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

Theorem scmatel 22511
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 22510 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑆 = {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )})
87eleq2d 2827 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆𝑀 ∈ {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )}))
9 eqeq1 2741 . . . 4 (𝑚 = 𝑀 → (𝑚 = (𝑐 · 1 ) ↔ 𝑀 = (𝑐 · 1 )))
109rexbidv 3179 . . 3 (𝑚 = 𝑀 → (∃𝑐𝐾 𝑚 = (𝑐 · 1 ) ↔ ∃𝑐𝐾 𝑀 = (𝑐 · 1 )))
1110elrab 3692 . 2 (𝑀 ∈ {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )} ↔ (𝑀𝐵 ∧ ∃𝑐𝐾 𝑀 = (𝑐 · 1 )))
128, 11bitrdi 287 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆 ↔ (𝑀𝐵 ∧ ∃𝑐𝐾 𝑀 = (𝑐 · 1 ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  {crab 3436  cfv 6561  (class class class)co 7431  Fincfn 8985  Basecbs 17247   ·𝑠 cvsca 17301  1rcur 20178   Mat cmat 22411   ScMat cscmat 22495
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-scmat 22497
This theorem is referenced by:  scmatscmid  22512  scmatmat  22515  scmatid  22520  scmataddcl  22522  scmatsubcl  22523  scmatmulcl  22524  smatvscl  22530  scmatrhmcl  22534  mat0scmat  22544  mat1scmat  22545  chmaidscmat  22854
  Copyright terms: Public domain W3C validator