Theorem imasval 16171

Description: Value of an image structure. (Contributed by Mario Carneiro, 23-Feb-2015.) (Revised by Mario Carneiro, 11-Jul-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.) (Revised by AV, 6-Oct-2020.)

Hypotheses

Ref	Expression
imasval.u	|- ( ph -> U = ( F "s R ) )
imasval.v	|- ( ph -> V = ( Base ` R ) )
imasval.p	|- .+ = ( +g ` R )
imasval.m	|- .X. = ( .r ` R )
imasval.g	|- G = (Scalar ` R )
imasval.k	|- K = ( Base ` G )
imasval.q	|- .x. = ( .s ` R )
imasval.i	|- ., = ( .i ` R )
imasval.j	|- J = ( TopOpen ` R )
imasval.e	|- E = ( dist ` R )
imasval.n	|- N = ( le ` R )
imasval.a	|- ( ph -> .+b = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } )
imasval.t	|- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. } )
imasval.s	|- ( ph -> .(x) = U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) )
imasval.w	|- ( ph -> I = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } )
imasval.o	|- ( ph -> O = ( J qTop F ) )
imasval.d	|- ( ph -> D = ( x e. B , y e. B |-> inf ( U_ n e. NN ran ( g e. { h e. ( ( V X. V ) ^m ( 1 ... n ) ) | ( ( F ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( F ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( F ` ( 2nd ` ( h ` i ) ) ) = ( F ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsumg ( E o. g ) ) ) , RR* , < ) ) )
imasval.l	|- ( ph -> .<_ = ( ( F o. N ) o. `' F ) )
imasval.f	|- ( ph -> F : V -onto-> B )
imasval.r	|- ( ph -> R e. Z )

Assertion

Ref	Expression
imasval	|- ( ph -> U = ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .xb >. } u. { <. (Scalar ` ndx ) , G >. , <. ( .s ` ndx ) , .(x) >. , <. ( .i ` ndx ) , I >. } ) u. { <. (TopSet ` ndx ) , O >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. } ) )

Distinct variable groups:   g, h, i, n, p, q, x, y, F   R, g, h, i, n, p, q, x, y   h, V, p, q   ph, g, h, i, n, p, q, x, y

Allowed substitution hints:   B( x, y, g, h, i, n, q, p)   D( x, y, g, h, i, n, q, p)   .+ ( x, y, g, h, i, n, q, p)   .+b ( x, y, g, h, i, n, q, p)   .xb ( x, y, g, h, i, n, q, p)   .x. ( x, y, g, h, i, n, q, p)   .X. ( x, y, g, h, i, n, q, p)   .(x) ( x, y, g, h, i, n, q, p)   U( x, y, g, h, i, n, q, p)   E( x, y, g, h, i, n, q, p)   G( x, y, g, h, i, n, q, p)   ., ( x, y, g, h, i, n, q, p)   I( x, y, g, h, i, n, q, p)   J( x, y, g, h, i, n, q, p)   K( x, y, g, h, i, n, q, p)   .<_ ( x, y, g, h, i, n, q, p)   N( x, y, g, h, i, n, q, p)   O( x, y, g, h, i, n, q, p)   V( x, y, g, i, n)   Z( x, y, g, h, i, n, q, p)

Proof of Theorem imasval

Dummy variables f r v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imasval.u	. 2 |- ( ph -> U = ( F "s R ) )
2		df-imas 16168	. . . 4 |- "s = ( f e. _V , r e. _V |-> [_ ( Base ` r ) / v ]_ ( ( { <. ( Base ` ndx ) , ran f >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } >. } u. { <. (Scalar ` ndx ) , (Scalar ` r ) >. , <. ( .s ` ndx ) , U_ q e. v ( p e. ( Base ` (Scalar ` r ) ) , x e. { ( f ` q ) } |-> ( f ` ( p ( .s ` r ) q ) ) ) >. , <. ( .i ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( p ( .i ` r ) q ) >. } >. } ) u. { <. (TopSet ` ndx ) , ( ( TopOpen ` r ) qTop f ) >. , <. ( le ` ndx ) , ( ( f o. ( le ` r ) ) o. `' f ) >. , <. ( dist ` ndx ) , ( x e. ran f , y e. ran f |-> inf ( U_ n e. NN ran ( g e. { h e. ( ( v X. v ) ^m ( 1 ... n ) ) | ( ( f ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( f ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( f ` ( 2nd ` ( h ` i ) ) ) = ( f ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsumg ( ( dist ` r ) o. g ) ) ) , RR* , < ) ) >. } ) )
3	2	a1i 11	. . 3 |- ( ph -> "s = ( f e. _V , r e. _V |-> [_ ( Base ` r ) / v ]_ ( ( { <. ( Base ` ndx ) , ran f >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } >. } u. { <. (Scalar ` ndx ) , (Scalar ` r ) >. , <. ( .s ` ndx ) , U_ q e. v ( p e. ( Base ` (Scalar ` r ) ) , x e. { ( f ` q ) } |-> ( f ` ( p ( .s ` r ) q ) ) ) >. , <. ( .i ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( p ( .i ` r ) q ) >. } >. } ) u. { <. (TopSet ` ndx ) , ( ( TopOpen ` r ) qTop f ) >. , <. ( le ` ndx ) , ( ( f o. ( le ` r ) ) o. `' f ) >. , <. ( dist ` ndx ) , ( x e. ran f , y e. ran f |-> inf ( U_ n e. NN ran ( g e. { h e. ( ( v X. v ) ^m ( 1 ... n ) ) | ( ( f ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( f ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( f ` ( 2nd ` ( h ` i ) ) ) = ( f ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsumg ( ( dist ` r ) o. g ) ) ) , RR* , < ) ) >. } ) ) )
4		fvexd 6203	. . . 4 |- ( ( ph /\ ( f = F /\ r = R ) ) -> ( Base ` r ) e. _V )
5		simplrl 800	. . . . . . . . . 10 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> f = F )
6	5	rneqd 5353	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ran f = ran F )
7		imasval.f	. . . . . . . . . . 11 |- ( ph -> F : V -onto-> B )
8		forn 6118	. . . . . . . . . . 11 |- ( F : V -onto-> B -> ran F = B )
9	7, 8	syl 17	. . . . . . . . . 10 |- ( ph -> ran F = B )
10	9	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ran F = B )
11	6, 10	eqtrd 2656	. . . . . . . 8 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ran f = B )
12	11	opeq2d 4409	. . . . . . 7 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> <. ( Base ` ndx ) , ran f >. = <. ( Base ` ndx ) , B >. )
13		simplrr 801	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> r = R )
14	13	fveq2d 6195	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( Base ` r ) = ( Base ` R ) )
15		simpr 477	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> v = ( Base ` r ) )
16		imasval.v	. . . . . . . . . . . 12 |- ( ph -> V = ( Base ` R ) )
17	16	ad2antrr 762	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> V = ( Base ` R ) )
18	14, 15, 17	3eqtr4d 2666	. . . . . . . . . 10 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> v = V )
19	5	fveq1d 6193	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( f ` p ) = ( F ` p ) )
20	5	fveq1d 6193	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( f ` q ) = ( F ` q ) )
21	19, 20	opeq12d 4410	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> <. ( f ` p ) , ( f ` q ) >. = <. ( F ` p ) , ( F ` q ) >. )
22	13	fveq2d 6195	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( +g ` r ) = ( +g ` R ) )
23		imasval.p	. . . . . . . . . . . . . . . 16 |- .+ = ( +g ` R )
24	22, 23	syl6eqr 2674	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( +g ` r ) = .+ )
25	24	oveqd 6667	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( p ( +g ` r ) q ) = ( p .+ q ) )
26	5, 25	fveq12d 6197	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( f ` ( p ( +g ` r ) q ) ) = ( F ` ( p .+ q ) ) )
27	21, 26	opeq12d 4410	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. = <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. )
28	27	sneqd 4189	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } = { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } )
29	18, 28	iuneq12d 4546	. . . . . . . . . 10 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } = U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } )
30	18, 29	iuneq12d 4546	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } )
31		imasval.a	. . . . . . . . . 10 |- ( ph -> .+b = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } )
32	31	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> .+b = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .+ q ) ) >. } )
33	30, 32	eqtr4d 2659	. . . . . . . 8 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } = .+b )
34	33	opeq2d 4409	. . . . . . 7 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } >. = <. ( +g ` ndx ) , .+b >. )
35	13	fveq2d 6195	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( .r ` r ) = ( .r ` R ) )
36		imasval.m	. . . . . . . . . . . . . . . 16 |- .X. = ( .r ` R )
37	35, 36	syl6eqr 2674	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( .r ` r ) = .X. )
38	37	oveqd 6667	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( p ( .r ` r ) q ) = ( p .X. q ) )
39	5, 38	fveq12d 6197	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( f ` ( p ( .r ` r ) q ) ) = ( F ` ( p .X. q ) ) )
40	21, 39	opeq12d 4410	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. = <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. )
41	40	sneqd 4189	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } = { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. } )
42	18, 41	iuneq12d 4546	. . . . . . . . . 10 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } = U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. } )
43	18, 42	iuneq12d 4546	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. } )
44		imasval.t	. . . . . . . . . 10 |- ( ph -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. } )
45	44	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> .xb = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( F ` ( p .X. q ) ) >. } )
46	43, 45	eqtr4d 2659	. . . . . . . 8 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } = .xb )
47	46	opeq2d 4409	. . . . . . 7 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } >. = <. ( .r ` ndx ) , .xb >. )
48	12, 34, 47	tpeq123d 4283	. . . . . 6 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> { <. ( Base ` ndx ) , ran f >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } >. } = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .xb >. } )
49	13	fveq2d 6195	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> (Scalar ` r ) = (Scalar ` R ) )
50		imasval.g	. . . . . . . . 9 |- G = (Scalar ` R )
51	49, 50	syl6eqr 2674	. . . . . . . 8 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> (Scalar ` r ) = G )
52	51	opeq2d 4409	. . . . . . 7 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> <. (Scalar ` ndx ) , (Scalar ` r ) >. = <. (Scalar ` ndx ) , G >. )
53	51	fveq2d 6195	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( Base ` (Scalar ` r ) ) = ( Base ` G ) )
54		imasval.k	. . . . . . . . . . . 12 |- K = ( Base ` G )
55	53, 54	syl6eqr 2674	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( Base ` (Scalar ` r ) ) = K )
56	20	sneqd 4189	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> { ( f ` q ) } = { ( F ` q ) } )
57	13	fveq2d 6195	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( .s ` r ) = ( .s ` R ) )
58		imasval.q	. . . . . . . . . . . . . 14 |- .x. = ( .s ` R )
59	57, 58	syl6eqr 2674	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( .s ` r ) = .x. )
60	59	oveqd 6667	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( p ( .s ` r ) q ) = ( p .x. q ) )
61	5, 60	fveq12d 6197	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( f ` ( p ( .s ` r ) q ) ) = ( F ` ( p .x. q ) ) )
62	55, 56, 61	mpt2eq123dv 6717	. . . . . . . . . 10 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( p e. ( Base ` (Scalar ` r ) ) , x e. { ( f ` q ) } |-> ( f ` ( p ( .s ` r ) q ) ) ) = ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) )
63	62	iuneq2d 4547	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> U_ q e. V ( p e. ( Base ` (Scalar ` r ) ) , x e. { ( f ` q ) } |-> ( f ` ( p ( .s ` r ) q ) ) ) = U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) )
64	18	iuneq1d 4545	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> U_ q e. v ( p e. ( Base ` (Scalar ` r ) ) , x e. { ( f ` q ) } |-> ( f ` ( p ( .s ` r ) q ) ) ) = U_ q e. V ( p e. ( Base ` (Scalar ` r ) ) , x e. { ( f ` q ) } |-> ( f ` ( p ( .s ` r ) q ) ) ) )
65		imasval.s	. . . . . . . . . 10 |- ( ph -> .(x) = U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) )
66	65	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> .(x) = U_ q e. V ( p e. K , x e. { ( F ` q ) } |-> ( F ` ( p .x. q ) ) ) )
67	63, 64, 66	3eqtr4d 2666	. . . . . . . 8 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> U_ q e. v ( p e. ( Base ` (Scalar ` r ) ) , x e. { ( f ` q ) } |-> ( f ` ( p ( .s ` r ) q ) ) ) = .(x) )
68	67	opeq2d 4409	. . . . . . 7 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> <. ( .s ` ndx ) , U_ q e. v ( p e. ( Base ` (Scalar ` r ) ) , x e. { ( f ` q ) } |-> ( f ` ( p ( .s ` r ) q ) ) ) >. = <. ( .s ` ndx ) , .(x) >. )
69	13	fveq2d 6195	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( .i ` r ) = ( .i ` R ) )
70		imasval.i	. . . . . . . . . . . . . . 15 |- ., = ( .i ` R )
71	69, 70	syl6eqr 2674	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( .i ` r ) = ., )
72	71	oveqd 6667	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( p ( .i ` r ) q ) = ( p ., q ) )
73	21, 72	opeq12d 4410	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> <. <. ( f ` p ) , ( f ` q ) >. , ( p ( .i ` r ) q ) >. = <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. )
74	73	sneqd 4189	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> { <. <. ( f ` p ) , ( f ` q ) >. , ( p ( .i ` r ) q ) >. } = { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } )
75	18, 74	iuneq12d 4546	. . . . . . . . . 10 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( p ( .i ` r ) q ) >. } = U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } )
76	18, 75	iuneq12d 4546	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( p ( .i ` r ) q ) >. } = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } )
77		imasval.w	. . . . . . . . . 10 |- ( ph -> I = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } )
78	77	ad2antrr 762	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> I = U_ p e. V U_ q e. V { <. <. ( F ` p ) , ( F ` q ) >. , ( p ., q ) >. } )
79	76, 78	eqtr4d 2659	. . . . . . . 8 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( p ( .i ` r ) q ) >. } = I )
80	79	opeq2d 4409	. . . . . . 7 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> <. ( .i ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( p ( .i ` r ) q ) >. } >. = <. ( .i ` ndx ) , I >. )
81	52, 68, 80	tpeq123d 4283	. . . . . 6 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> { <. (Scalar ` ndx ) , (Scalar ` r ) >. , <. ( .s ` ndx ) , U_ q e. v ( p e. ( Base ` (Scalar ` r ) ) , x e. { ( f ` q ) } |-> ( f ` ( p ( .s ` r ) q ) ) ) >. , <. ( .i ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( p ( .i ` r ) q ) >. } >. } = { <. (Scalar ` ndx ) , G >. , <. ( .s ` ndx ) , .(x) >. , <. ( .i ` ndx ) , I >. } )
82	48, 81	uneq12d 3768	. . . . 5 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( { <. ( Base ` ndx ) , ran f >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } >. } u. { <. (Scalar ` ndx ) , (Scalar ` r ) >. , <. ( .s ` ndx ) , U_ q e. v ( p e. ( Base ` (Scalar ` r ) ) , x e. { ( f ` q ) } |-> ( f ` ( p ( .s ` r ) q ) ) ) >. , <. ( .i ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( p ( .i ` r ) q ) >. } >. } ) = ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .xb >. } u. { <. (Scalar ` ndx ) , G >. , <. ( .s ` ndx ) , .(x) >. , <. ( .i ` ndx ) , I >. } ) )
83	13	fveq2d 6195	. . . . . . . . . 10 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( TopOpen ` r ) = ( TopOpen ` R ) )
84		imasval.j	. . . . . . . . . 10 |- J = ( TopOpen ` R )
85	83, 84	syl6eqr 2674	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( TopOpen ` r ) = J )
86	85, 5	oveq12d 6668	. . . . . . . 8 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( ( TopOpen ` r ) qTop f ) = ( J qTop F ) )
87		imasval.o	. . . . . . . . 9 |- ( ph -> O = ( J qTop F ) )
88	87	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> O = ( J qTop F ) )
89	86, 88	eqtr4d 2659	. . . . . . 7 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( ( TopOpen ` r ) qTop f ) = O )
90	89	opeq2d 4409	. . . . . 6 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> <. (TopSet ` ndx ) , ( ( TopOpen ` r ) qTop f ) >. = <. (TopSet ` ndx ) , O >. )
91	13	fveq2d 6195	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( le ` r ) = ( le ` R ) )
92		imasval.n	. . . . . . . . . . 11 |- N = ( le ` R )
93	91, 92	syl6eqr 2674	. . . . . . . . . 10 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( le ` r ) = N )
94	5, 93	coeq12d 5286	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( f o. ( le ` r ) ) = ( F o. N ) )
95	5	cnveqd 5298	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> `' f = `' F )
96	94, 95	coeq12d 5286	. . . . . . . 8 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( ( f o. ( le ` r ) ) o. `' f ) = ( ( F o. N ) o. `' F ) )
97		imasval.l	. . . . . . . . 9 |- ( ph -> .<_ = ( ( F o. N ) o. `' F ) )
98	97	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> .<_ = ( ( F o. N ) o. `' F ) )
99	96, 98	eqtr4d 2659	. . . . . . 7 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( ( f o. ( le ` r ) ) o. `' f ) = .<_ )
100	99	opeq2d 4409	. . . . . 6 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> <. ( le ` ndx ) , ( ( f o. ( le ` r ) ) o. `' f ) >. = <. ( le ` ndx ) , .<_ >. )
101	18	sqxpeqd 5141	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( v X. v ) = ( V X. V ) )
102	101	oveq1d 6665	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( ( v X. v ) ^m ( 1 ... n ) ) = ( ( V X. V ) ^m ( 1 ... n ) ) )
103	5	fveq1d 6193	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( f ` ( 1st ` ( h ` 1 ) ) ) = ( F ` ( 1st ` ( h ` 1 ) ) ) )
104	103	eqeq1d 2624	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( ( f ` ( 1st ` ( h ` 1 ) ) ) = x <-> ( F ` ( 1st ` ( h ` 1 ) ) ) = x ) )
105	5	fveq1d 6193	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( f ` ( 2nd ` ( h ` n ) ) ) = ( F ` ( 2nd ` ( h ` n ) ) ) )
106	105	eqeq1d 2624	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( ( f ` ( 2nd ` ( h ` n ) ) ) = y <-> ( F ` ( 2nd ` ( h ` n ) ) ) = y ) )
107	5	fveq1d 6193	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( f ` ( 2nd ` ( h ` i ) ) ) = ( F ` ( 2nd ` ( h ` i ) ) ) )
108	5	fveq1d 6193	. . . . . . . . . . . . . . . . 17 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( f ` ( 1st ` ( h ` ( i + 1 ) ) ) ) = ( F ` ( 1st ` ( h ` ( i + 1 ) ) ) ) )
109	107, 108	eqeq12d 2637	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( ( f ` ( 2nd ` ( h ` i ) ) ) = ( f ` ( 1st ` ( h ` ( i + 1 ) ) ) ) <-> ( F ` ( 2nd ` ( h ` i ) ) ) = ( F ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) )
110	109	ralbidv 2986	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( A. i e. ( 1 ... ( n - 1 ) ) ( f ` ( 2nd ` ( h ` i ) ) ) = ( f ` ( 1st ` ( h ` ( i + 1 ) ) ) ) <-> A. i e. ( 1 ... ( n - 1 ) ) ( F ` ( 2nd ` ( h ` i ) ) ) = ( F ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) )
111	104, 106, 110	3anbi123d 1399	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( ( ( f ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( f ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( f ` ( 2nd ` ( h ` i ) ) ) = ( f ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) <-> ( ( F ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( F ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( F ` ( 2nd ` ( h ` i ) ) ) = ( F ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) ) )
112	102, 111	rabeqbidv 3195	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> { h e. ( ( v X. v ) ^m ( 1 ... n ) ) | ( ( f ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( f ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( f ` ( 2nd ` ( h ` i ) ) ) = ( f ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } = { h e. ( ( V X. V ) ^m ( 1 ... n ) ) | ( ( F ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( F ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( F ` ( 2nd ` ( h ` i ) ) ) = ( F ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } )
113	13	fveq2d 6195	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( dist ` r ) = ( dist ` R ) )
114		imasval.e	. . . . . . . . . . . . . . . 16 |- E = ( dist ` R )
115	113, 114	syl6eqr 2674	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( dist ` r ) = E )
116	115	coeq1d 5283	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( ( dist ` r ) o. g ) = ( E o. g ) )
117	116	oveq2d 6666	. . . . . . . . . . . . 13 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( RR*s gsumg ( ( dist ` r ) o. g ) ) = ( RR*s gsumg ( E o. g ) ) )
118	112, 117	mpteq12dv 4733	. . . . . . . . . . . 12 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( g e. { h e. ( ( v X. v ) ^m ( 1 ... n ) ) | ( ( f ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( f ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( f ` ( 2nd ` ( h ` i ) ) ) = ( f ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsumg ( ( dist ` r ) o. g ) ) ) = ( g e. { h e. ( ( V X. V ) ^m ( 1 ... n ) ) | ( ( F ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( F ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( F ` ( 2nd ` ( h ` i ) ) ) = ( F ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsumg ( E o. g ) ) ) )
119	118	rneqd 5353	. . . . . . . . . . 11 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ran ( g e. { h e. ( ( v X. v ) ^m ( 1 ... n ) ) | ( ( f ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( f ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( f ` ( 2nd ` ( h ` i ) ) ) = ( f ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsumg ( ( dist ` r ) o. g ) ) ) = ran ( g e. { h e. ( ( V X. V ) ^m ( 1 ... n ) ) | ( ( F ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( F ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( F ` ( 2nd ` ( h ` i ) ) ) = ( F ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsumg ( E o. g ) ) ) )
120	119	iuneq2d 4547	. . . . . . . . . 10 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> U_ n e. NN ran ( g e. { h e. ( ( v X. v ) ^m ( 1 ... n ) ) | ( ( f ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( f ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( f ` ( 2nd ` ( h ` i ) ) ) = ( f ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsumg ( ( dist ` r ) o. g ) ) ) = U_ n e. NN ran ( g e. { h e. ( ( V X. V ) ^m ( 1 ... n ) ) | ( ( F ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( F ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( F ` ( 2nd ` ( h ` i ) ) ) = ( F ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsumg ( E o. g ) ) ) )
121	120	infeq1d 8383	. . . . . . . . 9 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> inf ( U_ n e. NN ran ( g e. { h e. ( ( v X. v ) ^m ( 1 ... n ) ) | ( ( f ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( f ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( f ` ( 2nd ` ( h ` i ) ) ) = ( f ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsumg ( ( dist ` r ) o. g ) ) ) , RR* , < ) = inf ( U_ n e. NN ran ( g e. { h e. ( ( V X. V ) ^m ( 1 ... n ) ) | ( ( F ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( F ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( F ` ( 2nd ` ( h ` i ) ) ) = ( F ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsumg ( E o. g ) ) ) , RR* , < ) )
122	11, 11, 121	mpt2eq123dv 6717	. . . . . . . 8 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( x e. ran f , y e. ran f |-> inf ( U_ n e. NN ran ( g e. { h e. ( ( v X. v ) ^m ( 1 ... n ) ) | ( ( f ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( f ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( f ` ( 2nd ` ( h ` i ) ) ) = ( f ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsumg ( ( dist ` r ) o. g ) ) ) , RR* , < ) ) = ( x e. B , y e. B |-> inf ( U_ n e. NN ran ( g e. { h e. ( ( V X. V ) ^m ( 1 ... n ) ) | ( ( F ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( F ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( F ` ( 2nd ` ( h ` i ) ) ) = ( F ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsumg ( E o. g ) ) ) , RR* , < ) ) )
123		imasval.d	. . . . . . . . 9 |- ( ph -> D = ( x e. B , y e. B |-> inf ( U_ n e. NN ran ( g e. { h e. ( ( V X. V ) ^m ( 1 ... n ) ) | ( ( F ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( F ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( F ` ( 2nd ` ( h ` i ) ) ) = ( F ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsumg ( E o. g ) ) ) , RR* , < ) ) )
124	123	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> D = ( x e. B , y e. B |-> inf ( U_ n e. NN ran ( g e. { h e. ( ( V X. V ) ^m ( 1 ... n ) ) | ( ( F ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( F ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( F ` ( 2nd ` ( h ` i ) ) ) = ( F ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsumg ( E o. g ) ) ) , RR* , < ) ) )
125	122, 124	eqtr4d 2659	. . . . . . 7 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( x e. ran f , y e. ran f |-> inf ( U_ n e. NN ran ( g e. { h e. ( ( v X. v ) ^m ( 1 ... n ) ) | ( ( f ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( f ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( f ` ( 2nd ` ( h ` i ) ) ) = ( f ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsumg ( ( dist ` r ) o. g ) ) ) , RR* , < ) ) = D )
126	125	opeq2d 4409	. . . . . 6 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> <. ( dist ` ndx ) , ( x e. ran f , y e. ran f |-> inf ( U_ n e. NN ran ( g e. { h e. ( ( v X. v ) ^m ( 1 ... n ) ) | ( ( f ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( f ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( f ` ( 2nd ` ( h ` i ) ) ) = ( f ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsumg ( ( dist ` r ) o. g ) ) ) , RR* , < ) ) >. = <. ( dist ` ndx ) , D >. )
127	90, 100, 126	tpeq123d 4283	. . . . 5 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> { <. (TopSet ` ndx ) , ( ( TopOpen ` r ) qTop f ) >. , <. ( le ` ndx ) , ( ( f o. ( le ` r ) ) o. `' f ) >. , <. ( dist ` ndx ) , ( x e. ran f , y e. ran f |-> inf ( U_ n e. NN ran ( g e. { h e. ( ( v X. v ) ^m ( 1 ... n ) ) | ( ( f ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( f ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( f ` ( 2nd ` ( h ` i ) ) ) = ( f ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsumg ( ( dist ` r ) o. g ) ) ) , RR* , < ) ) >. } = { <. (TopSet ` ndx ) , O >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. } )
128	82, 127	uneq12d 3768	. . . 4 |- ( ( ( ph /\ ( f = F /\ r = R ) ) /\ v = ( Base ` r ) ) -> ( ( { <. ( Base ` ndx ) , ran f >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } >. } u. { <. (Scalar ` ndx ) , (Scalar ` r ) >. , <. ( .s ` ndx ) , U_ q e. v ( p e. ( Base ` (Scalar ` r ) ) , x e. { ( f ` q ) } |-> ( f ` ( p ( .s ` r ) q ) ) ) >. , <. ( .i ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( p ( .i ` r ) q ) >. } >. } ) u. { <. (TopSet ` ndx ) , ( ( TopOpen ` r ) qTop f ) >. , <. ( le ` ndx ) , ( ( f o. ( le ` r ) ) o. `' f ) >. , <. ( dist ` ndx ) , ( x e. ran f , y e. ran f |-> inf ( U_ n e. NN ran ( g e. { h e. ( ( v X. v ) ^m ( 1 ... n ) ) | ( ( f ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( f ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( f ` ( 2nd ` ( h ` i ) ) ) = ( f ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsumg ( ( dist ` r ) o. g ) ) ) , RR* , < ) ) >. } ) = ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .xb >. } u. { <. (Scalar ` ndx ) , G >. , <. ( .s ` ndx ) , .(x) >. , <. ( .i ` ndx ) , I >. } ) u. { <. (TopSet ` ndx ) , O >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. } ) )
129	4, 128	csbied 3560	. . 3 |- ( ( ph /\ ( f = F /\ r = R ) ) -> [_ ( Base ` r ) / v ]_ ( ( { <. ( Base ` ndx ) , ran f >. , <. ( +g ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( +g ` r ) q ) ) >. } >. , <. ( .r ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( f ` ( p ( .r ` r ) q ) ) >. } >. } u. { <. (Scalar ` ndx ) , (Scalar ` r ) >. , <. ( .s ` ndx ) , U_ q e. v ( p e. ( Base ` (Scalar ` r ) ) , x e. { ( f ` q ) } |-> ( f ` ( p ( .s ` r ) q ) ) ) >. , <. ( .i ` ndx ) , U_ p e. v U_ q e. v { <. <. ( f ` p ) , ( f ` q ) >. , ( p ( .i ` r ) q ) >. } >. } ) u. { <. (TopSet ` ndx ) , ( ( TopOpen ` r ) qTop f ) >. , <. ( le ` ndx ) , ( ( f o. ( le ` r ) ) o. `' f ) >. , <. ( dist ` ndx ) , ( x e. ran f , y e. ran f |-> inf ( U_ n e. NN ran ( g e. { h e. ( ( v X. v ) ^m ( 1 ... n ) ) | ( ( f ` ( 1st ` ( h ` 1 ) ) ) = x /\ ( f ` ( 2nd ` ( h ` n ) ) ) = y /\ A. i e. ( 1 ... ( n - 1 ) ) ( f ` ( 2nd ` ( h ` i ) ) ) = ( f ` ( 1st ` ( h ` ( i + 1 ) ) ) ) ) } |-> ( RR*s gsumg ( ( dist ` r ) o. g ) ) ) , RR* , < ) ) >. } ) = ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .xb >. } u. { <. (Scalar ` ndx ) , G >. , <. ( .s ` ndx ) , .(x) >. , <. ( .i ` ndx ) , I >. } ) u. { <. (TopSet ` ndx ) , O >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. } ) )
130		fof 6115	. . . . 5 |- ( F : V -onto-> B -> F : V --> B )
131	7, 130	syl 17	. . . 4 |- ( ph -> F : V --> B )
132		fvex 6201	. . . . 5 |- ( Base ` R ) e. _V
133	16, 132	syl6eqel 2709	. . . 4 |- ( ph -> V e. _V )
134		fex 6490	. . . 4 |- ( ( F : V --> B /\ V e. _V ) -> F e. _V )
135	131, 133, 134	syl2anc 693	. . 3 |- ( ph -> F e. _V )
136		imasval.r	. . . 4 |- ( ph -> R e. Z )
137		elex 3212	. . . 4 |- ( R e. Z -> R e. _V )
138	136, 137	syl 17	. . 3 |- ( ph -> R e. _V )
139		tpex 6957	. . . . . 6 |- { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .xb >. } e. _V
140		tpex 6957	. . . . . 6 |- { <. (Scalar ` ndx ) , G >. , <. ( .s ` ndx ) , .(x) >. , <. ( .i ` ndx ) , I >. } e. _V
141	139, 140	unex 6956	. . . . 5 |- ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .xb >. } u. { <. (Scalar ` ndx ) , G >. , <. ( .s ` ndx ) , .(x) >. , <. ( .i ` ndx ) , I >. } ) e. _V
142		tpex 6957	. . . . 5 |- { <. (TopSet ` ndx ) , O >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. } e. _V
143	141, 142	unex 6956	. . . 4 |- ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .xb >. } u. { <. (Scalar ` ndx ) , G >. , <. ( .s ` ndx ) , .(x) >. , <. ( .i ` ndx ) , I >. } ) u. { <. (TopSet ` ndx ) , O >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. } ) e. _V
144	143	a1i 11	. . 3 |- ( ph -> ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .xb >. } u. { <. (Scalar ` ndx ) , G >. , <. ( .s ` ndx ) , .(x) >. , <. ( .i ` ndx ) , I >. } ) u. { <. (TopSet ` ndx ) , O >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. } ) e. _V )
145	3, 129, 135, 138, 144	ovmpt2d 6788	. 2 |- ( ph -> ( F "s R ) = ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .xb >. } u. { <. (Scalar ` ndx ) , G >. , <. ( .s ` ndx ) , .(x) >. , <. ( .i ` ndx ) , I >. } ) u. { <. (TopSet ` ndx ) , O >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. } ) )
146	1, 145	eqtrd 2656	1 |- ( ph -> U = ( ( { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+b >. , <. ( .r ` ndx ) , .xb >. } u. { <. (Scalar ` ndx ) , G >. , <. ( .s ` ndx ) , .(x) >. , <. ( .i ` ndx ) , I >. } ) u. { <. (TopSet ` ndx ) , O >. , <. ( le ` ndx ) , .<_ >. , <. ( dist ` ndx ) , D >. } ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   [_csb 3533   u. cun 3572   {csn 4177   {ctp 4181   <.cop 4183   U_ciun 4520   |-> cmpt 4729   X. cxp 5112   `'ccnv 5113   ran crn 5115   o. ccom 5118   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1stc1st 7166   2ndc2nd 7167   ^m cmap 7857  infcinf 8347   1c1 9937   + caddc 9939   RR*cxr 10073   < clt 10074   - cmin 10266   NNcn 11020   ...cfz 12326   ndxcnx 15854   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   .icip 15946  TopSetcts 15947   lecple 15948   distcds 15950   TopOpenctopn 16082   gsum_g cgsu 16101   RR*scxrs 16160   qTop cqtop 16163   "_s cimas 16164

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-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-reu 2919  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-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-sup 8348  df-inf 8349  df-imas 16168

This theorem is referenced by:  imasbas  16172  imasds  16173  imasplusg  16177  imasmulr  16178  imassca  16179  imasvsca  16180  imasip  16181  imastset  16182  imasle  16183