Theorem axcaucvglemcau 7965

Description: Lemma for axcaucvg 7967. 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 7922	. . . . . . . . . 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 4037	. . . . . . . . . . 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 5558	. . . . . . . . . . . . . 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 5937	. . . . . . . . . . . . 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 4045	. . . . . . . . . . . 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 4043	. . . . . . . . . . . 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 4036	. . . . . . . . . . . . 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 5558	. . . . . . . . . . . . . . 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 5929	. . . . . . . . . . . . . . . . . 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 5879	. . . . . . . . . . . . . . . 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 5938	. . . . . . . . . . . . . . 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 4046	. . . . . . . . . . . . . 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 5940	. . . . . . . . . . . . . . 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 4045	. . . . . . . . . . . . . 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 4036	. . . . . . . . . . . . . . 15 |- ( n = a -> ( n <RR k <-> a <RR k ) )
24		fveq2 5558	. . . . . . . . . . . . . . . . 17 |- ( n = a -> ( F ` n ) = ( F ` a ) )
25		oveq1 5929	. . . . . . . . . . . . . . . . . . . 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 5879	. . . . . . . . . . . . . . . . . 18 |- ( n = a -> ( iota_ r e. RR ( n x. r ) = 1 ) = ( iota_ r e. RR ( a x. r ) = 1 ) )
28	27	oveq2d 5938	. . . . . . . . . . . . . . . . 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 4046	. . . . . . . . . . . . . . . 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 5940	. . . . . . . . . . . . . . . . 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 4045	. . . . . . . . . . . . . . . 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 4037	. . . . . . . . . . . . . . 15 |- ( k = b -> ( a <RR k <-> a <RR b ) )
35		fveq2 5558	. . . . . . . . . . . . . . . . . 18 |- ( k = b -> ( F ` k ) = ( F ` b ) )
36	35	oveq1d 5937	. . . . . . . . . . . . . . . . 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 4045	. . . . . . . . . . . . . . . 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 4043	. . . . . . . . . . . . . . . 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 7915	. . . . . . . . . . . . 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 7915	. . . . . . . . . . . 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 7964	. . . . . . . 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 7964	. . . . . . . . . . 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 7957	. . . . . . . . . 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 5940	. . . . . . . 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 7963	. . . . . . . . . . 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 5698	. . . . . . . . 9 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( G ` k ) e. R. )
69		recnnpr 7615	. . . . . . . . . . 11 |- ( n e. N. -> <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. e. P. )
70		prsrcl 7851	. . . . . . . . . . 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 7904	. . . . . . . . 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 4064	. . . . . 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 7906	. . . . . 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 5940	. . . . . . . 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 5698	. . . . . . . . 9 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( G ` n ) e. R. )
83		addresr 7904	. . . . . . . . 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 4064	. . . . . 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 7906	. . . . . 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 3625   |^|cint 3874   class class class wbr 4033   |-> cmpt 4094   -->wf 5254   ` cfv 5258   iota_crio 5876  (class class class)co 5922   1oc1o 6467   [cec 6590   N.cnpi 7339   <N clti 7342   ~Q ceq 7346   *Qcrq 7351   <Q cltq 7352   P.cnp 7358   1Pc1p 7359   +P. cpp 7360   ~R cer 7363   R.cnr 7364   0Rc0r 7365   +R cplr 7368   <R cltr 7370   RRcr 7878   1c1 7880   + caddc 7882   <RR cltrr 7883   x. cmul 7884

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 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624

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 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-eprel 4324  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-1o 6474  df-2o 6475  df-oadd 6478  df-omul 6479  df-er 6592  df-ec 6594  df-qs 6598  df-ni 7371  df-pli 7372  df-mi 7373  df-lti 7374  df-plpq 7411  df-mpq 7412  df-enq 7414  df-nqqs 7415  df-plqqs 7416  df-mqqs 7417  df-1nqqs 7418  df-rq 7419  df-ltnqqs 7420  df-enq0 7491  df-nq0 7492  df-0nq0 7493  df-plq0 7494  df-mq0 7495  df-inp 7533  df-i1p 7534  df-iplp 7535  df-imp 7536  df-iltp 7537  df-enr 7793  df-nr 7794  df-plr 7795  df-mr 7796  df-ltr 7797  df-0r 7798  df-1r 7799  df-m1r 7800  df-c 7885  df-0 7886  df-1 7887  df-r 7889  df-add 7890  df-mul 7891  df-lt 7892

This theorem is referenced by:  axcaucvglemres  7966