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Theorem asclfval 19528
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 6344 . . . . . . . 8 (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
3 asclfval.f . . . . . . . 8 𝐹 = (Scalar‘𝑊)
42, 3syl6eqr 2804 . . . . . . 7 (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
54fveq2d 6348 . . . . . 6 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
6 asclfval.k . . . . . 6 𝐾 = (Base‘𝐹)
75, 6syl6eqr 2804 . . . . 5 (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
8 fveq2 6344 . . . . . . 7 (𝑤 = 𝑊 → ( ·𝑠𝑤) = ( ·𝑠𝑊))
9 asclfval.s . . . . . . 7 · = ( ·𝑠𝑊)
108, 9syl6eqr 2804 . . . . . 6 (𝑤 = 𝑊 → ( ·𝑠𝑤) = · )
11 eqidd 2753 . . . . . 6 (𝑤 = 𝑊𝑥 = 𝑥)
12 fveq2 6344 . . . . . . 7 (𝑤 = 𝑊 → (1r𝑤) = (1r𝑊))
13 asclfval.o . . . . . . 7 1 = (1r𝑊)
1412, 13syl6eqr 2804 . . . . . 6 (𝑤 = 𝑊 → (1r𝑤) = 1 )
1510, 11, 14oveq123d 6826 . . . . 5 (𝑤 = 𝑊 → (𝑥( ·𝑠𝑤)(1r𝑤)) = (𝑥 · 1 ))
167, 15mpteq12dv 4877 . . . 4 (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠𝑤)(1r𝑤))) = (𝑥𝐾 ↦ (𝑥 · 1 )))
17 df-ascl 19508 . . . 4 algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠𝑤)(1r𝑤))))
183fveq2i 6347 . . . . . . 7 (Base‘𝐹) = (Base‘(Scalar‘𝑊))
196, 18eqtri 2774 . . . . . 6 𝐾 = (Base‘(Scalar‘𝑊))
20 fvex 6354 . . . . . 6 (Base‘(Scalar‘𝑊)) ∈ V
2119, 20eqeltri 2827 . . . . 5 𝐾 ∈ V
2221mptex 6642 . . . 4 (𝑥𝐾 ↦ (𝑥 · 1 )) ∈ V
2316, 17, 22fvmpt 6436 . . 3 (𝑊 ∈ V → (algSc‘𝑊) = (𝑥𝐾 ↦ (𝑥 · 1 )))
24 fvprc 6338 . . . . 5 𝑊 ∈ V → (algSc‘𝑊) = ∅)
25 mpt0 6174 . . . . 5 (𝑥 ∈ ∅ ↦ (𝑥 · 1 )) = ∅
2624, 25syl6eqr 2804 . . . 4 𝑊 ∈ V → (algSc‘𝑊) = (𝑥 ∈ ∅ ↦ (𝑥 · 1 )))
27 fvprc 6338 . . . . . . . . 9 𝑊 ∈ V → (Scalar‘𝑊) = ∅)
283, 27syl5eq 2798 . . . . . . . 8 𝑊 ∈ V → 𝐹 = ∅)
2928fveq2d 6348 . . . . . . 7 𝑊 ∈ V → (Base‘𝐹) = (Base‘∅))
30 base0 16106 . . . . . . 7 ∅ = (Base‘∅)
3129, 30syl6eqr 2804 . . . . . 6 𝑊 ∈ V → (Base‘𝐹) = ∅)
326, 31syl5eq 2798 . . . . 5 𝑊 ∈ V → 𝐾 = ∅)
3332mpteq1d 4882 . . . 4 𝑊 ∈ V → (𝑥𝐾 ↦ (𝑥 · 1 )) = (𝑥 ∈ ∅ ↦ (𝑥 · 1 )))
3426, 33eqtr4d 2789 . . 3 𝑊 ∈ V → (algSc‘𝑊) = (𝑥𝐾 ↦ (𝑥 · 1 )))
3523, 34pm2.61i 176 . 2 (algSc‘𝑊) = (𝑥𝐾 ↦ (𝑥 · 1 ))
361, 35eqtri 2774 1 𝐴 = (𝑥𝐾 ↦ (𝑥 · 1 ))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1624  wcel 2131  Vcvv 3332  c0 4050  cmpt 4873  cfv 6041  (class class class)co 6805  Basecbs 16051  Scalarcsca 16138   ·𝑠 cvsca 16139  1rcur 18693  algSccascl 19505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-rep 4915  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-reu 3049  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-iun 4666  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-slot 16055  df-base 16057  df-ascl 19508
This theorem is referenced by:  asclval  19529  asclfn  19530  asclf  19531  rnascl  19537  ressascl  19538  asclpropd  19540
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