Theorem asclfval 20111

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	⊢ · = ( ·𝑠 ‘𝑊)
asclfval.o	⊢ 1 = (1r‘𝑊)

Assertion

Ref	Expression
asclfval	⊢ 𝐴 = (𝑥 ∈ 𝐾 ↦ (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.

Step	Hyp	Ref	Expression
1		asclfval.a	. 2 ⊢ 𝐴 = (algSc‘𝑊)
2		fveq2 6673	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
3		asclfval.f	. . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊)
4	2, 3	syl6eqr 2877	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
5	4	fveq2d 6677	. . . . . 6 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
6		asclfval.k	. . . . . 6 ⊢ 𝐾 = (Base‘𝐹)
7	5, 6	syl6eqr 2877	. . . . 5 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
8		fveq2 6673	. . . . . . 7 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
9		asclfval.s	. . . . . . 7 ⊢ · = ( ·𝑠 ‘𝑊)
10	8, 9	syl6eqr 2877	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
11		eqidd 2825	. . . . . 6 ⊢ (𝑤 = 𝑊 → 𝑥 = 𝑥)
12		fveq2 6673	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (1r‘𝑤) = (1r‘𝑊))
13		asclfval.o	. . . . . . 7 ⊢ 1 = (1r‘𝑊)
14	12, 13	syl6eqr 2877	. . . . . 6 ⊢ (𝑤 = 𝑊 → (1r‘𝑤) = 1 )
15	10, 11, 14	oveq123d 7180	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤)) = (𝑥 · 1 ))
16	7, 15	mpteq12dv 5154	. . . 4 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤))) = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )))
17		df-ascl 20090	. . . 4 ⊢ algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤))))
18	16, 17, 6	mptfvmpt 6993	. . 3 ⊢ (𝑊 ∈ V → (algSc‘𝑊) = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )))
19		fvprc 6666	. . . . 5 ⊢ (¬ 𝑊 ∈ V → (algSc‘𝑊) = ∅)
20		mpt0 6493	. . . . 5 ⊢ (𝑥 ∈ ∅ ↦ (𝑥 · 1 )) = ∅
21	19, 20	syl6eqr 2877	. . . 4 ⊢ (¬ 𝑊 ∈ V → (algSc‘𝑊) = (𝑥 ∈ ∅ ↦ (𝑥 · 1 )))
22		fvprc 6666	. . . . . . . . 9 ⊢ (¬ 𝑊 ∈ V → (Scalar‘𝑊) = ∅)
23	3, 22	syl5eq 2871	. . . . . . . 8 ⊢ (¬ 𝑊 ∈ V → 𝐹 = ∅)
24	23	fveq2d 6677	. . . . . . 7 ⊢ (¬ 𝑊 ∈ V → (Base‘𝐹) = (Base‘∅))
25		base0 16539	. . . . . . 7 ⊢ ∅ = (Base‘∅)
26	24, 25	syl6eqr 2877	. . . . . 6 ⊢ (¬ 𝑊 ∈ V → (Base‘𝐹) = ∅)
27	6, 26	syl5eq 2871	. . . . 5 ⊢ (¬ 𝑊 ∈ V → 𝐾 = ∅)
28	27	mpteq1d 5158	. . . 4 ⊢ (¬ 𝑊 ∈ V → (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )) = (𝑥 ∈ ∅ ↦ (𝑥 · 1 )))
29	21, 28	eqtr4d 2862	. . 3 ⊢ (¬ 𝑊 ∈ V → (algSc‘𝑊) = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )))
30	18, 29	pm2.61i 184	. 2 ⊢ (algSc‘𝑊) = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 ))
31	1, 30	eqtri 2847	1 ⊢ 𝐴 = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 ))

Syntax hints:  ¬ wn 3   = wceq 1536   ∈ wcel 2113  Vcvv 3497  ∅c0 4294   ↦ cmpt 5149  ‘cfv 6358  (class class class)co 7159  Basecbs 16486  Scalarcsca 16571   ·_𝑠 cvsca 16572  1_rcur 19254  algSccascl 20087

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-slot 16490  df-base 16492  df-ascl 20090

This theorem is referenced by:  asclval  20112  asclfn  20113  asclf  20114  rnascl  20123  ressascl  20128  asclpropd  20129  rnasclg  39137