mavmulcl - Metamath Proof Explorer

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Theorem mavmulcl 22574

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 22572	. 2 ⊢ (𝜑 → (𝑋 × 𝑌) = (𝑖 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))))
10		ringcmn 20305	. . . . . . 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 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑅 ∈ Ring)
15	1, 3	matbas2 22448	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝐵 ↑_m (𝑁 × 𝑁)) = (Base‘𝐴))
16	6, 5, 15	syl2anc 583	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ↑_m (𝑁 × 𝑁)) = (Base‘𝐴))
17	7, 16	eleqtrrd 2847	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ (𝐵 ↑_m (𝑁 × 𝑁)))
18		elmapi 8907	. . . . . . . . . 10 ⊢ (𝑋 ∈ (𝐵 ↑_m (𝑁 × 𝑁)) → 𝑋:(𝑁 × 𝑁)⟶𝐵)
19	17, 18	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑋:(𝑁 × 𝑁)⟶𝐵)
20	19	ad2antrr 725	. . . . . . . 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 7622	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑖𝑋𝑗) ∈ 𝐵)
25		elmapi 8907	. . . . . . . . . 10 ⊢ (𝑌 ∈ (𝐵 ↑_m 𝑁) → 𝑌:𝑁⟶𝐵)
26	8, 25	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑌:𝑁⟶𝐵)
27	26	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → 𝑌:𝑁⟶𝐵)
28	27, 23	ffvelcdmd 7119	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → (𝑌‘𝑗) ∈ 𝐵)
29	3, 4	ringcl 20277	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (𝑖𝑋𝑗) ∈ 𝐵 ∧ (𝑌‘𝑗) ∈ 𝐵) → ((𝑖𝑋𝑗) · (𝑌‘𝑗)) ∈ 𝐵)
30	14, 24, 28, 29	syl3anc 1371	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑁) ∧ 𝑗 ∈ 𝑁) → ((𝑖𝑋𝑗) · (𝑌‘𝑗)) ∈ 𝐵)
31	30	ralrimiva 3152	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → ∀𝑗 ∈ 𝑁 ((𝑖𝑋𝑗) · (𝑌‘𝑗)) ∈ 𝐵)
32	3, 12, 13, 31	gsummptcl 20009	. . . 4 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁) → (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) ∈ 𝐵)
33	32	ralrimiva 3152	. . 3 ⊢ (𝜑 → ∀𝑖 ∈ 𝑁 (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))) ∈ 𝐵)
34		eqid 2740	. . . . 5 ⊢ (𝑖 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗))))) = (𝑖 ∈ 𝑁 ↦ (𝑅 Σ_g (𝑗 ∈ 𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌‘𝑗)))))
35	34	fmpt 7144	. . . 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 6934	. . . . 5 ⊢ 𝐵 ∈ V
37		elmapg 8897	. . . . 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 586	. . . 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 2844	1 ⊢ (𝜑 → (𝑋 × 𝑌) ∈ (𝐵 ↑_m 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ∀wral 3067 Vcvv 3488 ⟨cop 4654 ↦ cmpt 5249 × cxp 5698 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ↑_m cmap 8884 Fincfn 9003 Basecbs 17258 ._rcmulr 17312 Σ_g cgsu 17500 CMndccmn 19822 Ringcrg 20260 Mat cmat 22432 maVecMul cmvmul 22567

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-ot 4657 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-map 8886 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-sup 9511 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-fz 13568 df-fzo 13712 df-seq 14053 df-hash 14380 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-hom 17335 df-cco 17336 df-0g 17501 df-gsum 17502 df-prds 17507 df-pws 17509 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-grp 18976 df-minusg 18977 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-ur 20209 df-ring 20262 df-sra 21195 df-rgmod 21196 df-dsmm 21775 df-frlm 21790 df-mat 22433 df-mvmul 22568

This theorem is referenced by: mavmulass 22576 slesolex 22709

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