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Theorem scmatval 22510
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 22497 . . . 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 7466 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → (𝑛 Mat 𝑟) ∈ V)
5 fveq2 6906 . . . . . . 7 (𝑎 = (𝑛 Mat 𝑟) → (Base‘𝑎) = (Base‘(𝑛 Mat 𝑟)))
6 fveq2 6906 . . . . . . . . . 10 (𝑎 = (𝑛 Mat 𝑟) → ( ·𝑠𝑎) = ( ·𝑠 ‘(𝑛 Mat 𝑟)))
7 eqidd 2738 . . . . . . . . . 10 (𝑎 = (𝑛 Mat 𝑟) → 𝑐 = 𝑐)
8 fveq2 6906 . . . . . . . . . 10 (𝑎 = (𝑛 Mat 𝑟) → (1r𝑎) = (1r‘(𝑛 Mat 𝑟)))
96, 7, 8oveq123d 7452 . . . . . . . . 9 (𝑎 = (𝑛 Mat 𝑟) → (𝑐( ·𝑠𝑎)(1r𝑎)) = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟))))
109eqeq2d 2748 . . . . . . . 8 (𝑎 = (𝑛 Mat 𝑟) → (𝑚 = (𝑐( ·𝑠𝑎)(1r𝑎)) ↔ 𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟)))))
1110rexbidv 3179 . . . . . . 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 3455 . . . . . 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 3935 . . . 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 7440 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
1615fveq2d 6910 . . . . . . 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 6909 . . . . . . . 8 (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
2017, 19eqtri 2765 . . . . . . 7 𝐵 = (Base‘(𝑁 Mat 𝑅))
2116, 20eqtr4di 2795 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
22 fveq2 6906 . . . . . . . . 9 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
23 scmatval.k . . . . . . . . 9 𝐾 = (Base‘𝑅)
2422, 23eqtr4di 2795 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐾)
2524adantl 481 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘𝑟) = 𝐾)
2615fveq2d 6910 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat 𝑟)) = ( ·𝑠 ‘(𝑁 Mat 𝑅)))
27 scmatval.t . . . . . . . . . . 11 · = ( ·𝑠𝐴)
2818fveq2i 6909 . . . . . . . . . . 11 ( ·𝑠𝐴) = ( ·𝑠 ‘(𝑁 Mat 𝑅))
2927, 28eqtri 2765 . . . . . . . . . 10 · = ( ·𝑠 ‘(𝑁 Mat 𝑅))
3026, 29eqtr4di 2795 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat 𝑟)) = · )
31 eqidd 2738 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑐 = 𝑐)
3215fveq2d 6910 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (1r‘(𝑛 Mat 𝑟)) = (1r‘(𝑁 Mat 𝑅)))
33 scmatval.1 . . . . . . . . . . 11 1 = (1r𝐴)
3418fveq2i 6909 . . . . . . . . . . 11 (1r𝐴) = (1r‘(𝑁 Mat 𝑅))
3533, 34eqtri 2765 . . . . . . . . . 10 1 = (1r‘(𝑁 Mat 𝑅))
3632, 35eqtr4di 2795 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (1r‘(𝑛 Mat 𝑟)) = 1 )
3730, 31, 36oveq123d 7452 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟))) = (𝑐 · 1 ))
3837eqeq2d 2748 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟))) ↔ 𝑚 = (𝑐 · 1 )))
3925, 38rexeqbidv 3347 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟))) ↔ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )))
4021, 39rabeqbidv 3455 . . . . 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 2777 . . 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 3501 . . . 4 (𝑅𝑉𝑅 ∈ V)
4544adantl 481 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
4617fvexi 6920 . . . . 5 𝐵 ∈ V
4746rabex 5339 . . . 4 {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )} ∈ V
4847a1i 11 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )} ∈ V)
493, 42, 43, 45, 48ovmpod 7585 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 ScMat 𝑅) = {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )})
501, 49eqtrid 2789 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑆 = {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wrex 3070  {crab 3436  Vcvv 3480  csb 3899  cfv 6561  (class class class)co 7431  cmpo 7433  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:  scmatel  22511  scmatmats  22517  scmatlss  22531
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