Theorem caucvgprprlemcbv 6877

Description: Lemma for caucvgprpr 6902. Change bound variables in Cauchy condition. (Contributed by Jim Kingdon, 12-Feb-2021.)

Hypotheses

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

Assertion

Ref	Expression
caucvgprprlemcbv	|- ( ph -> A. a e. N. A. b e. N. ( a <N b -> ( ( F ` a ) <P ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` b ) <P ( ( F ` a ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) ) ) )

Distinct variable groups:   F, a, b, k   n, F, a, k   a, l, b, k   u, a, b, k   n, l   u, n

Allowed substitution hints:   ph( u, k, n, a, b, l)   F( u, l)

Proof of Theorem caucvgprprlemcbv

Step	Hyp	Ref	Expression
1		caucvgprpr.cau	. 2 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )
2		breq1 3788	. . . 4 |- ( n = a -> ( n <N k <-> a <N k ) )
3		fveq2 5198	. . . . . 6 |- ( n = a -> ( F ` n ) = ( F ` a ) )
4		opeq1 3570	. . . . . . . . . . . 12 |- ( n = a -> <. n , 1o >. = <. a , 1o >. )
5	4	eceq1d 6165	. . . . . . . . . . 11 |- ( n = a -> [ <. n , 1o >. ] ~Q = [ <. a , 1o >. ] ~Q )
6	5	fveq2d 5202	. . . . . . . . . 10 |- ( n = a -> ( *Q ` [ <. n , 1o >. ] ~Q ) = ( *Q ` [ <. a , 1o >. ] ~Q ) )
7	6	breq2d 3797	. . . . . . . . 9 |- ( n = a -> ( l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) <-> l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
8	7	abbidv 2196	. . . . . . . 8 |- ( n = a -> { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } = { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } )
9	6	breq1d 3795	. . . . . . . . 9 |- ( n = a -> ( ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u <-> ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u ) )
10	9	abbidv 2196	. . . . . . . 8 |- ( n = a -> { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } = { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } )
11	8, 10	opeq12d 3578	. . . . . . 7 |- ( n = a -> <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. = <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. )
12	11	oveq2d 5548	. . . . . 6 |- ( n = a -> ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) = ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) )
13	3, 12	breq12d 3798	. . . . 5 |- ( n = a -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) <-> ( F ` a ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) ) )
14	3, 11	oveq12d 5550	. . . . . 6 |- ( n = a -> ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) = ( ( F ` a ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) )
15	14	breq2d 3797	. . . . 5 |- ( n = a -> ( ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) <-> ( F ` k ) <P ( ( F ` a ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) ) )
16	13, 15	anbi12d 456	. . . 4 |- ( n = a -> ( ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) <-> ( ( F ` a ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` a ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) ) ) )
17	2, 16	imbi12d 232	. . 3 |- ( n = a -> ( ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) <-> ( a <N k -> ( ( F ` a ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` a ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) ) ) ) )
18		breq2 3789	. . . 4 |- ( k = b -> ( a <N k <-> a <N b ) )
19		fveq2 5198	. . . . . . 7 |- ( k = b -> ( F ` k ) = ( F ` b ) )
20	19	oveq1d 5547	. . . . . 6 |- ( k = b -> ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) = ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) )
21	20	breq2d 3797	. . . . 5 |- ( k = b -> ( ( F ` a ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) <-> ( F ` a ) <P ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) ) )
22	19	breq1d 3795	. . . . 5 |- ( k = b -> ( ( F ` k ) <P ( ( F ` a ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) <-> ( F ` b ) <P ( ( F ` a ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) ) )
23	21, 22	anbi12d 456	. . . 4 |- ( k = b -> ( ( ( F ` a ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` a ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) ) <-> ( ( F ` a ) <P ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` b ) <P ( ( F ` a ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) ) ) )
24	18, 23	imbi12d 232	. . 3 |- ( k = b -> ( ( a <N k -> ( ( F ` a ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` a ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) ) ) <-> ( a <N b -> ( ( F ` a ) <P ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` b ) <P ( ( F ` a ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) ) ) ) )
25	17, 24	cbvral2v 2585	. 2 |- ( A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) <-> A. a e. N. A. b e. N. ( a <N b -> ( ( F ` a ) <P ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` b ) <P ( ( F ` a ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) ) ) )
26	1, 25	sylib 120	1 |- ( ph -> A. a e. N. A. b e. N. ( a <N b -> ( ( F ` a ) <P ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` b ) <P ( ( F ` a ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) ) ) )

Syntax hints:   -> wi 4   /\ wa 102   {cab 2067   A.wral 2348   <.cop 3401   class class class wbr 3785   -->wf 4918   ` cfv 4922  (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   +P. cpp 6483   <P cltp 6485

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-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-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fv 4930  df-ov 5535  df-ec 6131

This theorem is referenced by:  caucvgprprlemval  6878