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Theorem scmatval 22526
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 22513 . . . 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 6907 . . . . . . 7 (𝑎 = (𝑛 Mat 𝑟) → (Base‘𝑎) = (Base‘(𝑛 Mat 𝑟)))
6 fveq2 6907 . . . . . . . . . 10 (𝑎 = (𝑛 Mat 𝑟) → ( ·𝑠𝑎) = ( ·𝑠 ‘(𝑛 Mat 𝑟)))
7 eqidd 2736 . . . . . . . . . 10 (𝑎 = (𝑛 Mat 𝑟) → 𝑐 = 𝑐)
8 fveq2 6907 . . . . . . . . . 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 2746 . . . . . . . 8 (𝑎 = (𝑛 Mat 𝑟) → (𝑚 = (𝑐( ·𝑠𝑎)(1r𝑎)) ↔ 𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟)))))
1110rexbidv 3177 . . . . . . 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 3452 . . . . . 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 3946 . . . 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 6911 . . . . . . 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 6910 . . . . . . . 8 (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
2017, 19eqtri 2763 . . . . . . 7 𝐵 = (Base‘(𝑁 Mat 𝑅))
2116, 20eqtr4di 2793 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
22 fveq2 6907 . . . . . . . . 9 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
23 scmatval.k . . . . . . . . 9 𝐾 = (Base‘𝑅)
2422, 23eqtr4di 2793 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐾)
2524adantl 481 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘𝑟) = 𝐾)
2615fveq2d 6911 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat 𝑟)) = ( ·𝑠 ‘(𝑁 Mat 𝑅)))
27 scmatval.t . . . . . . . . . . 11 · = ( ·𝑠𝐴)
2818fveq2i 6910 . . . . . . . . . . 11 ( ·𝑠𝐴) = ( ·𝑠 ‘(𝑁 Mat 𝑅))
2927, 28eqtri 2763 . . . . . . . . . 10 · = ( ·𝑠 ‘(𝑁 Mat 𝑅))
3026, 29eqtr4di 2793 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat 𝑟)) = · )
31 eqidd 2736 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑐 = 𝑐)
3215fveq2d 6911 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (1r‘(𝑛 Mat 𝑟)) = (1r‘(𝑁 Mat 𝑅)))
33 scmatval.1 . . . . . . . . . . 11 1 = (1r𝐴)
3418fveq2i 6910 . . . . . . . . . . 11 (1r𝐴) = (1r‘(𝑁 Mat 𝑅))
3533, 34eqtri 2763 . . . . . . . . . 10 1 = (1r‘(𝑁 Mat 𝑅))
3632, 35eqtr4di 2793 . . . . . . . . 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 2746 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟))) ↔ 𝑚 = (𝑐 · 1 )))
3925, 38rexeqbidv 3345 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟))) ↔ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )))
4021, 39rabeqbidv 3452 . . . . 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 2775 . . 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 3499 . . . 4 (𝑅𝑉𝑅 ∈ V)
4544adantl 481 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
4617fvexi 6921 . . . . 5 𝐵 ∈ V
4746rabex 5345 . . . 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 2787 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑆 = {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wrex 3068  {crab 3433  Vcvv 3478  csb 3908  cfv 6563  (class class class)co 7431  cmpo 7433  Fincfn 8984  Basecbs 17245   ·𝑠 cvsca 17302  1rcur 20199   Mat cmat 22427   ScMat cscmat 22511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-scmat 22513
This theorem is referenced by:  scmatel  22527  scmatmats  22533  scmatlss  22547
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