MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  scmatel Structured version   Visualization version   GIF version

Theorem scmatel 21108
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 21107 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑆 = {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )})
87eleq2d 2899 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆𝑀 ∈ {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )}))
9 eqeq1 2826 . . . 4 (𝑚 = 𝑀 → (𝑚 = (𝑐 · 1 ) ↔ 𝑀 = (𝑐 · 1 )))
109rexbidv 3283 . . 3 (𝑚 = 𝑀 → (∃𝑐𝐾 𝑚 = (𝑐 · 1 ) ↔ ∃𝑐𝐾 𝑀 = (𝑐 · 1 )))
1110elrab 3655 . 2 (𝑀 ∈ {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )} ↔ (𝑀𝐵 ∧ ∃𝑐𝐾 𝑀 = (𝑐 · 1 )))
128, 11syl6bb 290 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆 ↔ (𝑀𝐵 ∧ ∃𝑐𝐾 𝑀 = (𝑐 · 1 ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2114  wrex 3131  {crab 3134  cfv 6334  (class class class)co 7140  Fincfn 8496  Basecbs 16474   ·𝑠 cvsca 16560  1rcur 19242   Mat cmat 21010   ScMat cscmat 21092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-scmat 21094
This theorem is referenced by:  scmatscmid  21109  scmatmat  21112  scmatid  21117  scmataddcl  21119  scmatsubcl  21120  scmatmulcl  21121  smatvscl  21127  scmatrhmcl  21131  mat0scmat  21141  mat1scmat  21142  chmaidscmat  21451
  Copyright terms: Public domain W3C validator