Theorem caucvgsrlemgt1 7825

Description: Lemma for caucvgsr 7832. 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 2189	. . . 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 7822	. . 3 |- ( ph -> ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) : N. --> P. )
6	1, 2, 3, 4	caucvgsrlemcau 7823	. . 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 7824	. . 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 7742	. 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 7814	. . . 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 5905	. . . . . . . . . . . 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 4030	. . . . . . . . . . 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 5905	. . . . . . . . . . . 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 4030	. . . . . . . . . . 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 2516	. . . . . . . 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 536	. . . . . . . . 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 7813	. . . . . . . . . 10 |- ( ( x e. R. /\ 0R <R x ) -> E! c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x )
21		riotacl 5867	. . . . . . . . . 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 2861	. . . . . . 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 1539	. . . . . . . . . . 11 |- F/ j ph
26		nfv 1539	. . . . . . . . . . . 12 |- F/ j a e. P.
27		nfcv 2332	. . . . . . . . . . . . 13 |- F/_ j P.
28		nfre1 2533	. . . . . . . . . . . . 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 2530	. . . . . . . . . . . 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 1576	. . . . . . . . . . 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 1576	. . . . . . . . . 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 1539	. . . . . . . . . 10 |- F/ j x e. R.
33	31, 32	nfan 1576	. . . . . . . . 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 1539	. . . . . . . . 9 |- F/ j 0R <R x
35	33, 34	nfan 1576	. . . . . . . 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 1539	. . . . . . . . . . . 12 |- F/ k ph
37		nfv 1539	. . . . . . . . . . . . 13 |- F/ k a e. P.
38		nfcv 2332	. . . . . . . . . . . . . 14 |- F/_ k P.
39		nfcv 2332	. . . . . . . . . . . . . . 15 |- F/_ k N.
40		nfra1 2521	. . . . . . . . . . . . . . 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 2531	. . . . . . . . . . . . . 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 2530	. . . . . . . . . . . . 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 1576	. . . . . . . . . . . 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 1576	. . . . . . . . . . 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 1539	. . . . . . . . . . 11 |- F/ k x e. R.
46	44, 45	nfan 1576	. . . . . . . . . 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 1539	. . . . . . . . . 10 |- F/ k 0R <R x
48	46, 47	nfan 1576	. . . . . . . . 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 5673	. . . . . . . . . . . . 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 535	. . . . . . . . . . . . . . . 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 7567	. . . . . . . . . . . . . . 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 7817	. . . . . . . . . . . . 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 7821	. . . . . . . . . . . . . . . 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 7816	. . . . . . . . . . . . . . . 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 7818	. . . . . . . . . . . . . . . . 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 5913	. . . . . . . . . . . . . . . 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 2222	. . . . . . . . . . . . . 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 4031	. . . . . . . . . . . 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 7567	. . . . . . . . . . . . . 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 7817	. . . . . . . . . . . . 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 7816	. . . . . . . . . . . . . 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 4030	. . . . . . . . . . . 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 5915	. . . . . . . . . . . . 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 4030	. . . . . . . . . . . 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 2484	. . . . . . . 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 2489	. . . . . . 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 4022	. . . . . . . . 9 |- ( k = i -> ( j <N k <-> j <N i ) )
92		fveq2 5534	. . . . . . . . . . 11 |- ( k = i -> ( F ` k ) = ( F ` i ) )
93	92	breq1d 4028	. . . . . . . . . 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 5912	. . . . . . . . . . 11 |- ( k = i -> ( ( F ` k ) +R x ) = ( ( F ` i ) +R x ) )
95	94	breq2d 4030	. . . . . . . . . 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 2718	. . . . . . 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 2497	. . . . . 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 2563	. . 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 5904	. . . . . . . . . 10 |- ( y = [ <. ( a +P. 1P ) , 1P >. ] ~R -> ( y +R x ) = ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) )
104	103	breq2d 4030	. . . . . . . . 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 4021	. . . . . . . . 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 2516	. . . . . 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 2490	. . . 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 2856	. . 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 2612	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 1364   e. wcel 2160   {cab 2175   A.wral 2468   E.wrex 2469   E!wreu 2470   <.cop 3610   class class class wbr 4018   |-> cmpt 4079   -->wf 5231   ` cfv 5235   iota_crio 5851  (class class class)co 5897   1oc1o 6435   [cec 6558   N.cnpi 7302   <N clti 7305   ~Q ceq 7309   *Qcrq 7314   <Q cltq 7315   P.cnp 7321   1Pc1p 7322   +P. cpp 7323   <P cltp 7325   ~R cer 7326   R.cnr 7327   0Rc0r 7328   1Rc1r 7329   +R cplr 7331   <R cltr 7333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-eprel 4307  df-id 4311  df-po 4314  df-iso 4315  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-recs 6331  df-irdg 6396  df-1o 6442  df-2o 6443  df-oadd 6446  df-omul 6447  df-er 6560  df-ec 6562  df-qs 6566  df-ni 7334  df-pli 7335  df-mi 7336  df-lti 7337  df-plpq 7374  df-mpq 7375  df-enq 7377  df-nqqs 7378  df-plqqs 7379  df-mqqs 7380  df-1nqqs 7381  df-rq 7382  df-ltnqqs 7383  df-enq0 7454  df-nq0 7455  df-0nq0 7456  df-plq0 7457  df-mq0 7458  df-inp 7496  df-i1p 7497  df-iplp 7498  df-iltp 7500  df-enr 7756  df-nr 7757  df-plr 7758  df-ltr 7760  df-0r 7761  df-1r 7762

This theorem is referenced by:  caucvgsrlemoffres  7830