Theorem asclfval 15021

Description: Function value of the algebra scalar lifting 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, 1   𝑥, ·   𝑥,𝐾   𝑥,𝑊

Allowed substitution hints:   𝐴(𝑥)   𝐹(𝑥)

Proof of Theorem asclfval

Dummy variables 𝑞 𝑤 𝑗 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		asclfval.a	. 2 ⊢ 𝐴 = (algSc‘𝑊)
2		df-ascl 15001	. . . . 5 ⊢ algSc = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤))))
3	2	mptrcl 5788	. . . 4 ⊢ (𝑞 ∈ (algSc‘𝑊) → 𝑊 ∈ V)
4		mptmex 5945	. . . . 5 ⊢ (𝑞 ∈ (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )) → ∃𝑗 𝑗 ∈ 𝐾)
5		asclfval.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝐹)
6	5	basm 13414	. . . . . 6 ⊢ (𝑗 ∈ 𝐾 → ∃𝑘 𝑘 ∈ 𝐹)
7	6	exlimiv 1651	. . . . 5 ⊢ (∃𝑗 𝑗 ∈ 𝐾 → ∃𝑘 𝑘 ∈ 𝐹)
8		mptrel 4908	. . . . . . . . . . 11 ⊢ Rel (𝑢 ∈ V ↦ (𝑢‘(Scalar‘ndx)))
9		df-slot 13356	. . . . . . . . . . . 12 ⊢ Slot (Scalar‘ndx) = (𝑢 ∈ V ↦ (𝑢‘(Scalar‘ndx)))
10	9	releqi 4858	. . . . . . . . . . 11 ⊢ (Rel Slot (Scalar‘ndx) ↔ Rel (𝑢 ∈ V ↦ (𝑢‘(Scalar‘ndx))))
11	8, 10	mpbir 146	. . . . . . . . . 10 ⊢ Rel Slot (Scalar‘ndx)
12		scaid 13506	. . . . . . . . . . 11 ⊢ Scalar = Slot (Scalar‘ndx)
13	12	releqi 4858	. . . . . . . . . 10 ⊢ (Rel Scalar ↔ Rel Slot (Scalar‘ndx))
14	11, 13	mpbir 146	. . . . . . . . 9 ⊢ Rel Scalar
15		relelfvdm 5727	. . . . . . . . 9 ⊢ ((Rel Scalar ∧ 𝑘 ∈ (Scalar‘𝑊)) → 𝑊 ∈ dom Scalar)
16	14, 15	mpan 428	. . . . . . . 8 ⊢ (𝑘 ∈ (Scalar‘𝑊) → 𝑊 ∈ dom Scalar)
17		asclfval.f	. . . . . . . 8 ⊢ 𝐹 = (Scalar‘𝑊)
18	16, 17	eleq2s 2333	. . . . . . 7 ⊢ (𝑘 ∈ 𝐹 → 𝑊 ∈ dom Scalar)
19	18	exlimiv 1651	. . . . . 6 ⊢ (∃𝑘 𝑘 ∈ 𝐹 → 𝑊 ∈ dom Scalar)
20	19	elexd 2835	. . . . 5 ⊢ (∃𝑘 𝑘 ∈ 𝐹 → 𝑊 ∈ V)
21	4, 7, 20	3syl 17	. . . 4 ⊢ (𝑞 ∈ (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )) → 𝑊 ∈ V)
22		fveq2 5695	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = (Scalar‘𝑊))
23	22, 17	eqtr4di 2289	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (Scalar‘𝑤) = 𝐹)
24	23	fveq2d 5699	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = (Base‘𝐹))
25	24, 5	eqtr4di 2289	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (Base‘(Scalar‘𝑤)) = 𝐾)
26		fveq2 5695	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = ( ·𝑠 ‘𝑊))
27		asclfval.s	. . . . . . . . 9 ⊢ · = ( ·𝑠 ‘𝑊)
28	26, 27	eqtr4di 2289	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ( ·𝑠 ‘𝑤) = · )
29		eqidd 2239	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → 𝑥 = 𝑥)
30		fveq2 5695	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (1r‘𝑤) = (1r‘𝑊))
31		asclfval.o	. . . . . . . . 9 ⊢ 1 = (1r‘𝑊)
32	30, 31	eqtr4di 2289	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → (1r‘𝑤) = 1 )
33	28, 29, 32	oveq123d 6106	. . . . . . 7 ⊢ (𝑤 = 𝑊 → (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤)) = (𝑥 · 1 ))
34	25, 33	mpteq12dv 4213	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝑥 ∈ (Base‘(Scalar‘𝑤)) ↦ (𝑥( ·𝑠 ‘𝑤)(1r‘𝑤))) = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )))
35		id 19	. . . . . 6 ⊢ (𝑊 ∈ V → 𝑊 ∈ V)
36		basfn 13411	. . . . . . . . 9 ⊢ Base Fn V
37		scaslid 13507	. . . . . . . . . . 11 ⊢ (Scalar = Slot (Scalar‘ndx) ∧ (Scalar‘ndx) ∈ ℕ)
38	37	slotex 13379	. . . . . . . . . 10 ⊢ (𝑊 ∈ V → (Scalar‘𝑊) ∈ V)
39	17, 38	eqeltrid 2325	. . . . . . . . 9 ⊢ (𝑊 ∈ V → 𝐹 ∈ V)
40		funfvex 5712	. . . . . . . . . 10 ⊢ ((Fun Base ∧ 𝐹 ∈ dom Base) → (Base‘𝐹) ∈ V)
41	40	funfni 5483	. . . . . . . . 9 ⊢ ((Base Fn V ∧ 𝐹 ∈ V) → (Base‘𝐹) ∈ V)
42	36, 39, 41	sylancr 418	. . . . . . . 8 ⊢ (𝑊 ∈ V → (Base‘𝐹) ∈ V)
43	5, 42	eqeltrid 2325	. . . . . . 7 ⊢ (𝑊 ∈ V → 𝐾 ∈ V)
44	43	mptexd 5944	. . . . . 6 ⊢ (𝑊 ∈ V → (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )) ∈ V)
45	2, 34, 35, 44	fvmptd3 5799	. . . . 5 ⊢ (𝑊 ∈ V → (algSc‘𝑊) = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )))
46	45	eleq2d 2308	. . . 4 ⊢ (𝑊 ∈ V → (𝑞 ∈ (algSc‘𝑊) ↔ 𝑞 ∈ (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 ))))
47	3, 21, 46	pm5.21nii 716	. . 3 ⊢ (𝑞 ∈ (algSc‘𝑊) ↔ 𝑞 ∈ (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 )))
48	47	eqriv 2235	. 2 ⊢ (algSc‘𝑊) = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 ))
49	1, 48	eqtri 2259	1 ⊢ 𝐴 = (𝑥 ∈ 𝐾 ↦ (𝑥 · 1 ))

This proof depends on syntax axioms:   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821   ↦ cmpt 4192  dom cdm 4774  Rel wrel 4779   Fn wfn 5372  ‘cfv 5377  (class class class)co 6085  ndxcnx 13349  Slot cslot 13351  Basecbs 13352  Scalarcsca 13434   ·_𝑠 cvsca 13435  1_rcur 14262  algSccascl 14998

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-cnex 8270  ax-resscn 8271  ax-1re 8273  ax-addrcl 8276

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-ndx 13355  df-slot 13356  df-base 13358  df-sca 13447  df-ascl 15001

This theorem is used by:  asclvald  15022  asclfnd  15023  asclf  15024  rnascl  15034  ressascl  15039  asclpropd  15040