Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > slesolvec | Structured version Visualization version GIF version |
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.) |
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〉) |
Ref | Expression |
---|---|
slesolvec | ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑍) = 𝑌 → 𝑍 ∈ 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | slesolex.a | . . . . . . 7 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
2 | slesolex.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐴) | |
3 | 1, 2 | matrcl 21015 | . . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V)) |
4 | 3 | simpld 497 | . . . . 5 ⊢ (𝑋 ∈ 𝐵 → 𝑁 ∈ Fin) |
5 | simpr 487 | . . . . . . . 8 ⊢ ((𝑁 ≠ ∅ ∧ 𝑁 ∈ Fin) → 𝑁 ∈ Fin) | |
6 | simpl 485 | . . . . . . . 8 ⊢ ((𝑁 ≠ ∅ ∧ 𝑁 ∈ Fin) → 𝑁 ≠ ∅) | |
7 | 5, 5, 6 | 3jca 1124 | . . . . . . 7 ⊢ ((𝑁 ≠ ∅ ∧ 𝑁 ∈ Fin) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅)) |
8 | 7 | ex 415 | . . . . . 6 ⊢ (𝑁 ≠ ∅ → (𝑁 ∈ Fin → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅))) |
9 | 8 | adantr 483 | . . . . 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 483 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) → ((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅))) |
12 | 11 | impcom 410 | . 2 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅)) |
13 | simpr 487 | . . 3 ⊢ ((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝑅 ∈ Ring) | |
14 | simpr 487 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) → 𝑌 ∈ 𝑉) | |
15 | 13, 14 | anim12i 614 | . 2 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → (𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑉)) |
16 | eqid 2821 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
17 | eqid 2821 | . . 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 21154 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑁 ≠ ∅) ∧ (𝑅 ∈ Ring ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑍) = 𝑌 → 𝑍 ∈ 𝑉)) |
21 | 12, 15, 20 | syl2anc 586 | 1 ⊢ (((𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉)) → ((𝑋 · 𝑍) = 𝑌 → 𝑍 ∈ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 Vcvv 3494 ∅c0 4290 〈cop 4566 × cxp 5547 ‘cfv 6349 (class class class)co 7150 ↑m cmap 8400 Fincfn 8503 Basecbs 16477 Ringcrg 19291 Mat cmat 21010 maVecMul cmvmul 21143 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-map 8402 df-slot 16481 df-base 16483 df-mat 21011 df-mvmul 21144 |
This theorem is referenced by: slesolinv 21283 cramerimplem3 21288 cramerimp 21289 cramer 21294 |
Copyright terms: Public domain | W3C validator |