Theorem scmatscmid 22393

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 = (1r‘𝐴)
scmatval.t	⊢ · = ( ·𝑠 ‘𝐴)
scmatval.s	⊢ 𝑆 = (𝑁 ScMat 𝑅)

Assertion

Ref	Expression
scmatscmid	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝑆) → ∃𝑐 ∈ 𝐾 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

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 22392	. . 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 ))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3053  ‘cfv 6511  (class class class)co 7387  Fincfn 8918  Basecbs 17179   ·_𝑠 cvsca 17224  1_rcur 20090   Mat cmat 22294   ScMat cscmat 22376

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-scmat 22378

This theorem is referenced by:  scmate  22397  scmatscm  22400  scmataddcl  22403  scmatsubcl  22404  scmatfo  22417