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| 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 22416 | . . . . . 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 1129 | . . . . . . 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 2737 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
| 17 | eqid 2737 | . . 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 22557 | . 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 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 Vcvv 3480 ∅c0 4333 〈cop 4632 × cxp 5683 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 Fincfn 8985 Basecbs 17247 Ringcrg 20230 Mat cmat 22411 maVecMul cmvmul 22546 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addcl 11215 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-map 8868 df-nn 12267 df-slot 17219 df-ndx 17231 df-base 17248 df-mat 22412 df-mvmul 22547 | 
| This theorem is referenced by: slesolinv 22686 cramerimplem3 22691 cramerimp 22692 cramer 22697 | 
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