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Theorem scmatel 22394
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 22393 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑆 = {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )})
87eleq2d 2814 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆𝑀 ∈ {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )}))
9 eqeq1 2731 . . . 4 (𝑚 = 𝑀 → (𝑚 = (𝑐 · 1 ) ↔ 𝑀 = (𝑐 · 1 )))
109rexbidv 3173 . . 3 (𝑚 = 𝑀 → (∃𝑐𝐾 𝑚 = (𝑐 · 1 ) ↔ ∃𝑐𝐾 𝑀 = (𝑐 · 1 )))
1110elrab 3680 . 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 205  wa 395   = wceq 1534  wcel 2099  wrex 3065  {crab 3427  cfv 6542  (class class class)co 7414  Fincfn 8955  Basecbs 17171   ·𝑠 cvsca 17228  1rcur 20112   Mat cmat 22294   ScMat cscmat 22378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-scmat 22380
This theorem is referenced by:  scmatscmid  22395  scmatmat  22398  scmatid  22403  scmataddcl  22405  scmatsubcl  22406  scmatmulcl  22407  smatvscl  22413  scmatrhmcl  22417  mat0scmat  22427  mat1scmat  22428  chmaidscmat  22737
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