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Theorem slesolvec 22706

Description: Every solution of a system of linear equations represented by a matrix and a vector is a vector. (Contributed by AV, 10-Feb-2019.) (Revised by AV, 27-Feb-2019.)

**Hypotheses**
Ref	Expression
slesolex.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
slesolex.b	⊢ 𝐵 = (Base‘𝐴)
slesolex.v	⊢ 𝑉 = ((Base‘𝑅) ↑_m 𝑁)
slesolex.x	⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)

**Assertion**
Ref	Expression
slesolvec	⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑍) = 𝑌 → 𝑍 ∈ 𝑉))

**Proof of Theorem slesolvec**
Step	Hyp	Ref	Expression
1		slesolex.a	. . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		slesolex.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐴)
3	1, 2	matrcl 22437	. . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))
4	3	simpld 494	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝑁 ∈ Fin)
5		simpr 484	. . . . . . . 8 ⊢ ((𝑁 ≠ ∅ ∧ 𝑁 ∈ Fin) → 𝑁 ∈ Fin)
6		simpl 482	. . . . . . . 8 ⊢ ((𝑁 ≠ ∅ ∧ 𝑁 ∈ Fin) → 𝑁 ≠ ∅)
7	5, 5, 6	3jca 1128	. . . . . . 7 ⊢ ((𝑁 ≠ ∅ ∧ 𝑁 ∈ Fin) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅))
8	7	ex 412	. . . . . 6 ⊢ (𝑁 ≠ ∅ → (𝑁 ∈ Fin → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅)))
9	8	adantr 480	. . . . 5 ⊢ ((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅)))
10	4, 9	syl5com 31	. . . 4 ⊢ (𝑋 ∈ 𝐵 → ((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅)))
11	10	adantr 480	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) → ((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅)))
12	11	impcom 407	. 2 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅))
13		simpr 484	. . 3 ⊢ ((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring)
14		simpr 484	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ 𝑉)
15	13, 14	anim12i 612	. 2 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → (𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑉))
16		eqid 2740	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
17		eqid 2740	. . 3 ⊢ ((Base‘𝑅) ↑_m (𝑁 × 𝑁)) = ((Base‘𝑅) ↑_m (𝑁 × 𝑁))
18		slesolex.v	. . 3 ⊢ 𝑉 = ((Base‘𝑅) ↑_m 𝑁)
19		slesolex.x	. . 3 ⊢ · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)
20	16, 17, 18, 19, 18	mavmulsolcl 22578	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅) ∧ (𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑍) = 𝑌 → 𝑍 ∈ 𝑉))
21	12, 15, 20	syl2anc 583	1 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑍) = 𝑌 → 𝑍 ∈ 𝑉))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 Vcvv 3488 ∅c0 4352 ⟨cop 4654 × cxp 5698 ‘cfv 6573 (class class class)co 7448 ↑_m cmap 8884 Fincfn 9003 Basecbs 17258 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-1cn 11242 ax-addcl 11244

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-ral 3068 df-rex 3077 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-op 4655 df-uni 4932 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-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-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-map 8886 df-nn 12294 df-slot 17229 df-ndx 17241 df-base 17259 df-mat 22433 df-mvmul 22568

This theorem is referenced by: slesolinv 22707 cramerimplem3 22712 cramerimp 22713 cramer 22718

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