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Mirrors > Home > MPE Home > Th. List > scmatscmid | Structured version Visualization version GIF version |
Description: A scalar matrix can be expressed as a multiplication of a scalar with the identity matrix. (Contributed by AV, 30-Oct-2019.) (Revised by AV, 18-Dec-2019.) |
Ref | Expression |
---|---|
scmatval.k | ⊢ 𝐾 = (Base‘𝑅) |
scmatval.a | ⊢ 𝐴 = (𝑁 Mat 𝑅) |
scmatval.b | ⊢ 𝐵 = (Base‘𝐴) |
scmatval.1 | ⊢ 1 = (1r‘𝐴) |
scmatval.t | ⊢ · = ( ·𝑠 ‘𝐴) |
scmatval.s | ⊢ 𝑆 = (𝑁 ScMat 𝑅) |
Ref | Expression |
---|---|
scmatscmid | ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) → ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scmatval.k | . . . 4 ⊢ 𝐾 = (Base‘𝑅) | |
2 | scmatval.a | . . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅) | |
3 | scmatval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐴) | |
4 | scmatval.1 | . . . 4 ⊢ 1 = (1r‘𝐴) | |
5 | scmatval.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝐴) | |
6 | scmatval.s | . . . 4 ⊢ 𝑆 = (𝑁 ScMat 𝑅) | |
7 | 1, 2, 3, 4, 5, 6 | scmatel 21806 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 ↔ (𝑀 ∈ 𝐵 ∧ ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 )))) |
8 | 7 | simplbda 501 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) ∧ 𝑀 ∈ 𝑆) → ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 )) |
9 | 8 | 3impa 1111 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) → ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∃wrex 3072 ‘cfv 6494 (class class class)co 7352 Fincfn 8842 Basecbs 17043 ·𝑠 cvsca 17097 1rcur 19872 Mat cmat 21706 ScMat cscmat 21790 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-id 5530 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-iota 6446 df-fun 6496 df-fv 6502 df-ov 7355 df-oprab 7356 df-mpo 7357 df-scmat 21792 |
This theorem is referenced by: scmate 21811 scmatscm 21814 scmataddcl 21817 scmatsubcl 21818 scmatfo 21831 |
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