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Theorem scmatval 22405
Description: The set of 𝑁 x 𝑁 scalar matrices 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
scmatval ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑆 = {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )})
Distinct variable groups:   𝐵,𝑚   𝐾,𝑐   𝑁,𝑐,𝑚   𝑅,𝑐,𝑚
Allowed substitution hints:   𝐴(𝑚,𝑐)   𝐵(𝑐)   𝑆(𝑚,𝑐)   · (𝑚,𝑐)   1 (𝑚,𝑐)   𝐾(𝑚)   𝑉(𝑚,𝑐)

Proof of Theorem scmatval
Dummy variables 𝑛 𝑟 𝑎 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 scmatval.s . 2 𝑆 = (𝑁 ScMat 𝑅)
2 df-scmat 22392 . . . 4 ScMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑛 Mat 𝑟) / 𝑎{𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠𝑎)(1r𝑎))})
32a1i 11 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → ScMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑛 Mat 𝑟) / 𝑎{𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠𝑎)(1r𝑎))}))
4 ovexd 7455 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → (𝑛 Mat 𝑟) ∈ V)
5 fveq2 6897 . . . . . . 7 (𝑎 = (𝑛 Mat 𝑟) → (Base‘𝑎) = (Base‘(𝑛 Mat 𝑟)))
6 fveq2 6897 . . . . . . . . . 10 (𝑎 = (𝑛 Mat 𝑟) → ( ·𝑠𝑎) = ( ·𝑠 ‘(𝑛 Mat 𝑟)))
7 eqidd 2729 . . . . . . . . . 10 (𝑎 = (𝑛 Mat 𝑟) → 𝑐 = 𝑐)
8 fveq2 6897 . . . . . . . . . 10 (𝑎 = (𝑛 Mat 𝑟) → (1r𝑎) = (1r‘(𝑛 Mat 𝑟)))
96, 7, 8oveq123d 7441 . . . . . . . . 9 (𝑎 = (𝑛 Mat 𝑟) → (𝑐( ·𝑠𝑎)(1r𝑎)) = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟))))
109eqeq2d 2739 . . . . . . . 8 (𝑎 = (𝑛 Mat 𝑟) → (𝑚 = (𝑐( ·𝑠𝑎)(1r𝑎)) ↔ 𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟)))))
1110rexbidv 3175 . . . . . . 7 (𝑎 = (𝑛 Mat 𝑟) → (∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠𝑎)(1r𝑎)) ↔ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟)))))
125, 11rabeqbidv 3446 . . . . . 6 (𝑎 = (𝑛 Mat 𝑟) → {𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠𝑎)(1r𝑎))} = {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟)))})
1312adantl 481 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) ∧ 𝑎 = (𝑛 Mat 𝑟)) → {𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠𝑎)(1r𝑎))} = {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟)))})
144, 13csbied 3930 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → (𝑛 Mat 𝑟) / 𝑎{𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠𝑎)(1r𝑎))} = {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟)))})
15 oveq12 7429 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
1615fveq2d 6901 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘(𝑁 Mat 𝑅)))
17 scmatval.b . . . . . . . 8 𝐵 = (Base‘𝐴)
18 scmatval.a . . . . . . . . 9 𝐴 = (𝑁 Mat 𝑅)
1918fveq2i 6900 . . . . . . . 8 (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
2017, 19eqtri 2756 . . . . . . 7 𝐵 = (Base‘(𝑁 Mat 𝑅))
2116, 20eqtr4di 2786 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
22 fveq2 6897 . . . . . . . . 9 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
23 scmatval.k . . . . . . . . 9 𝐾 = (Base‘𝑅)
2422, 23eqtr4di 2786 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐾)
2524adantl 481 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘𝑟) = 𝐾)
2615fveq2d 6901 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat 𝑟)) = ( ·𝑠 ‘(𝑁 Mat 𝑅)))
27 scmatval.t . . . . . . . . . . 11 · = ( ·𝑠𝐴)
2818fveq2i 6900 . . . . . . . . . . 11 ( ·𝑠𝐴) = ( ·𝑠 ‘(𝑁 Mat 𝑅))
2927, 28eqtri 2756 . . . . . . . . . 10 · = ( ·𝑠 ‘(𝑁 Mat 𝑅))
3026, 29eqtr4di 2786 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat 𝑟)) = · )
31 eqidd 2729 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑐 = 𝑐)
3215fveq2d 6901 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (1r‘(𝑛 Mat 𝑟)) = (1r‘(𝑁 Mat 𝑅)))
33 scmatval.1 . . . . . . . . . . 11 1 = (1r𝐴)
3418fveq2i 6900 . . . . . . . . . . 11 (1r𝐴) = (1r‘(𝑁 Mat 𝑅))
3533, 34eqtri 2756 . . . . . . . . . 10 1 = (1r‘(𝑁 Mat 𝑅))
3632, 35eqtr4di 2786 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (1r‘(𝑛 Mat 𝑟)) = 1 )
3730, 31, 36oveq123d 7441 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟))) = (𝑐 · 1 ))
3837eqeq2d 2739 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟))) ↔ 𝑚 = (𝑐 · 1 )))
3925, 38rexeqbidv 3340 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟))) ↔ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )))
4021, 39rabeqbidv 3446 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟)))} = {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )})
4140adantl 481 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟)))} = {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )})
4214, 41eqtrd 2768 . . 3 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → (𝑛 Mat 𝑟) / 𝑎{𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠𝑎)(1r𝑎))} = {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )})
43 simpl 482 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑁 ∈ Fin)
44 elex 3490 . . . 4 (𝑅𝑉𝑅 ∈ V)
4544adantl 481 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
4617fvexi 6911 . . . . 5 𝐵 ∈ V
4746rabex 5334 . . . 4 {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )} ∈ V
4847a1i 11 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )} ∈ V)
493, 42, 43, 45, 48ovmpod 7573 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 ScMat 𝑅) = {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )})
501, 49eqtrid 2780 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑆 = {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wrex 3067  {crab 3429  Vcvv 3471  csb 3892  cfv 6548  (class class class)co 7420  cmpo 7422  Fincfn 8963  Basecbs 17179   ·𝑠 cvsca 17236  1rcur 20120   Mat cmat 22306   ScMat cscmat 22390
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 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
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 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-scmat 22392
This theorem is referenced by:  scmatel  22406  scmatmats  22412  scmatlss  22426
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