Theorem caucvgsrlemgt1 8075

Description: Lemma for caucvgsr 8082. A Cauchy sequence whose terms are greater than one converges. (Contributed by Jim Kingdon, 22-Jun-2021.)

Hypotheses

Ref	Expression
caucvgsr.f	|- ( ph -> F : N. --> R. )
caucvgsr.cau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )
caucvgsrlemgt1.gt1	|- ( ph -> A. m e. N. 1R <R ( F ` m ) )

Assertion

Ref	Expression
caucvgsrlemgt1	|- ( ph -> E. y e. R. A. x e. R. ( 0R <R x -> E. j e. N. A. i e. N. ( j <N i -> ( ( F ` i ) <R ( y +R x ) /\ y <R ( ( F ` i ) +R x ) ) ) ) )

Distinct variable groups:   j, F, k, l, u   i, F, x, j, k   m, F, n, k   n, l, u   y, F, i, j, x   ph, j, k, x   ph, n   k, m, n

Allowed substitution hints:   ph( y, u, i, m, l)

Proof of Theorem caucvgsrlemgt1

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

Step	Hyp	Ref	Expression
1		caucvgsr.f	. . . 4 |- ( ph -> F : N. --> R. )
2		caucvgsr.cau	. . . 4 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )
3		caucvgsrlemgt1.gt1	. . . 4 |- ( ph -> A. m e. N. 1R <R ( F ` m ) )
4		eqid 2231	. . . 4 |- ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) = ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) )
5	1, 2, 3, 4	caucvgsrlemf 8072	. . 3 |- ( ph -> ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) : N. --> P. )
6	1, 2, 3, 4	caucvgsrlemcau 8073	. . 3 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` n ) <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )
7	1, 2, 3, 4	caucvgsrlembound 8074	. . 3 |- ( ph -> A. m e. N. 1P <P ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` m ) )
8	5, 6, 7	caucvgprpr 7992	. 2 |- ( ph -> E. a e. P. A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) )
9		prsrcl 8064	. . . 4 |- ( a e. P. -> [ <. ( a +P. 1P ) , 1P >. ] ~R e. R. )
10	9	ad2antrl 490	. . 3 |- ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) -> [ <. ( a +P. 1P ) , 1P >. ] ~R e. R. )
11		oveq2 6036	. . . . . . . . . . . 12 |- ( b = ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) -> ( a +P. b ) = ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) )
12	11	breq2d 4105	. . . . . . . . . . 11 |- ( b = ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) <-> ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) ) )
13		oveq2 6036	. . . . . . . . . . . 12 |- ( b = ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) = ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) )
14	13	breq2d 4105	. . . . . . . . . . 11 |- ( b = ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) -> ( a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) <-> a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) ) )
15	12, 14	anbi12d 473	. . . . . . . . . 10 |- ( b = ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) -> ( ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) <-> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) ) ) )
16	15	imbi2d 230	. . . . . . . . 9 |- ( b = ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) -> ( ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) <-> ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) ) ) ) )
17	16	rexralbidv 2559	. . . . . . . 8 |- ( b = ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) -> ( E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) <-> E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) ) ) ) )
18		simplrr 538	. . . . . . . . 9 |- ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) -> A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) )
19	18	adantr 276	. . . . . . . 8 |- ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) -> A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) )
20		srpospr 8063	. . . . . . . . . 10 |- ( ( x e. R. /\ 0R <R x ) -> E! c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x )
21		riotacl 5997	. . . . . . . . . 10 |- ( E! c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x -> ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) e. P. )
22	20, 21	syl 14	. . . . . . . . 9 |- ( ( x e. R. /\ 0R <R x ) -> ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) e. P. )
23	22	adantll 476	. . . . . . . 8 |- ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) -> ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) e. P. )
24	17, 19, 23	rspcdva 2916	. . . . . . 7 |- ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) -> E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) ) ) )
25		nfv 1577	. . . . . . . . . . 11 |- F/ j ph
26		nfv 1577	. . . . . . . . . . . 12 |- F/ j a e. P.
27		nfcv 2375	. . . . . . . . . . . . 13 |- F/_ j P.
28		nfre1 2576	. . . . . . . . . . . . 13 |- F/ j E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) )
29	27, 28	nfralya 2573	. . . . . . . . . . . 12 |- F/ j A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) )
30	26, 29	nfan 1614	. . . . . . . . . . 11 |- F/ j ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) )
31	25, 30	nfan 1614	. . . . . . . . . 10 |- F/ j ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) )
32		nfv 1577	. . . . . . . . . 10 |- F/ j x e. R.
33	31, 32	nfan 1614	. . . . . . . . 9 |- F/ j ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. )
34		nfv 1577	. . . . . . . . 9 |- F/ j 0R <R x
35	33, 34	nfan 1614	. . . . . . . 8 |- F/ j ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x )
36		nfv 1577	. . . . . . . . . . . 12 |- F/ k ph
37		nfv 1577	. . . . . . . . . . . . 13 |- F/ k a e. P.
38		nfcv 2375	. . . . . . . . . . . . . 14 |- F/_ k P.
39		nfcv 2375	. . . . . . . . . . . . . . 15 |- F/_ k N.
40		nfra1 2564	. . . . . . . . . . . . . . 15 |- F/ k A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) )
41	39, 40	nfrexya 2574	. . . . . . . . . . . . . 14 |- F/ k E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) )
42	38, 41	nfralya 2573	. . . . . . . . . . . . 13 |- F/ k A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) )
43	37, 42	nfan 1614	. . . . . . . . . . . 12 |- F/ k ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) )
44	36, 43	nfan 1614	. . . . . . . . . . 11 |- F/ k ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) )
45		nfv 1577	. . . . . . . . . . 11 |- F/ k x e. R.
46	44, 45	nfan 1614	. . . . . . . . . 10 |- F/ k ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. )
47		nfv 1577	. . . . . . . . . 10 |- F/ k 0R <R x
48	46, 47	nfan 1614	. . . . . . . . 9 |- F/ k ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x )
49	5	ad4antr 494	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) /\ k e. N. ) -> ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) : N. --> P. )
50		simpr 110	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) /\ k e. N. ) -> k e. N. )
51	49, 50	ffvelcdmd 5791	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) /\ k e. N. ) -> ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) e. P. )
52		simplrl 537	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) -> a e. P. )
53	52	adantr 276	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) -> a e. P. )
54		addclpr 7817	. . . . . . . . . . . . . . 15 |- ( ( a e. P. /\ ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) e. P. ) -> ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) e. P. )
55	53, 23, 54	syl2anc 411	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) -> ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) e. P. )
56	55	adantr 276	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) /\ k e. N. ) -> ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) e. P. )
57		prsrlt 8067	. . . . . . . . . . . . 13 |- ( ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) e. P. /\ ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) e. P. ) -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) <-> [ <. ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. 1P ) , 1P >. ] ~R <R [ <. ( ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) +P. 1P ) , 1P >. ] ~R ) )
58	51, 56, 57	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) /\ k e. N. ) -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) <-> [ <. ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. 1P ) , 1P >. ] ~R <R [ <. ( ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) +P. 1P ) , 1P >. ] ~R ) )
59	1, 2, 3, 4	caucvgsrlemfv 8071	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ k e. N. ) -> [ <. ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. 1P ) , 1P >. ] ~R = ( F ` k ) )
60	59	adantlr 477	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ k e. N. ) -> [ <. ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. 1P ) , 1P >. ] ~R = ( F ` k ) )
61	60	adantlr 477	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ k e. N. ) -> [ <. ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. 1P ) , 1P >. ] ~R = ( F ` k ) )
62	61	adantlr 477	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) /\ k e. N. ) -> [ <. ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. 1P ) , 1P >. ] ~R = ( F ` k ) )
63		prsradd 8066	. . . . . . . . . . . . . . . 16 |- ( ( a e. P. /\ ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) e. P. ) -> [ <. ( ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) +P. 1P ) , 1P >. ] ~R = ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R [ <. ( ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) +P. 1P ) , 1P >. ] ~R ) )
64	53, 23, 63	syl2anc 411	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) -> [ <. ( ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) +P. 1P ) , 1P >. ] ~R = ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R [ <. ( ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) +P. 1P ) , 1P >. ] ~R ) )
65		prsrriota 8068	. . . . . . . . . . . . . . . . 17 |- ( ( x e. R. /\ 0R <R x ) -> [ <. ( ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) +P. 1P ) , 1P >. ] ~R = x )
66	65	oveq2d 6044	. . . . . . . . . . . . . . . 16 |- ( ( x e. R. /\ 0R <R x ) -> ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R [ <. ( ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) +P. 1P ) , 1P >. ] ~R ) = ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) )
67	66	adantll 476	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) -> ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R [ <. ( ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) +P. 1P ) , 1P >. ] ~R ) = ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) )
68	64, 67	eqtrd 2264	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) -> [ <. ( ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) +P. 1P ) , 1P >. ] ~R = ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) )
69	68	adantr 276	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) /\ k e. N. ) -> [ <. ( ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) +P. 1P ) , 1P >. ] ~R = ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) )
70	62, 69	breq12d 4106	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) /\ k e. N. ) -> ( [ <. ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. 1P ) , 1P >. ] ~R <R [ <. ( ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) +P. 1P ) , 1P >. ] ~R <-> ( F ` k ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) ) )
71	58, 70	bitrd 188	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) /\ k e. N. ) -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) <-> ( F ` k ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) ) )
72	53	adantr 276	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) /\ k e. N. ) -> a e. P. )
73	23	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) /\ k e. N. ) -> ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) e. P. )
74		addclpr 7817	. . . . . . . . . . . . . 14 |- ( ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) e. P. /\ ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) e. P. ) -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) e. P. )
75	51, 73, 74	syl2anc 411	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) /\ k e. N. ) -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) e. P. )
76		prsrlt 8067	. . . . . . . . . . . . 13 |- ( ( a e. P. /\ ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) e. P. ) -> ( a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) <-> [ <. ( a +P. 1P ) , 1P >. ] ~R <R [ <. ( ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) +P. 1P ) , 1P >. ] ~R ) )
77	72, 75, 76	syl2anc 411	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) /\ k e. N. ) -> ( a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) <-> [ <. ( a +P. 1P ) , 1P >. ] ~R <R [ <. ( ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) +P. 1P ) , 1P >. ] ~R ) )
78		prsradd 8066	. . . . . . . . . . . . . 14 |- ( ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) e. P. /\ ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) e. P. ) -> [ <. ( ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) +P. 1P ) , 1P >. ] ~R = ( [ <. ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. 1P ) , 1P >. ] ~R +R [ <. ( ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) +P. 1P ) , 1P >. ] ~R ) )
79	51, 73, 78	syl2anc 411	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) /\ k e. N. ) -> [ <. ( ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) +P. 1P ) , 1P >. ] ~R = ( [ <. ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. 1P ) , 1P >. ] ~R +R [ <. ( ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) +P. 1P ) , 1P >. ] ~R ) )
80	79	breq2d 4105	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) /\ k e. N. ) -> ( [ <. ( a +P. 1P ) , 1P >. ] ~R <R [ <. ( ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) +P. 1P ) , 1P >. ] ~R <-> [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( [ <. ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. 1P ) , 1P >. ] ~R +R [ <. ( ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) +P. 1P ) , 1P >. ] ~R ) ) )
81	65	adantll 476	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) -> [ <. ( ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) +P. 1P ) , 1P >. ] ~R = x )
82	81	adantr 276	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) /\ k e. N. ) -> [ <. ( ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) +P. 1P ) , 1P >. ] ~R = x )
83	62, 82	oveq12d 6046	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) /\ k e. N. ) -> ( [ <. ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. 1P ) , 1P >. ] ~R +R [ <. ( ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) +P. 1P ) , 1P >. ] ~R ) = ( ( F ` k ) +R x ) )
84	83	breq2d 4105	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) /\ k e. N. ) -> ( [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( [ <. ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. 1P ) , 1P >. ] ~R +R [ <. ( ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) +P. 1P ) , 1P >. ] ~R ) <-> [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` k ) +R x ) ) )
85	77, 80, 84	3bitrd 214	. . . . . . . . . . 11 |- ( ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) /\ k e. N. ) -> ( a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) <-> [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` k ) +R x ) ) )
86	71, 85	anbi12d 473	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) /\ k e. N. ) -> ( ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) ) <-> ( ( F ` k ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` k ) +R x ) ) ) )
87	86	imbi2d 230	. . . . . . . . 9 |- ( ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) /\ k e. N. ) -> ( ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) ) ) <-> ( j <N k -> ( ( F ` k ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` k ) +R x ) ) ) ) )
88	48, 87	ralbida 2527	. . . . . . . 8 |- ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) -> ( A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) ) ) <-> A. k e. N. ( j <N k -> ( ( F ` k ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` k ) +R x ) ) ) ) )
89	35, 88	rexbid 2532	. . . . . . 7 |- ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) -> ( E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. ( iota_ c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x ) ) ) ) <-> E. j e. N. A. k e. N. ( j <N k -> ( ( F ` k ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` k ) +R x ) ) ) ) )
90	24, 89	mpbid 147	. . . . . 6 |- ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) -> E. j e. N. A. k e. N. ( j <N k -> ( ( F ` k ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` k ) +R x ) ) ) )
91		breq2 4097	. . . . . . . . 9 |- ( k = i -> ( j <N k <-> j <N i ) )
92		fveq2 5648	. . . . . . . . . . 11 |- ( k = i -> ( F ` k ) = ( F ` i ) )
93	92	breq1d 4103	. . . . . . . . . 10 |- ( k = i -> ( ( F ` k ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) <-> ( F ` i ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) ) )
94	92	oveq1d 6043	. . . . . . . . . . 11 |- ( k = i -> ( ( F ` k ) +R x ) = ( ( F ` i ) +R x ) )
95	94	breq2d 4105	. . . . . . . . . 10 |- ( k = i -> ( [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` k ) +R x ) <-> [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` i ) +R x ) ) )
96	93, 95	anbi12d 473	. . . . . . . . 9 |- ( k = i -> ( ( ( F ` k ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` k ) +R x ) ) <-> ( ( F ` i ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` i ) +R x ) ) ) )
97	91, 96	imbi12d 234	. . . . . . . 8 |- ( k = i -> ( ( j <N k -> ( ( F ` k ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` k ) +R x ) ) ) <-> ( j <N i -> ( ( F ` i ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` i ) +R x ) ) ) ) )
98	97	cbvralv 2768	. . . . . . 7 |- ( A. k e. N. ( j <N k -> ( ( F ` k ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` k ) +R x ) ) ) <-> A. i e. N. ( j <N i -> ( ( F ` i ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` i ) +R x ) ) ) )
99	98	rexbii 2540	. . . . . 6 |- ( E. j e. N. A. k e. N. ( j <N k -> ( ( F ` k ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` k ) +R x ) ) ) <-> E. j e. N. A. i e. N. ( j <N i -> ( ( F ` i ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` i ) +R x ) ) ) )
100	90, 99	sylib 122	. . . . 5 |- ( ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) /\ 0R <R x ) -> E. j e. N. A. i e. N. ( j <N i -> ( ( F ` i ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` i ) +R x ) ) ) )
101	100	ex 115	. . . 4 |- ( ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) /\ x e. R. ) -> ( 0R <R x -> E. j e. N. A. i e. N. ( j <N i -> ( ( F ` i ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` i ) +R x ) ) ) ) )
102	101	ralrimiva 2606	. . 3 |- ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) -> A. x e. R. ( 0R <R x -> E. j e. N. A. i e. N. ( j <N i -> ( ( F ` i ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` i ) +R x ) ) ) ) )
103		oveq1 6035	. . . . . . . . . 10 |- ( y = [ <. ( a +P. 1P ) , 1P >. ] ~R -> ( y +R x ) = ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) )
104	103	breq2d 4105	. . . . . . . . 9 |- ( y = [ <. ( a +P. 1P ) , 1P >. ] ~R -> ( ( F ` i ) <R ( y +R x ) <-> ( F ` i ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) ) )
105		breq1 4096	. . . . . . . . 9 |- ( y = [ <. ( a +P. 1P ) , 1P >. ] ~R -> ( y <R ( ( F ` i ) +R x ) <-> [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` i ) +R x ) ) )
106	104, 105	anbi12d 473	. . . . . . . 8 |- ( y = [ <. ( a +P. 1P ) , 1P >. ] ~R -> ( ( ( F ` i ) <R ( y +R x ) /\ y <R ( ( F ` i ) +R x ) ) <-> ( ( F ` i ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` i ) +R x ) ) ) )
107	106	imbi2d 230	. . . . . . 7 |- ( y = [ <. ( a +P. 1P ) , 1P >. ] ~R -> ( ( j <N i -> ( ( F ` i ) <R ( y +R x ) /\ y <R ( ( F ` i ) +R x ) ) ) <-> ( j <N i -> ( ( F ` i ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` i ) +R x ) ) ) ) )
108	107	rexralbidv 2559	. . . . . 6 |- ( y = [ <. ( a +P. 1P ) , 1P >. ] ~R -> ( E. j e. N. A. i e. N. ( j <N i -> ( ( F ` i ) <R ( y +R x ) /\ y <R ( ( F ` i ) +R x ) ) ) <-> E. j e. N. A. i e. N. ( j <N i -> ( ( F ` i ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` i ) +R x ) ) ) ) )
109	108	imbi2d 230	. . . . 5 |- ( y = [ <. ( a +P. 1P ) , 1P >. ] ~R -> ( ( 0R <R x -> E. j e. N. A. i e. N. ( j <N i -> ( ( F ` i ) <R ( y +R x ) /\ y <R ( ( F ` i ) +R x ) ) ) ) <-> ( 0R <R x -> E. j e. N. A. i e. N. ( j <N i -> ( ( F ` i ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` i ) +R x ) ) ) ) ) )
110	109	ralbidv 2533	. . . 4 |- ( y = [ <. ( a +P. 1P ) , 1P >. ] ~R -> ( A. x e. R. ( 0R <R x -> E. j e. N. A. i e. N. ( j <N i -> ( ( F ` i ) <R ( y +R x ) /\ y <R ( ( F ` i ) +R x ) ) ) ) <-> A. x e. R. ( 0R <R x -> E. j e. N. A. i e. N. ( j <N i -> ( ( F ` i ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` i ) +R x ) ) ) ) ) )
111	110	rspcev 2911	. . 3 |- ( ( [ <. ( a +P. 1P ) , 1P >. ] ~R e. R. /\ A. x e. R. ( 0R <R x -> E. j e. N. A. i e. N. ( j <N i -> ( ( F ` i ) <R ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) /\ [ <. ( a +P. 1P ) , 1P >. ] ~R <R ( ( F ` i ) +R x ) ) ) ) ) -> E. y e. R. A. x e. R. ( 0R <R x -> E. j e. N. A. i e. N. ( j <N i -> ( ( F ` i ) <R ( y +R x ) /\ y <R ( ( F ` i ) +R x ) ) ) ) )
112	10, 102, 111	syl2anc 411	. 2 |- ( ( ph /\ ( a e. P. /\ A. b e. P. E. j e. N. A. k e. N. ( j <N k -> ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) <P ( a +P. b ) /\ a <P ( ( ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) ` k ) +P. b ) ) ) ) ) -> E. y e. R. A. x e. R. ( 0R <R x -> E. j e. N. A. i e. N. ( j <N i -> ( ( F ` i ) <R ( y +R x ) /\ y <R ( ( F ` i ) +R x ) ) ) ) )
113	8, 112	rexlimddv 2656	1 |- ( ph -> E. y e. R. A. x e. R. ( 0R <R x -> E. j e. N. A. i e. N. ( j <N i -> ( ( F ` i ) <R ( y +R x ) /\ y <R ( ( F ` i ) +R x ) ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   {cab 2217   A.wral 2511   E.wrex 2512   E!wreu 2513   <.cop 3676   class class class wbr 4093   |-> cmpt 4155   -->wf 5329   ` cfv 5333   iota_crio 5980  (class class class)co 6028   1oc1o 6618   [cec 6743   N.cnpi 7552   <N clti 7555   ~Q ceq 7559   *Qcrq 7564   <Q cltq 7565   P.cnp 7571   1Pc1p 7572   +P. cpp 7573   <P cltp 7575   ~R cer 7576   R.cnr 7577   0Rc0r 7578   1Rc1r 7579   +R cplr 7581   <R cltr 7583

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7584  df-pli 7585  df-mi 7586  df-lti 7587  df-plpq 7624  df-mpq 7625  df-enq 7627  df-nqqs 7628  df-plqqs 7629  df-mqqs 7630  df-1nqqs 7631  df-rq 7632  df-ltnqqs 7633  df-enq0 7704  df-nq0 7705  df-0nq0 7706  df-plq0 7707  df-mq0 7708  df-inp 7746  df-i1p 7747  df-iplp 7748  df-iltp 7750  df-enr 8006  df-nr 8007  df-plr 8008  df-ltr 8010  df-0r 8011  df-1r 8012

This theorem is referenced by:  caucvgsrlemoffres  8080