Theorem cnpwstotbnd 33596

Description: A subset of A ^ I, where A C_ CC, is totally bounded iff it is bounded. (Contributed by Mario Carneiro, 14-Sep-2015.)

Hypotheses

Ref	Expression
cnpwstotbnd.y	|- Y = ( (ℂfld ↾s A ) ^s I )
cnpwstotbnd.d	|- D = ( ( dist ` Y ) |` ( X X. X ) )

Assertion

Ref	Expression
cnpwstotbnd	|- ( ( A C_ CC /\ I e. Fin ) -> ( D e. ( TotBnd ` X ) <-> D e. ( Bnd ` X ) ) )

Proof of Theorem cnpwstotbnd

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- ( (Scalar ` (ℂfld ↾s A ) ) X_s ( I X. { (ℂfld ↾s A ) } ) ) = ( (Scalar ` (ℂfld ↾s A ) ) X_s ( I X. { (ℂfld ↾s A ) } ) )
2		eqid 2622	. . 3 |- ( Base ` ( (Scalar ` (ℂfld ↾s A ) ) X_s ( I X. { (ℂfld ↾s A ) } ) ) ) = ( Base ` ( (Scalar ` (ℂfld ↾s A ) ) X_s ( I X. { (ℂfld ↾s A ) } ) ) )
3		eqid 2622	. . 3 |- ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) = ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) )
4		eqid 2622	. . 3 |- ( ( dist ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) |` ( ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) X. ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) ) ) = ( ( dist ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) |` ( ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) X. ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) ) )
5		eqid 2622	. . 3 |- ( dist ` ( (Scalar ` (ℂfld ↾s A ) ) X_s ( I X. { (ℂfld ↾s A ) } ) ) ) = ( dist ` ( (Scalar ` (ℂfld ↾s A ) ) X_s ( I X. { (ℂfld ↾s A ) } ) ) )
6		fvexd 6203	. . 3 |- ( ( A C_ CC /\ I e. Fin ) -> (Scalar ` (ℂfld ↾s A ) ) e. _V )
7		simpr 477	. . 3 |- ( ( A C_ CC /\ I e. Fin ) -> I e. Fin )
8		ovex 6678	. . . 4 |- (ℂfld ↾s A ) e. _V
9		fnconstg 6093	. . . 4 |- ( (ℂfld ↾s A ) e. _V -> ( I X. { (ℂfld ↾s A ) } ) Fn I )
10	8, 9	mp1i 13	. . 3 |- ( ( A C_ CC /\ I e. Fin ) -> ( I X. { (ℂfld ↾s A ) } ) Fn I )
11		eqid 2622	. . 3 |- ( ( dist ` ( (Scalar ` (ℂfld ↾s A ) ) X_s ( I X. { (ℂfld ↾s A ) } ) ) ) |` ( X X. X ) ) = ( ( dist ` ( (Scalar ` (ℂfld ↾s A ) ) X_s ( I X. { (ℂfld ↾s A ) } ) ) ) |` ( X X. X ) )
12		cnfldms 22579	. . . . . 6 |- ℂfld e. MetSp
13		cnex 10017	. . . . . . . 8 |- CC e. _V
14	13	ssex 4802	. . . . . . 7 |- ( A C_ CC -> A e. _V )
15	14	ad2antrr 762	. . . . . 6 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> A e. _V )
16		ressms 22331	. . . . . 6 |- ( (ℂfld e. MetSp /\ A e. _V ) -> (ℂfld ↾s A ) e. MetSp )
17	12, 15, 16	sylancr 695	. . . . 5 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> (ℂfld ↾s A ) e. MetSp )
18		eqid 2622	. . . . . 6 |- ( Base ` (ℂfld ↾s A ) ) = ( Base ` (ℂfld ↾s A ) )
19		eqid 2622	. . . . . 6 |- ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) = ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) )
20	18, 19	msmet 22262	. . . . 5 |- ( (ℂfld ↾s A ) e. MetSp -> ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) e. ( Met ` ( Base ` (ℂfld ↾s A ) ) ) )
21	17, 20	syl 17	. . . 4 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) e. ( Met ` ( Base ` (ℂfld ↾s A ) ) ) )
22	8	fvconst2 6469	. . . . . . 7 |- ( x e. I -> ( ( I X. { (ℂfld ↾s A ) } ) ` x ) = (ℂfld ↾s A ) )
23	22	adantl 482	. . . . . 6 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> ( ( I X. { (ℂfld ↾s A ) } ) ` x ) = (ℂfld ↾s A ) )
24	23	fveq2d 6195	. . . . 5 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> ( dist ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) = ( dist ` (ℂfld ↾s A ) ) )
25	23	fveq2d 6195	. . . . . 6 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) = ( Base ` (ℂfld ↾s A ) ) )
26	25	sqxpeqd 5141	. . . . 5 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> ( ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) X. ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) ) = ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) )
27	24, 26	reseq12d 5397	. . . 4 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> ( ( dist ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) |` ( ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) X. ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) ) ) = ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) )
28	25	fveq2d 6195	. . . 4 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> ( Met ` ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) ) = ( Met ` ( Base ` (ℂfld ↾s A ) ) ) )
29	21, 27, 28	3eltr4d 2716	. . 3 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> ( ( dist ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) |` ( ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) X. ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) ) ) e. ( Met ` ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) ) )
30		totbndbnd 33588	. . . . . 6 |- ( ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) |` ( y X. y ) ) e. ( TotBnd ` y ) -> ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) |` ( y X. y ) ) e. ( Bnd ` y ) )
31		eqid 2622	. . . . . . . . . . 11 |- (ℂfld ↾s A ) = (ℂfld ↾s A )
32		cnfldbas 19750	. . . . . . . . . . 11 |- CC = ( Base ` ℂfld )
33	31, 32	ressbas2 15931	. . . . . . . . . 10 |- ( A C_ CC -> A = ( Base ` (ℂfld ↾s A ) ) )
34	33	ad2antrr 762	. . . . . . . . 9 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> A = ( Base ` (ℂfld ↾s A ) ) )
35	34	fveq2d 6195	. . . . . . . 8 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> ( Met ` A ) = ( Met ` ( Base ` (ℂfld ↾s A ) ) ) )
36	21, 35	eleqtrrd 2704	. . . . . . 7 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) e. ( Met ` A ) )
37		eqid 2622	. . . . . . . . 9 |- ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) |` ( y X. y ) ) = ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) |` ( y X. y ) )
38	37	bnd2lem 33590	. . . . . . . 8 |- ( ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) e. ( Met ` A ) /\ ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) |` ( y X. y ) ) e. ( Bnd ` y ) ) -> y C_ A )
39	38	ex 450	. . . . . . 7 |- ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) e. ( Met ` A ) -> ( ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) |` ( y X. y ) ) e. ( Bnd ` y ) -> y C_ A ) )
40	36, 39	syl 17	. . . . . 6 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> ( ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) |` ( y X. y ) ) e. ( Bnd ` y ) -> y C_ A ) )
41	30, 40	syl5 34	. . . . 5 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> ( ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) |` ( y X. y ) ) e. ( TotBnd ` y ) -> y C_ A ) )
42		eqid 2622	. . . . . . . . 9 |- ( ( abs o. - ) |` ( y X. y ) ) = ( ( abs o. - ) |` ( y X. y ) )
43	42	cntotbnd 33595	. . . . . . . 8 |- ( ( ( abs o. - ) |` ( y X. y ) ) e. ( TotBnd ` y ) <-> ( ( abs o. - ) |` ( y X. y ) ) e. ( Bnd ` y ) )
44	43	a1i 11	. . . . . . 7 |- ( ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) /\ y C_ A ) -> ( ( ( abs o. - ) |` ( y X. y ) ) e. ( TotBnd ` y ) <-> ( ( abs o. - ) |` ( y X. y ) ) e. ( Bnd ` y ) ) )
45	34	sseq2d 3633	. . . . . . . . . . . 12 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> ( y C_ A <-> y C_ ( Base ` (ℂfld ↾s A ) ) ) )
46	45	biimpa 501	. . . . . . . . . . 11 |- ( ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) /\ y C_ A ) -> y C_ ( Base ` (ℂfld ↾s A ) ) )
47		xpss12 5225	. . . . . . . . . . 11 |- ( ( y C_ ( Base ` (ℂfld ↾s A ) ) /\ y C_ ( Base ` (ℂfld ↾s A ) ) ) -> ( y X. y ) C_ ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) )
48	46, 46, 47	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) /\ y C_ A ) -> ( y X. y ) C_ ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) )
49	48	resabs1d 5428	. . . . . . . . 9 |- ( ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) /\ y C_ A ) -> ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) |` ( y X. y ) ) = ( ( dist ` (ℂfld ↾s A ) ) |` ( y X. y ) ) )
50	15	adantr 481	. . . . . . . . . . 11 |- ( ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) /\ y C_ A ) -> A e. _V )
51		cnfldds 19756	. . . . . . . . . . . 12 |- ( abs o. - ) = ( dist ` ℂfld )
52	31, 51	ressds 16073	. . . . . . . . . . 11 |- ( A e. _V -> ( abs o. - ) = ( dist ` (ℂfld ↾s A ) ) )
53	50, 52	syl 17	. . . . . . . . . 10 |- ( ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) /\ y C_ A ) -> ( abs o. - ) = ( dist ` (ℂfld ↾s A ) ) )
54	53	reseq1d 5395	. . . . . . . . 9 |- ( ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) /\ y C_ A ) -> ( ( abs o. - ) |` ( y X. y ) ) = ( ( dist ` (ℂfld ↾s A ) ) |` ( y X. y ) ) )
55	49, 54	eqtr4d 2659	. . . . . . . 8 |- ( ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) /\ y C_ A ) -> ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) |` ( y X. y ) ) = ( ( abs o. - ) |` ( y X. y ) ) )
56	55	eleq1d 2686	. . . . . . 7 |- ( ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) /\ y C_ A ) -> ( ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) |` ( y X. y ) ) e. ( TotBnd ` y ) <-> ( ( abs o. - ) |` ( y X. y ) ) e. ( TotBnd ` y ) ) )
57	55	eleq1d 2686	. . . . . . 7 |- ( ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) /\ y C_ A ) -> ( ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) |` ( y X. y ) ) e. ( Bnd ` y ) <-> ( ( abs o. - ) |` ( y X. y ) ) e. ( Bnd ` y ) ) )
58	44, 56, 57	3bitr4d 300	. . . . . 6 |- ( ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) /\ y C_ A ) -> ( ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) |` ( y X. y ) ) e. ( TotBnd ` y ) <-> ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) |` ( y X. y ) ) e. ( Bnd ` y ) ) )
59	58	ex 450	. . . . 5 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> ( y C_ A -> ( ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) |` ( y X. y ) ) e. ( TotBnd ` y ) <-> ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) |` ( y X. y ) ) e. ( Bnd ` y ) ) ) )
60	41, 40, 59	pm5.21ndd 369	. . . 4 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> ( ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) |` ( y X. y ) ) e. ( TotBnd ` y ) <-> ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) |` ( y X. y ) ) e. ( Bnd ` y ) ) )
61	27	reseq1d 5395	. . . . 5 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> ( ( ( dist ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) |` ( ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) X. ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) ) ) |` ( y X. y ) ) = ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) |` ( y X. y ) ) )
62	61	eleq1d 2686	. . . 4 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> ( ( ( ( dist ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) |` ( ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) X. ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) ) ) |` ( y X. y ) ) e. ( TotBnd ` y ) <-> ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) |` ( y X. y ) ) e. ( TotBnd ` y ) ) )
63	61	eleq1d 2686	. . . 4 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> ( ( ( ( dist ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) |` ( ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) X. ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) ) ) |` ( y X. y ) ) e. ( Bnd ` y ) <-> ( ( ( dist ` (ℂfld ↾s A ) ) |` ( ( Base ` (ℂfld ↾s A ) ) X. ( Base ` (ℂfld ↾s A ) ) ) ) |` ( y X. y ) ) e. ( Bnd ` y ) ) )
64	60, 62, 63	3bitr4d 300	. . 3 |- ( ( ( A C_ CC /\ I e. Fin ) /\ x e. I ) -> ( ( ( ( dist ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) |` ( ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) X. ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) ) ) |` ( y X. y ) ) e. ( TotBnd ` y ) <-> ( ( ( dist ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) |` ( ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) X. ( Base ` ( ( I X. { (ℂfld ↾s A ) } ) ` x ) ) ) ) |` ( y X. y ) ) e. ( Bnd ` y ) ) )
65	1, 2, 3, 4, 5, 6, 7, 10, 11, 29, 64	prdsbnd2 33594	. 2 |- ( ( A C_ CC /\ I e. Fin ) -> ( ( ( dist ` ( (Scalar ` (ℂfld ↾s A ) ) X_s ( I X. { (ℂfld ↾s A ) } ) ) ) |` ( X X. X ) ) e. ( TotBnd ` X ) <-> ( ( dist ` ( (Scalar ` (ℂfld ↾s A ) ) X_s ( I X. { (ℂfld ↾s A ) } ) ) ) |` ( X X. X ) ) e. ( Bnd ` X ) ) )
66		cnpwstotbnd.d	. . . 4 |- D = ( ( dist ` Y ) |` ( X X. X ) )
67		cnpwstotbnd.y	. . . . . . . 8 |- Y = ( (ℂfld ↾s A ) ^s I )
68		eqid 2622	. . . . . . . 8 |- (Scalar ` (ℂfld ↾s A ) ) = (Scalar ` (ℂfld ↾s A ) )
69	67, 68	pwsval 16146	. . . . . . 7 |- ( ( (ℂfld ↾s A ) e. _V /\ I e. Fin ) -> Y = ( (Scalar ` (ℂfld ↾s A ) ) X_s ( I X. { (ℂfld ↾s A ) } ) ) )
70	8, 7, 69	sylancr 695	. . . . . 6 |- ( ( A C_ CC /\ I e. Fin ) -> Y = ( (Scalar ` (ℂfld ↾s A ) ) X_s ( I X. { (ℂfld ↾s A ) } ) ) )
71	70	fveq2d 6195	. . . . 5 |- ( ( A C_ CC /\ I e. Fin ) -> ( dist ` Y ) = ( dist ` ( (Scalar ` (ℂfld ↾s A ) ) X_s ( I X. { (ℂfld ↾s A ) } ) ) ) )
72	71	reseq1d 5395	. . . 4 |- ( ( A C_ CC /\ I e. Fin ) -> ( ( dist ` Y ) |` ( X X. X ) ) = ( ( dist ` ( (Scalar ` (ℂfld ↾s A ) ) X_s ( I X. { (ℂfld ↾s A ) } ) ) ) |` ( X X. X ) ) )
73	66, 72	syl5eq 2668	. . 3 |- ( ( A C_ CC /\ I e. Fin ) -> D = ( ( dist ` ( (Scalar ` (ℂfld ↾s A ) ) X_s ( I X. { (ℂfld ↾s A ) } ) ) ) |` ( X X. X ) ) )
74	73	eleq1d 2686	. 2 |- ( ( A C_ CC /\ I e. Fin ) -> ( D e. ( TotBnd ` X ) <-> ( ( dist ` ( (Scalar ` (ℂfld ↾s A ) ) X_s ( I X. { (ℂfld ↾s A ) } ) ) ) |` ( X X. X ) ) e. ( TotBnd ` X ) ) )
75	73	eleq1d 2686	. 2 |- ( ( A C_ CC /\ I e. Fin ) -> ( D e. ( Bnd ` X ) <-> ( ( dist ` ( (Scalar ` (ℂfld ↾s A ) ) X_s ( I X. { (ℂfld ↾s A ) } ) ) ) |` ( X X. X ) ) e. ( Bnd ` X ) ) )
76	65, 74, 75	3bitr4d 300	1 |- ( ( A C_ CC /\ I e. Fin ) -> ( D e. ( TotBnd ` X ) <-> D e. ( Bnd ` X ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   {csn 4177   X. cxp 5112   |` cres 5116   o. ccom 5118   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   - cmin 10266   abscabs 13974   Basecbs 15857   ↾_s cress 15858  Scalarcsca 15944   distcds 15950   X__scprds 16106   ^_s cpws 16107   Metcme 19732  ℂ_fldccnfld 19746   MetSpcmt 22123   TotBndctotbnd 33565   Bndcbnd 33566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-ec 7744  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-fz 12327  df-fl 12593  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-gz 15634  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-topgen 16104  df-prds 16108  df-pws 16110  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-xms 22125  df-ms 22126  df-totbnd 33567  df-bnd 33578

This theorem is referenced by:  rrntotbnd  33635