mavmulcl - Metamath Proof Explorer

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Theorem mavmulcl 22463

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 22461	. 2 ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))))
10		ringcmn 20202	. . . . . . 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 22337	. . . . . . . . . . . 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 2836	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑_m (𝑁 × 𝑁)))
18		elmapi 8779	. . . . . . . . . 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 7524	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑖𝑋𝑗) ∈ 𝐵)
25		elmapi 8779	. . . . . . . . . 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 7024	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑌‘𝑗) ∈ 𝐵)
29	3, 4	ringcl 20170	. . . . . . 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 3125	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → ∀𝑗 ∈ 𝑁 ((𝑖𝑋𝑗) · (𝑌‘𝑗)) ∈ 𝐵)
32	3, 12, 13, 31	gsummptcl 19881	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) ∈ 𝐵)
33	32	ralrimiva 3125	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ 𝑁 (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) ∈ 𝐵)
34		eqid 2733	. . . . 5 ⊢ (𝑖 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) = (𝑖 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))))
35	34	fmpt 7049	. . . 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 6842	. . . . 5 ⊢ 𝐵 ∈ V
37		elmapg 8769	. . . . 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 2833	1 ⊢ (𝜑 → (𝑋 × 𝑌) ∈ (𝐵 ↑_m 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3048 Vcvv 3437 ⟨cop 4581 ↦ cmpt 5174 × cxp 5617 ⟶wf 6482 ‘cfv 6486 (class class class)co 7352 ↑_m cmap 8756 Fincfn 8875 Basecbs 17122 ._rcmulr 17164 Σ_g cgsu 17346 CMndccmn 19694 Ringcrg 20153 Mat cmat 22323 maVecMul cmvmul 22456

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-ot 4584 df-uni 4859 df-int 4898 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-er 8628 df-map 8758 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9253 df-sup 9333 df-oi 9403 df-card 9839 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-z 12476 df-dec 12595 df-uz 12739 df-fz 13410 df-fzo 13557 df-seq 13911 df-hash 14240 df-struct 17060 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-ress 17144 df-plusg 17176 df-mulr 17177 df-sca 17179 df-vsca 17180 df-ip 17181 df-tset 17182 df-ple 17183 df-ds 17185 df-hom 17187 df-cco 17188 df-0g 17347 df-gsum 17348 df-prds 17353 df-pws 17355 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-grp 18851 df-minusg 18852 df-cntz 19231 df-cmn 19696 df-abl 19697 df-mgp 20061 df-ur 20102 df-ring 20155 df-sra 21109 df-rgmod 21110 df-dsmm 21671 df-frlm 21686 df-mat 22324 df-mvmul 22457

This theorem is referenced by: mavmulass 22465 slesolex 22598

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