Theorem axcaucvglemcau 7064

Description: Lemma for axcaucvg 7066. 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 7023	. . . . . . . . . 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 271	. . . . . . . . 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		pitonn 7016	. . . . . . . . . . . 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 ) } )
4		axcaucvg.n	. . . . . . . . . . . 12 |- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }
5	3, 4	syl6eleqr 2172	. . . . . . . . . . 11 |- ( k e. N. -> <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. N )
6	5	ad2antlr 472	. . . . . . . . . 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 )
7		pitonn 7016	. . . . . . . . . . . . 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 ) } )
8	7, 4	syl6eleqr 2172	. . . . . . . . . . . 12 |- ( n e. N. -> <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. N )
9	8	ad3antlr 476	. . . . . . . . . . 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 )
10		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 ) ) ) ) )
11		breq1 3788	. . . . . . . . . . . . . . 15 |- ( n = a -> ( n <RR k <-> a <RR k ) )
12		fveq2 5198	. . . . . . . . . . . . . . . . 17 |- ( n = a -> ( F ` n ) = ( F ` a ) )
13		oveq1 5539	. . . . . . . . . . . . . . . . . . . 20 |- ( n = a -> ( n x. r ) = ( a x. r ) )
14	13	eqeq1d 2089	. . . . . . . . . . . . . . . . . . 19 |- ( n = a -> ( ( n x. r ) = 1 <-> ( a x. r ) = 1 ) )
15	14	riotabidv 5490	. . . . . . . . . . . . . . . . . 18 |- ( n = a -> ( iota_ r e. RR ( n x. r ) = 1 ) = ( iota_ r e. RR ( a x. r ) = 1 ) )
16	15	oveq2d 5548	. . . . . . . . . . . . . . . . 17 |- ( n = a -> ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) = ( ( F ` k ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) )
17	12, 16	breq12d 3798	. . . . . . . . . . . . . . . 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 ) ) ) )
18	12, 15	oveq12d 5550	. . . . . . . . . . . . . . . . 17 |- ( n = a -> ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) = ( ( F ` a ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) )
19	18	breq2d 3797	. . . . . . . . . . . . . . . 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 ) ) ) )
20	17, 19	anbi12d 456	. . . . . . . . . . . . . . 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 ) ) ) ) )
21	11, 20	imbi12d 232	. . . . . . . . . . . . . 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 ) ) ) ) ) )
22		breq2 3789	. . . . . . . . . . . . . . 15 |- ( k = b -> ( a <RR k <-> a <RR b ) )
23		fveq2 5198	. . . . . . . . . . . . . . . . . 18 |- ( k = b -> ( F ` k ) = ( F ` b ) )
24	23	oveq1d 5547	. . . . . . . . . . . . . . . . 17 |- ( k = b -> ( ( F ` k ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) = ( ( F ` b ) + ( iota_ r e. RR ( a x. r ) = 1 ) ) )
25	24	breq2d 3797	. . . . . . . . . . . . . . . 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 ) ) ) )
26	23	breq1d 3795	. . . . . . . . . . . . . . . 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 ) ) ) )
27	25, 26	anbi12d 456	. . . . . . . . . . . . . . 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 ) ) ) ) )
28	22, 27	imbi12d 232	. . . . . . . . . . . . . 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 ) ) ) ) ) )
29	21, 28	cbvral2v 2585	. . . . . . . . . . . . 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 ) ) ) ) )
30	10, 29	sylib 120	. . . . . . . . . . . 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 ) ) ) ) )
31	30	ad3antrrr 475	. . . . . . . . . . 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 ) ) ) ) )
32		breq1 3788	. . . . . . . . . . . . . 14 |- ( 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 ) )
33		fveq2 5198	. . . . . . . . . . . . . . . 16 |- ( 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 >. ) )
34		oveq1 5539	. . . . . . . . . . . . . . . . . . 19 |- ( 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 ) )
35	34	eqeq1d 2089	. . . . . . . . . . . . . . . . . 18 |- ( 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 ) )
36	35	riotabidv 5490	. . . . . . . . . . . . . . . . 17 |- ( 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 ) )
37	36	oveq2d 5548	. . . . . . . . . . . . . . . 16 |- ( 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 ) ) )
38	33, 37	breq12d 3798	. . . . . . . . . . . . . . 15 |- ( 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 ) ) ) )
39	33, 36	oveq12d 5550	. . . . . . . . . . . . . . . 16 |- ( 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 ) ) )
40	39	breq2d 3797	. . . . . . . . . . . . . . 15 |- ( 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 ) ) ) )
41	38, 40	anbi12d 456	. . . . . . . . . . . . . 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 ` 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 ) ) ) ) )
42	32, 41	imbi12d 232	. . . . . . . . . . . . 13 |- ( 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 ) ) ) ) ) )
43	42	ralbidv 2368	. . . . . . . . . . . 12 |- ( 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 ) ) ) ) ) )
44	43	rspcva 2699	. . . . . . . . . . 11 |- ( ( <. [ <. ( <. { l | l <Q [ <. n , 1o >. ] ~Q } , { u | [ <. n , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. N /\ 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 ) ) ) ) ) -> 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 ) ) ) ) )
45	9, 31, 44	syl2anc 403	. . . . . . . . . 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 ) ) ) ) )
46		breq2 3789	. . . . . . . . . . . 12 |- ( 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 >. ) )
47		fveq2 5198	. . . . . . . . . . . . . . 15 |- ( 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 >. ) )
48	47	oveq1d 5547	. . . . . . . . . . . . . 14 |- ( 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 ) ) )
49	48	breq2d 3797	. . . . . . . . . . . . 13 |- ( 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 ) ) ) )
50	47	breq1d 3795	. . . . . . . . . . . . 13 |- ( 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 ) ) ) )
51	49, 50	anbi12d 456	. . . . . . . . . . . 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 ` 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 ) ) ) ) )
52	46, 51	imbi12d 232	. . . . . . . . . . 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 -> ( ( 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 ) ) ) ) ) )
53	52	rspcva 2699	. . . . . . . . . 10 |- ( ( <. [ <. ( <. { l | l <Q [ <. k , 1o >. ] ~Q } , { u | [ <. k , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. N /\ 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 ) ) ) ) ) -> ( <. [ <. ( <. { 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 ) ) ) ) )
54	6, 45, 53	syl2anc 403	. . . . . . . . 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 ) ) ) ) )
55	2, 54	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 ) ) ) )
56	55	simpld 110	. . . . . . 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 ) ) )
57		axcaucvg.f	. . . . . . . . 9 |- ( ph -> F : N --> RR )
58		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 >. ) )
59	4, 57, 10, 58	axcaucvglemval 7063	. . . . . . . 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 >. )
60	59	ad2antrr 471	. . . . . . 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 >. )
61	4, 57, 10, 58	axcaucvglemval 7063	. . . . . . . . . . 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 >. )
62	61	adantlr 460	. . . . . . . . . 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 >. )
63	62	adantr 270	. . . . . . . . 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 >. )
64		recriota 7056	. . . . . . . . . 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 >. )
65	64	ad3antlr 476	. . . . . . . . 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 >. )
66	63, 65	oveq12d 5550	. . . . . . . 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 >. ) )
67	4, 57, 10, 58	axcaucvglemf 7062	. . . . . . . . . . 11 |- ( ph -> G : N. --> R. )
68	67	ad3antrrr 475	. . . . . . . . . 10 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> G : N. --> R. )
69		simplr 496	. . . . . . . . . 10 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> k e. N. )
70	68, 69	ffvelrnd 5324	. . . . . . . . 9 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( G ` k ) e. R. )
71		recnnpr 6738	. . . . . . . . . . 11 |- ( n e. N. -> <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. e. P. )
72		prsrcl 6960	. . . . . . . . . . 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. )
73	71, 72	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. )
74	73	ad3antlr 476	. . . . . . . . 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. )
75		addresr 7005	. . . . . . . . 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 >. )
76	70, 74, 75	syl2anc 403	. . . . . . . 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 >. )
77	66, 76	eqtrd 2113	. . . . . . 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 >. )
78	56, 60, 77	3brtr3d 3814	. . . . . 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 >. )
79		ltresr 7007	. . . . . 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 ) )
80	78, 79	sylib 120	. . . . 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 ) )
81	55	simprd 112	. . . . . . 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 ) ) )
82	60, 65	oveq12d 5550	. . . . . . . 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 >. ) )
83		simpllr 500	. . . . . . . . . 10 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> n e. N. )
84	68, 83	ffvelrnd 5324	. . . . . . . . 9 |- ( ( ( ( ph /\ n e. N. ) /\ k e. N. ) /\ n <N k ) -> ( G ` n ) e. R. )
85		addresr 7005	. . . . . . . . 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 >. )
86	84, 74, 85	syl2anc 403	. . . . . . . 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 >. )
87	82, 86	eqtrd 2113	. . . . . . 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 >. )
88	81, 63, 87	3brtr3d 3814	. . . . . 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 >. )
89		ltresr 7007	. . . . . 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 ) )
90	88, 89	sylib 120	. . . . 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 ) )
91	80, 90	jca 300	. . . 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 ) ) )
92	91	ex 113	. . 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 ) ) ) )
93	92	ralrimiva 2434	. 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 ) ) ) )
94	93	ralrimiva 2434	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 102   = wceq 1284   e. wcel 1433   {cab 2067   A.wral 2348   <.cop 3401   |^|cint 3636   class class class wbr 3785   |-> cmpt 3839   -->wf 4918   ` cfv 4922   iota_crio 5487  (class class class)co 5532   1oc1o 6017   [cec 6127   N.cnpi 6462   <N clti 6465   ~Q ceq 6469   *Qcrq 6474   <Q cltq 6475   P.cnp 6481   1Pc1p 6482   +P. cpp 6483   ~R cer 6486   R.cnr 6487   0Rc0r 6488   +R cplr 6491   <R cltr 6493   RRcr 6980   1c1 6982   + caddc 6984   <RR cltrr 6985   x. cmul 6986

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-eprel 4044  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-irdg 5980  df-1o 6024  df-2o 6025  df-oadd 6028  df-omul 6029  df-er 6129  df-ec 6131  df-qs 6135  df-ni 6494  df-pli 6495  df-mi 6496  df-lti 6497  df-plpq 6534  df-mpq 6535  df-enq 6537  df-nqqs 6538  df-plqqs 6539  df-mqqs 6540  df-1nqqs 6541  df-rq 6542  df-ltnqqs 6543  df-enq0 6614  df-nq0 6615  df-0nq0 6616  df-plq0 6617  df-mq0 6618  df-inp 6656  df-i1p 6657  df-iplp 6658  df-imp 6659  df-iltp 6660  df-enr 6903  df-nr 6904  df-plr 6905  df-mr 6906  df-ltr 6907  df-0r 6908  df-1r 6909  df-m1r 6910  df-c 6987  df-0 6988  df-1 6989  df-r 6991  df-add 6992  df-mul 6993  df-lt 6994

This theorem is referenced by:  axcaucvglemres  7065