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Theorem scmatscmid 21807
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.)
Hypotheses
Ref Expression
scmatval.k 𝐾 = (Base‘𝑅)
scmatval.a 𝐴 = (𝑁 Mat 𝑅)
scmatval.b 𝐵 = (Base‘𝐴)
scmatval.1 1 = (1r𝐴)
scmatval.t · = ( ·𝑠𝐴)
scmatval.s 𝑆 = (𝑁 ScMat 𝑅)
Assertion
Ref Expression
scmatscmid ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → ∃𝑐𝐾 𝑀 = (𝑐 · 1 ))
Distinct variable groups:   𝐾,𝑐   𝑁,𝑐   𝑅,𝑐   𝑀,𝑐
Allowed substitution hints:   𝐴(𝑐)   𝐵(𝑐)   𝑆(𝑐)   · (𝑐)   1 (𝑐)   𝑉(𝑐)

Proof of Theorem scmatscmid
StepHypRef 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 = (1r𝐴)
5 scmatval.t . . . 4 · = ( ·𝑠𝐴)
6 scmatval.s . . . 4 𝑆 = (𝑁 ScMat 𝑅)
71, 2, 3, 4, 5, 6scmatel 21806 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆 ↔ (𝑀𝐵 ∧ ∃𝑐𝐾 𝑀 = (𝑐 · 1 ))))
87simplbda 501 . 2 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ 𝑀𝑆) → ∃𝑐𝐾 𝑀 = (𝑐 · 1 ))
983impa 1111 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → ∃𝑐𝐾 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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