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Theorem scmatel 20686
Description: An 𝑁 x 𝑁 scalar matrix over (a ring) 𝑅. (Contributed 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
scmatel ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆 ↔ (𝑀𝐵 ∧ ∃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 scmatel
Dummy variable 𝑚 is distinct from all other variables.
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, 6scmatval 20685 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑆 = {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )})
87eleq2d 2892 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆𝑀 ∈ {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )}))
9 eqeq1 2829 . . . 4 (𝑚 = 𝑀 → (𝑚 = (𝑐 · 1 ) ↔ 𝑀 = (𝑐 · 1 )))
109rexbidv 3262 . . 3 (𝑚 = 𝑀 → (∃𝑐𝐾 𝑚 = (𝑐 · 1 ) ↔ ∃𝑐𝐾 𝑀 = (𝑐 · 1 )))
1110elrab 3585 . 2 (𝑀 ∈ {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )} ↔ (𝑀𝐵 ∧ ∃𝑐𝐾 𝑀 = (𝑐 · 1 )))
128, 11syl6bb 279 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆 ↔ (𝑀𝐵 ∧ ∃𝑐𝐾 𝑀 = (𝑐 · 1 ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1656  wcel 2164  wrex 3118  {crab 3121  cfv 6127  (class class class)co 6910  Fincfn 8228  Basecbs 16229   ·𝑠 cvsca 16316  1rcur 18862   Mat cmat 20587   ScMat cscmat 20670
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-iota 6090  df-fun 6129  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-scmat 20672
This theorem is referenced by:  scmatscmid  20687  scmatmat  20690  scmatid  20695  scmataddcl  20697  scmatsubcl  20698  scmatmulcl  20699  smatvscl  20705  scmatrhmcl  20709  mat0scmat  20719  mat1scmat  20720  chmaidscmat  21030
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