Theorem scmatscmid 22414

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 22413	. . 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 1541   ∈ wcel 2110  ∃wrex 3054  ‘cfv 6477  (class class class)co 7341  Fincfn 8864  Basecbs 17112   ·_𝑠 cvsca 17157  1_rcur 20092   Mat cmat 22315   ScMat cscmat 22397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-op 4581  df-uni 4858  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6433  df-fun 6479  df-fv 6485  df-ov 7344  df-oprab 7345  df-mpo 7346  df-scmat 22399

This theorem is referenced by:  scmate  22418  scmatscm  22421  scmataddcl  22424  scmatsubcl  22425  scmatfo  22438