ Ring)
6		mavmulcl.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ Fin)
7		mavmulcl.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (Base‘𝐴))
8		mavmulcl.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝐵 ↑_m 𝑁))
9	1, 2, 3, 4, 5, 6, 7, 8	mavmulval 22483	. 2 ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))))
10		ringcmn 20242	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ CMnd)
11	5, 10	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ CMnd)
12	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑅 ∈ CMnd)
13	6	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑁 ∈ Fin)
14	5	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
15	1, 3	matbas2 22359	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐵 ↑_m (𝑁 × 𝑁)) = (Base‘𝐴))
16	6, 5, 15	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ↑_m (𝑁 × 𝑁)) = (Base‘𝐴))
17	7, 16	eleqtrrd 2837	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑_m (𝑁 × 𝑁)))
18		elmapi 8863	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝐵 ↑_m (𝑁 × 𝑁)) → 𝑋:(𝑁 × 𝑁)⟶𝐵)
19	17, 18	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑋:(𝑁 × 𝑁)⟶𝐵)
20	19	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑋:(𝑁 × 𝑁)⟶𝐵)
21		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → 𝑖 ∈ 𝑁)
22	21	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑖 ∈ 𝑁)
23		simpr 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑗 ∈ 𝑁)
24	20, 22, 23	fovcdmd 7579	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑖𝑋𝑗) ∈ 𝐵)
25		elmapi 8863	. . . . . . . . . 10 ⊢ (𝑌 ∈ (𝐵 ↑_m 𝑁) → 𝑌:𝑁⟶𝐵)
26	8, 25	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑌:𝑁⟶𝐵)
27	26	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑌:𝑁⟶𝐵)
28	27, 23	ffvelcdmd 7075	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑌‘𝑗) ∈ 𝐵)
29	3, 4	ringcl 20210	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑖𝑋𝑗) ∈ 𝐵 ∧ (𝑌‘𝑗) ∈ 𝐵) → ((𝑖𝑋𝑗) · (𝑌‘𝑗)) ∈ 𝐵)
30	14, 24, 28, 29	syl3anc 1373	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → ((𝑖𝑋𝑗) · (𝑌‘𝑗)) ∈ 𝐵)
31	30	ralrimiva 3132	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → ∀𝑗 ∈ 𝑁 ((𝑖𝑋𝑗) · (𝑌‘𝑗)) ∈ 𝐵)
32	3, 12, 13, 31	gsummptcl 19948	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) ∈ 𝐵)
33	32	ralrimiva 3132	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ 𝑁 (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) ∈ 𝐵)
34		eqid 2735	. . . . 5 ⊢ (𝑖 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) = (𝑖 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))))
35	34	fmpt 7100	. . . 4 ⊢ (∀𝑖 ∈ 𝑁 (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) ∈ 𝐵 ↔ (𝑖 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))):𝑁⟶𝐵)
36	3	fvexi 6890	. . . . 5 ⊢ 𝐵 ∈ V
37		elmapg 8853	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝑁 ∈ Fin) → ((𝑖 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) ∈ (𝐵 ↑_m 𝑁) ↔ (𝑖 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))):𝑁⟶𝐵))
38	36, 6, 37	sylancr 587	. . . 4 ⊢ (𝜑 → ((𝑖 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) ∈ (𝐵 ↑_m 𝑁) ↔ (𝑖 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))):𝑁⟶𝐵))
39	35, 38	bitr4id 290	. . 3 ⊢ (𝜑 → (∀𝑖 ∈ 𝑁 (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) ∈ 𝐵 ↔ (𝑖 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) ∈ (𝐵 ↑_m 𝑁)))
40	33, 39	mpbid 232	. 2 ⊢ (𝜑 → (𝑖 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) ∈ (𝐵 ↑_m 𝑁))
41	9, 40	eqeltrd 2834	1 ⊢ (𝜑 → (𝑋 × 𝑌) ∈ (𝐵 ↑_m 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3051 Vcvv 3459 ⟨cop 4607 ↦ cmpt 5201 × cxp 5652 ⟶wf 6527 ‘cfv 6531 (class class class)co 7405 ↑_m cmap 8840 Fincfn 8959 Basecbs 17228 ._rcmulr 17272 Σ_g cgsu 17454 CMndccmn 19761 Ringcrg 20193 Mat cmat 22345 maVecMul cmvmul 22478

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-ot 4610 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-map 8842 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9374 df-sup 9454 df-oi 9524 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-fz 13525 df-fzo 13672 df-seq 14020 df-hash 14349 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-sca 17287 df-vsca 17288 df-ip 17289 df-tset 17290 df-ple 17291 df-ds 17293 df-hom 17295 df-cco 17296 df-0g 17455 df-gsum 17456 df-prds 17461 df-pws 17463 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-grp 18919 df-minusg 18920 df-cntz 19300 df-cmn 19763 df-abl 19764 df-mgp 20101 df-ur 20142 df-ring 20195 df-sra 21131 df-rgmod 21132 df-dsmm 21692 df-frlm 21707 df-mat 22346 df-mvmul 22479

This theorem is referenced by: mavmulass 22487 slesolex 22620

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