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Theorem slesolvec 22587

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 22320	. . . . . 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 613	. 2 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → (𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑉))
16		eqid 2730	. . 3 ⊢ (Base‘𝑅) = (Base‘𝑅)
17		eqid 2730	. . 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 22459	. 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅) ∧ (𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑍) = 𝑌 → 𝑍 ∈ 𝑉))
21	12, 15, 20	syl2anc 584	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 1086 = wceq 1541 ∈ wcel 2110 ≠ wne 2926 Vcvv 3434 ∅c0 4281 ⟨cop 4580 × cxp 5612 ‘cfv 6477 (class class class)co 7341 ↑_m cmap 8745 Fincfn 8864 Basecbs 17112 Ringcrg 20144 Mat cmat 22315 maVecMul cmvmul 22448

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 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-cnex 11054 ax-1cn 11056 ax-addcl 11058

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 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-pss 3920 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6244 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-ov 7344 df-oprab 7345 df-mpo 7346 df-om 7792 df-1st 7916 df-2nd 7917 df-frecs 8206 df-wrecs 8237 df-recs 8286 df-rdg 8324 df-map 8747 df-nn 12118 df-slot 17085 df-ndx 17097 df-base 17113 df-mat 22316 df-mvmul 22449

This theorem is referenced by: slesolinv 22588 cramerimplem3 22593 cramerimp 22594 cramer 22599

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