Theorem axcaucvglemcau 7982

Description: Lemma for axcaucvg 7984. The result of mapping to N. and R. satisfies the Cauchy condition. (Contributed by Jim Kingdon, 9-Jul-2021.)

Hypotheses

Ref	Expression
axcaucvg.n	|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }
axcaucvg.f	|- ( ph -> F : N --> RR )
axcaucvg.cau	|- ( ph -> A. n e. N A. k e. N ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) )
axcaucvg.g	|- G = ( j e. N. |-> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) )

Assertion

Ref	Expression
axcaucvglemcau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( G ` n ) <R ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( G ` k ) <R ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )

Distinct variable groups:   k, F, n, z, j   k, N, n   z, G   k, l, r, u, n   j, l, u, z   ph, j, k, n   y, l, u   x, y   j, n, z, k

Allowed substitution hints:   ph( x, y, z, u, r, l)   F( x, y, u, r, l)   G( x, y, u, j, k, n, r, l)   N( x, y, z, u, j, r, l)

Proof of Theorem axcaucvglemcau

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrenn 7939	. . . . . . . . . 10 |- ( n <N k -> <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. <RR <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
2	1	adantl 277	. . . . . . . . 9 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. <RR <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
3		breq2 4038	. . . . . . . . . . 11 |- ( b = <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. <RR b <-> <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. <RR <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) )
4		fveq2 5561	. . . . . . . . . . . . . 14 |- ( b = <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( F ` b ) = ( F ` <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) )
5	4	oveq1d 5940	. . . . . . . . . . . . 13 |- ( b = <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( ( F ` b ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) = ( ( F ` <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) )
6	5	breq2d 4046	. . . . . . . . . . . 12 |- ( b = <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <RR ( ( F ` b ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) <-> ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <RR ( ( F ` <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) ) )
7	4	breq1d 4044	. . . . . . . . . . . 12 |- ( b = <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( ( F ` b ) <RR ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) <-> ( F ` <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <RR ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) ) )
8	6, 7	anbi12d 473	. . . . . . . . . . 11 |- ( b = <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <RR ( ( F ` b ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) /\ ( F ` b ) <RR ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) ) <-> ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <RR ( ( F ` <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) /\ ( F ` <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <RR ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) ) ) )
9	3, 8	imbi12d 234	. . . . . . . . . 10 |- ( b = <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. <RR b -> ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <RR ( ( F ` b ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) /\ ( F ` b ) <RR ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) ) ) <-> ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. <RR <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <RR ( ( F ` <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) /\ ( F ` <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <RR ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) ) ) ) )
10		breq1 4037	. . . . . . . . . . . . 13 |- ( a = <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( a <RR b <-> <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. <RR b ) )
11		fveq2 5561	. . . . . . . . . . . . . . 15 |- ( a = <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( F ` a ) = ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) )
12		oveq1 5932	. . . . . . . . . . . . . . . . . 18 |- ( a = <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( a x. r ) = ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) )
13	12	eqeq1d 2205	. . . . . . . . . . . . . . . . 17 |- ( a = <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( ( a x. r ) = 1 <-> ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) )
14	13	riotabidv 5882	. . . . . . . . . . . . . . . 16 |- ( a = <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( iota_ r e. RR ( a x. r ) = 1 ) = ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) )
15	14	oveq2d 5941	. . . . . . . . . . . . . . 15 |- ( a = <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( ( F ` b ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) = ( ( F ` b ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) )
16	11, 15	breq12d 4047	. . . . . . . . . . . . . 14 |- ( a = <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( ( F ` a ) <RR ( ( F ` b ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) <-> ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <RR ( ( F ` b ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) ) )
17	11, 14	oveq12d 5943	. . . . . . . . . . . . . . 15 |- ( a = <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( ( F ` a ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) = ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) )
18	17	breq2d 4046	. . . . . . . . . . . . . 14 |- ( a = <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( ( F ` b ) <RR ( ( F ` a ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) <-> ( F ` b ) <RR ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) ) )
19	16, 18	anbi12d 473	. . . . . . . . . . . . 13 |- ( a = <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( ( ( F ` a ) <RR ( ( F ` b ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) /\ ( F ` b ) <RR ( ( F ` a ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) ) <-> ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <RR ( ( F ` b ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) /\ ( F ` b ) <RR ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) ) ) )
20	10, 19	imbi12d 234	. . . . . . . . . . . 12 |- ( a = <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( ( a <RR b -> ( ( F ` a ) <RR ( ( F ` b ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) /\ ( F ` b ) <RR ( ( F ` a ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) ) ) <-> ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. <RR b -> ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <RR ( ( F ` b ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) /\ ( F ` b ) <RR ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) ) ) ) )
21	20	ralbidv 2497	. . . . . . . . . . 11 |- ( a = <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( A. b e. N ( a <RR b -> ( ( F ` a ) <RR ( ( F ` b ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) /\ ( F ` b ) <RR ( ( F ` a ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) ) ) <-> A. b e. N ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. <RR b -> ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <RR ( ( F ` b ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) /\ ( F ` b ) <RR ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) ) ) ) )
22		axcaucvg.cau	. . . . . . . . . . . . 13 |- ( ph -> A. n e. N A. k e. N ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) )
23		breq1 4037	. . . . . . . . . . . . . . 15 |- ( n = a -> ( n <RR k <-> a <RR k ) )
24		fveq2 5561	. . . . . . . . . . . . . . . . 17 |- ( n = a -> ( F ` n ) = ( F ` a ) )
25		oveq1 5932	. . . . . . . . . . . . . . . . . . . 20 |- ( n = a -> ( n x. r ) = ( a x. r ) )
26	25	eqeq1d 2205	. . . . . . . . . . . . . . . . . . 19 |- ( n = a -> ( ( n x. r ) = 1 <-> ( a x. r ) = 1 ) )
27	26	riotabidv 5882	. . . . . . . . . . . . . . . . . 18 |- ( n = a -> ( iota_ r e. RR ( n x. r ) = 1 ) = ( iota_ r e. RR ( a x. r ) = 1 ) )
28	27	oveq2d 5941	. . . . . . . . . . . . . . . . 17 |- ( n = a -> ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) = ( ( F ` k ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) )
29	24, 28	breq12d 4047	. . . . . . . . . . . . . . . 16 |- ( n = a -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) <-> ( F ` a ) <RR ( ( F ` k ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) ) )
30	24, 27	oveq12d 5943	. . . . . . . . . . . . . . . . 17 |- ( n = a -> ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) = ( ( F ` a ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) )
31	30	breq2d 4046	. . . . . . . . . . . . . . . 16 |- ( n = a -> ( ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) <-> ( F ` k ) <RR ( ( F ` a ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) ) )
32	29, 31	anbi12d 473	. . . . . . . . . . . . . . 15 |- ( n = a -> ( ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) <-> ( ( F ` a ) <RR ( ( F ` k ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` a ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) ) ) )
33	23, 32	imbi12d 234	. . . . . . . . . . . . . 14 |- ( n = a -> ( ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) <-> ( a <RR k -> ( ( F ` a ) <RR ( ( F ` k ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` a ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) ) ) ) )
34		breq2 4038	. . . . . . . . . . . . . . 15 |- ( k = b -> ( a <RR k <-> a <RR b ) )
35		fveq2 5561	. . . . . . . . . . . . . . . . . 18 |- ( k = b -> ( F ` k ) = ( F ` b ) )
36	35	oveq1d 5940	. . . . . . . . . . . . . . . . 17 |- ( k = b -> ( ( F ` k ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) = ( ( F ` b ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) )
37	36	breq2d 4046	. . . . . . . . . . . . . . . 16 |- ( k = b -> ( ( F ` a ) <RR ( ( F ` k ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) <-> ( F ` a ) <RR ( ( F ` b ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) ) )
38	35	breq1d 4044	. . . . . . . . . . . . . . . 16 |- ( k = b -> ( ( F ` k ) <RR ( ( F ` a ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) <-> ( F ` b ) <RR ( ( F ` a ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) ) )
39	37, 38	anbi12d 473	. . . . . . . . . . . . . . 15 |- ( k = b -> ( ( ( F ` a ) <RR ( ( F ` k ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` a ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) ) <-> ( ( F ` a ) <RR ( ( F ` b ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) /\ ( F ` b ) <RR ( ( F ` a ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) ) ) )
40	34, 39	imbi12d 234	. . . . . . . . . . . . . 14 |- ( k = b -> ( ( a <RR k -> ( ( F ` a ) <RR ( ( F ` k ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` a ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) ) ) <-> ( a <RR b -> ( ( F ` a ) <RR ( ( F ` b ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) /\ ( F ` b ) <RR ( ( F ` a ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) ) ) ) )
41	33, 40	cbvral2v 2742	. . . . . . . . . . . . 13 |- ( A. n e. N A. k e. N ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) <-> A. a e. N A. b e. N ( a <RR b -> ( ( F ` a ) <RR ( ( F ` b ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) /\ ( F ` b ) <RR ( ( F ` a ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) ) ) )
42	22, 41	sylib 122	. . . . . . . . . . . 12 |- ( ph -> A. a e. N A. b e. N ( a <RR b -> ( ( F ` a ) <RR ( ( F ` b ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) /\ ( F ` b ) <RR ( ( F ` a ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) ) ) )
43	42	ad3antrrr 492	. . . . . . . . . . 11 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> A. a e. N A. b e. N ( a <RR b -> ( ( F ` a ) <RR ( ( F ` b ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) /\ ( F ` b ) <RR ( ( F ` a ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) ) ) )
44		pitonn 7932	. . . . . . . . . . . . 13 |- ( n e. N. -> <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } )
45		axcaucvg.n	. . . . . . . . . . . . 13 |- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }
46	44, 45	eleqtrrdi 2290	. . . . . . . . . . . 12 |- ( n e. N. -> <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. N )
47	46	ad3antlr 493	. . . . . . . . . . 11 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. N )
48	21, 43, 47	rspcdva 2873	. . . . . . . . . 10 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> A. b e. N ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. <RR b -> ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <RR ( ( F ` b ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) /\ ( F ` b ) <RR ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) ) ) )
49		pitonn 7932	. . . . . . . . . . . 12 |- ( k e. N. -> <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } )
50	49, 45	eleqtrrdi 2290	. . . . . . . . . . 11 |- ( k e. N. -> <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. N )
51	50	ad2antlr 489	. . . . . . . . . 10 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. N )
52	9, 48, 51	rspcdva 2873	. . . . . . . . 9 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. <RR <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. -> ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <RR ( ( F ` <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) /\ ( F ` <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <RR ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) ) ) )
53	2, 52	mpd 13	. . . . . . . 8 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <RR ( ( F ` <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) /\ ( F ` <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <RR ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) ) )
54	53	simpld 112	. . . . . . 7 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <RR ( ( F ` <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) )
55		axcaucvg.f	. . . . . . . . 9 |- ( ph -> F : N --> RR )
56		axcaucvg.g	. . . . . . . . 9 |- G = ( j e. N. |-> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) )
57	45, 55, 22, 56	axcaucvglemval 7981	. . . . . . . 8 |- ( ( ph /\ n e. N. ) -> ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. ( G ` n ) , 0R >. )
58	57	ad2antrr 488	. . . . . . 7 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. ( G ` n ) , 0R >. )
59	45, 55, 22, 56	axcaucvglemval 7981	. . . . . . . . . . 11 |- ( ( ph /\ k e. N. ) -> ( F ` <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. ( G ` k ) , 0R >. )
60	59	adantlr 477	. . . . . . . . . 10 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( F ` <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. ( G ` k ) , 0R >. )
61	60	adantr 276	. . . . . . . . 9 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( F ` <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. ( G ` k ) , 0R >. )
62		recriota 7974	. . . . . . . . . 10 |- ( n e. N. -> ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) = <. [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
63	62	ad3antlr 493	. . . . . . . . 9 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) = <. [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
64	61, 63	oveq12d 5943	. . . . . . . 8 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( ( F ` <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) = ( <. ( G ` k ) , 0R >. + <. [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) )
65	45, 55, 22, 56	axcaucvglemf 7980	. . . . . . . . . . 11 |- ( ph -> G : N. --> R. )
66	65	ad3antrrr 492	. . . . . . . . . 10 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> G : N. --> R. )
67		simplr 528	. . . . . . . . . 10 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> k e. N. )
68	66, 67	ffvelcdmd 5701	. . . . . . . . 9 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( G ` k ) e. R. )
69		recnnpr 7632	. . . . . . . . . . 11 |- ( n e. N. -> <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. e. P. )
70		prsrcl 7868	. . . . . . . . . . 11 |- ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. e. P. -> [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R e. R. )
71	69, 70	syl 14	. . . . . . . . . 10 |- ( n e. N. -> [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R e. R. )
72	71	ad3antlr 493	. . . . . . . . 9 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R e. R. )
73		addresr 7921	. . . . . . . . 9 |- ( ( ( G ` k ) e. R. /\ [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R e. R. ) -> ( <. ( G ` k ) , 0R >. + <. [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) , 0R >. )
74	68, 72, 73	syl2anc 411	. . . . . . . 8 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( <. ( G ` k ) , 0R >. + <. [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) , 0R >. )
75	64, 74	eqtrd 2229	. . . . . . 7 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( ( F ` <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) = <. ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) , 0R >. )
76	54, 58, 75	3brtr3d 4065	. . . . . 6 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> <. ( G ` n ) , 0R >. <RR <. ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) , 0R >. )
77		ltresr 7923	. . . . . 6 |- ( <. ( G ` n ) , 0R >. <RR <. ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) , 0R >. <-> ( G ` n ) <R ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
78	76, 77	sylib 122	. . . . 5 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( G ` n ) <R ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
79	53	simprd 114	. . . . . . 7 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( F ` <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <RR ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) )
80	58, 63	oveq12d 5943	. . . . . . . 8 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) = ( <. ( G ` n ) , 0R >. + <. [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) )
81		simpllr 534	. . . . . . . . . 10 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> n e. N. )
82	66, 81	ffvelcdmd 5701	. . . . . . . . 9 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( G ` n ) e. R. )
83		addresr 7921	. . . . . . . . 9 |- ( ( ( G ` n ) e. R. /\ [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R e. R. ) -> ( <. ( G ` n ) , 0R >. + <. [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) , 0R >. )
84	82, 72, 83	syl2anc 411	. . . . . . . 8 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( <. ( G ` n ) , 0R >. + <. [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) , 0R >. )
85	80, 84	eqtrd 2229	. . . . . . 7 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( ( F ` <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) + ( iota_ r e. RR ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. x. r ) = 1 ) ) = <. ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) , 0R >. )
86	79, 61, 85	3brtr3d 4065	. . . . . 6 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> <. ( G ` k ) , 0R >. <RR <. ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) , 0R >. )
87		ltresr 7923	. . . . . 6 |- ( <. ( G ` k ) , 0R >. <RR <. ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) , 0R >. <-> ( G ` k ) <R ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
88	86, 87	sylib 122	. . . . 5 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( G ` k ) <R ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) )
89	78, 88	jca 306	. . . 4 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( ( G ` n ) <R ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( G ` k ) <R ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) )
90	89	ex 115	. . 3 |- ( ( ( ph /\ n e. N. ) /\ k e. N. ) -> ( n <N k -> ( ( G ` n ) <R ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( G ` k ) <R ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )
91	90	ralrimiva 2570	. 2 |- ( ( ph /\ n e. N. ) -> A. k e. N. ( n <N k -> ( ( G ` n ) <R ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( G ` k ) <R ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )
92	91	ralrimiva 2570	1 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( G ` n ) <R ( ( G ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( G ` k ) <R ( ( G ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   {cab 2182   A.wral 2475   <.cop 3626   |^|cint 3875   class class class wbr 4034   |-> cmpt 4095   -->wf 5255   ` cfv 5259   iota_crio 5879  (class class class)co 5925   1oc1o 6476   [cec 6599   N.cnpi 7356   <N clti 7359   ~Q ceq 7363   *Qcrq 7368   <Q cltq 7369   P.cnp 7375   1Pc1p 7376   +P. cpp 7377   ~R cer 7380   R.cnr 7381   0Rc0r 7382   +R cplr 7385   <R cltr 7387   RRcr 7895   1c1 7897   + caddc 7899   <RR cltrr 7900   x. cmul 7901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-1o 6483  df-2o 6484  df-oadd 6487  df-omul 6488  df-er 6601  df-ec 6603  df-qs 6607  df-ni 7388  df-pli 7389  df-mi 7390  df-lti 7391  df-plpq 7428  df-mpq 7429  df-enq 7431  df-nqqs 7432  df-plqqs 7433  df-mqqs 7434  df-1nqqs 7435  df-rq 7436  df-ltnqqs 7437  df-enq0 7508  df-nq0 7509  df-0nq0 7510  df-plq0 7511  df-mq0 7512  df-inp 7550  df-i1p 7551  df-iplp 7552  df-imp 7553  df-iltp 7554  df-enr 7810  df-nr 7811  df-plr 7812  df-mr 7813  df-ltr 7814  df-0r 7815  df-1r 7816  df-m1r 7817  df-c 7902  df-0 7903  df-1 7904  df-r 7906  df-add 7907  df-mul 7908  df-lt 7909

This theorem is referenced by:  axcaucvglemres  7983