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Theorem scmatval 21428
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 21415 . . . 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 7267 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → (𝑛 Mat 𝑟) ∈ V)
5 fveq2 6736 . . . . . . 7 (𝑎 = (𝑛 Mat 𝑟) → (Base‘𝑎) = (Base‘(𝑛 Mat 𝑟)))
6 fveq2 6736 . . . . . . . . . 10 (𝑎 = (𝑛 Mat 𝑟) → ( ·𝑠𝑎) = ( ·𝑠 ‘(𝑛 Mat 𝑟)))
7 eqidd 2739 . . . . . . . . . 10 (𝑎 = (𝑛 Mat 𝑟) → 𝑐 = 𝑐)
8 fveq2 6736 . . . . . . . . . 10 (𝑎 = (𝑛 Mat 𝑟) → (1r𝑎) = (1r‘(𝑛 Mat 𝑟)))
96, 7, 8oveq123d 7253 . . . . . . . . 9 (𝑎 = (𝑛 Mat 𝑟) → (𝑐( ·𝑠𝑎)(1r𝑎)) = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟))))
109eqeq2d 2749 . . . . . . . 8 (𝑎 = (𝑛 Mat 𝑟) → (𝑚 = (𝑐( ·𝑠𝑎)(1r𝑎)) ↔ 𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟)))))
1110rexbidv 3224 . . . . . . 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 3409 . . . . . 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 485 . . . . 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 3864 . . . 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 7241 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
1615fveq2d 6740 . . . . . . 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 6739 . . . . . . . 8 (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
2017, 19eqtri 2766 . . . . . . 7 𝐵 = (Base‘(𝑁 Mat 𝑅))
2116, 20eqtr4di 2797 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
22 fveq2 6736 . . . . . . . . 9 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
23 scmatval.k . . . . . . . . 9 𝐾 = (Base‘𝑅)
2422, 23eqtr4di 2797 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐾)
2524adantl 485 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘𝑟) = 𝐾)
2615fveq2d 6740 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat 𝑟)) = ( ·𝑠 ‘(𝑁 Mat 𝑅)))
27 scmatval.t . . . . . . . . . . 11 · = ( ·𝑠𝐴)
2818fveq2i 6739 . . . . . . . . . . 11 ( ·𝑠𝐴) = ( ·𝑠 ‘(𝑁 Mat 𝑅))
2927, 28eqtri 2766 . . . . . . . . . 10 · = ( ·𝑠 ‘(𝑁 Mat 𝑅))
3026, 29eqtr4di 2797 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat 𝑟)) = · )
31 eqidd 2739 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑐 = 𝑐)
3215fveq2d 6740 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (1r‘(𝑛 Mat 𝑟)) = (1r‘(𝑁 Mat 𝑅)))
33 scmatval.1 . . . . . . . . . . 11 1 = (1r𝐴)
3418fveq2i 6739 . . . . . . . . . . 11 (1r𝐴) = (1r‘(𝑁 Mat 𝑅))
3533, 34eqtri 2766 . . . . . . . . . 10 1 = (1r‘(𝑁 Mat 𝑅))
3632, 35eqtr4di 2797 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (1r‘(𝑛 Mat 𝑟)) = 1 )
3730, 31, 36oveq123d 7253 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟))) = (𝑐 · 1 ))
3837eqeq2d 2749 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟))) ↔ 𝑚 = (𝑐 · 1 )))
3925, 38rexeqbidv 3327 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟))) ↔ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )))
4021, 39rabeqbidv 3409 . . . . 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 485 . . . 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 2778 . . 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 486 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑁 ∈ Fin)
44 elex 3439 . . . 4 (𝑅𝑉𝑅 ∈ V)
4544adantl 485 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
4617fvexi 6750 . . . . 5 𝐵 ∈ V
4746rabex 5240 . . . 4 {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )} ∈ V
4847a1i 11 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )} ∈ V)
493, 42, 43, 45, 48ovmpod 7380 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 ScMat 𝑅) = {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )})
501, 49syl5eq 2791 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑆 = {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2111  wrex 3063  {crab 3066  Vcvv 3421  csb 3826  cfv 6398  (class class class)co 7232  cmpo 7234  Fincfn 8647  Basecbs 16788   ·𝑠 cvsca 16834  1rcur 19544   Mat cmat 21331   ScMat cscmat 21413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pr 5337
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3423  df-sbc 3710  df-csb 3827  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4835  df-br 5069  df-opab 5131  df-id 5470  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-iota 6356  df-fun 6400  df-fv 6406  df-ov 7235  df-oprab 7236  df-mpo 7237  df-scmat 21415
This theorem is referenced by:  scmatel  21429  scmatmats  21435  scmatlss  21449
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