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Theorem asclfval 19155
 Description: Function value of the algebraic scalars function. (Contributed by Mario Carneiro, 8-Mar-2015.)
Hypotheses
Ref Expression
asclfval.a 𝐴 = (algSc‘𝑊)
asclfval.f 𝐹 = (Scalar‘𝑊)
asclfval.k 𝐾 = (Base‘𝐹)
asclfval.s · = ( ·𝑠𝑊)
asclfval.o 1 = (1r𝑊)
Assertion
Ref Expression
asclfval 𝐴 = (𝑥𝐾 ↦ (𝑥 · 1 ))
Distinct variable groups:   𝑥,𝐾   𝑥, 1   𝑥, ·   𝑥,𝑊
Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)

Proof of Theorem asclfval
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 asclfval.a . 2 𝐴 = (algSc‘𝑊)
2 fveq2 6103 . . . . . . . 8 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
3 asclfval.f . . . . . . . 8 𝐹 = (Scalar‘𝑊)
42, 3syl6eqr 2662 . . . . . . 7 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
54fveq2d 6107 . . . . . 6 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
6 asclfval.k . . . . . 6 𝐾 = (Base‘𝐹)
75, 6syl6eqr 2662 . . . . 5 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
8 fveq2 6103 . . . . . . 7 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
9 asclfval.s . . . . . . 7 · = ( ·𝑠𝑊)
108, 9syl6eqr 2662 . . . . . 6 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
11 eqidd 2611 . . . . . 6 (𝑤 = 𝑊𝑥 = 𝑥)
12 fveq2 6103 . . . . . . 7 (𝑤 = 𝑊 → (1r𝑤) = (1r𝑊))
13 asclfval.o . . . . . . 7 1 = (1r𝑊)
1412, 13syl6eqr 2662 . . . . . 6 (𝑤 = 𝑊 → (1r𝑤) = 1 )
1510, 11, 14oveq123d 6570 . . . . 5 (𝑤 = 𝑊 → (𝑥( ·𝑠𝑤)(1r𝑤)) = (𝑥 · 1 ))
167, 15mpteq12dv 4663 . . . 4 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠𝑤)(1r𝑤))) = (𝑥𝐾 ↦ (𝑥 · 1 )))
17 df-ascl 19135 . . . 4 algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠𝑤)(1r𝑤))))
183fveq2i 6106 . . . . . . 7 (Base‘𝐹) = (Base‘(Scalar‘𝑊))
196, 18eqtri 2632 . . . . . 6 𝐾 = (Base‘(Scalar‘𝑊))
20 fvex 6113 . . . . . 6 (Base‘(Scalar‘𝑊)) ∈ V
2119, 20eqeltri 2684 . . . . 5 𝐾 ∈ V
2221mptex 6390 . . . 4 (𝑥𝐾 ↦ (𝑥 · 1 )) ∈ V
2316, 17, 22fvmpt 6191 . . 3 (𝑊 ∈ V → (algSc‘𝑊) = (𝑥𝐾 ↦ (𝑥 · 1 )))
24 fvprc 6097 . . . . 5 𝑊 ∈ V → (algSc‘𝑊) = ∅)
25 mpt0 5934 . . . . 5 (𝑥 ∈ ∅ ↦ (𝑥 · 1 )) = ∅
2624, 25syl6eqr 2662 . . . 4 𝑊 ∈ V → (algSc‘𝑊) = (𝑥 ∈ ∅ ↦ (𝑥 · 1 )))
27 fvprc 6097 . . . . . . . . 9 𝑊 ∈ V → (Scalar‘𝑊) = ∅)
283, 27syl5eq 2656 . . . . . . . 8 𝑊 ∈ V → 𝐹 = ∅)
2928fveq2d 6107 . . . . . . 7 𝑊 ∈ V → (Base‘𝐹) = (Base‘∅))
30 base0 15740 . . . . . . 7 ∅ = (Base‘∅)
3129, 30syl6eqr 2662 . . . . . 6 𝑊 ∈ V → (Base‘𝐹) = ∅)
326, 31syl5eq 2656 . . . . 5 𝑊 ∈ V → 𝐾 = ∅)
3332mpteq1d 4666 . . . 4 𝑊 ∈ V → (𝑥𝐾 ↦ (𝑥 · 1 )) = (𝑥 ∈ ∅ ↦ (𝑥 · 1 )))
3426, 33eqtr4d 2647 . . 3 𝑊 ∈ V → (algSc‘𝑊) = (𝑥𝐾 ↦ (𝑥 · 1 )))
3523, 34pm2.61i 175 . 2 (algSc‘𝑊) = (𝑥𝐾 ↦ (𝑥 · 1 ))
361, 35eqtri 2632 1 𝐴 = (𝑥𝐾 ↦ (𝑥 · 1 ))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874   ↦ cmpt 4643  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Scalarcsca 15771   ·𝑠 cvsca 15772  1rcur 18324  algSccascl 19132 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-slot 15699  df-base 15700  df-ascl 19135 This theorem is referenced by:  asclval  19156  asclfn  19157  asclf  19158  rnascl  19164  ressascl  19165  asclpropd  19167
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