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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 22527 | . . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 ↔ (𝑀 ∈ 𝐵 ∧ ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 )))) |
8 | 7 | simplbda 499 | . 2 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) ∧ 𝑀 ∈ 𝑆) → ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 )) |
9 | 8 | 3impa 1109 | 1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) → ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 Basecbs 17245 ·𝑠 cvsca 17302 1rcur 20199 Mat cmat 22427 ScMat cscmat 22511 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-scmat 22513 |
This theorem is referenced by: scmate 22532 scmatscm 22535 scmataddcl 22538 scmatsubcl 22539 scmatfo 22552 |
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