Theorem caucvgsrlemgt1 7537

Description: Lemma for caucvgsr 7544. 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 2115	. . . 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 7534	. . 3 |- ( ph -> ( z e. N. |-> ( iota_ w e. P. ( F ` z ) = [ <. ( w +P. 1P ) , 1P >. ] ~R ) ) : N. --> P. )
6	1, 2, 3, 4	caucvgsrlemcau 7535	. . 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 7536	. . 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 7468	. 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 7526	. . . 4 |- ( a e. P. -> [ <. ( a +P. 1P ) , 1P >. ] ~R e. R. )
10	9	ad2antrl 479	. . 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 5736	. . . . . . . . . . . 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 3907	. . . . . . . . . . 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 5736	. . . . . . . . . . . 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 3907	. . . . . . . . . . 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 462	. . . . . . . . . 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 229	. . . . . . . . 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 2435	. . . . . . . 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 508	. . . . . . . . 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 272	. . . . . . . 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 7525	. . . . . . . . . 10 |- ( ( x e. R. /\ 0R <R x ) -> E! c e. P. [ <. ( c +P. 1P ) , 1P >. ] ~R = x )
21		riotacl 5698	. . . . . . . . . 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 465	. . . . . . . 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 2765	. . . . . . 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 1491	. . . . . . . . . . 11 |- F/ j ph
26		nfv 1491	. . . . . . . . . . . 12 |- F/ j a e. P.
27		nfcv 2255	. . . . . . . . . . . . 13 |- F/_ j P.
28		nfre1 2450	. . . . . . . . . . . . 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 2447	. . . . . . . . . . . 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 1527	. . . . . . . . . . 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 1527	. . . . . . . . . 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 1491	. . . . . . . . . 10 |- F/ j x e. R.
33	31, 32	nfan 1527	. . . . . . . . 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 1491	. . . . . . . . 9 |- F/ j 0R <R x
35	33, 34	nfan 1527	. . . . . . . 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 1491	. . . . . . . . . . . 12 |- F/ k ph
37		nfv 1491	. . . . . . . . . . . . 13 |- F/ k a e. P.
38		nfcv 2255	. . . . . . . . . . . . . 14 |- F/_ k P.
39		nfcv 2255	. . . . . . . . . . . . . . 15 |- F/_ k N.
40		nfra1 2440	. . . . . . . . . . . . . . 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 2448	. . . . . . . . . . . . . 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 2447	. . . . . . . . . . . . 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 1527	. . . . . . . . . . . 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 1527	. . . . . . . . . . 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 1491	. . . . . . . . . . 11 |- F/ k x e. R.
46	44, 45	nfan 1527	. . . . . . . . . 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 1491	. . . . . . . . . 10 |- F/ k 0R <R x
48	46, 47	nfan 1527	. . . . . . . . 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 483	. . . . . . . . . . . . . 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 109	. . . . . . . . . . . . . 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	ffvelrnd 5510	. . . . . . . . . . . . 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 507	. . . . . . . . . . . . . . . 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 272	. . . . . . . . . . . . . . 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 7293	. . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . 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 272	. . . . . . . . . . . . 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 7529	. . . . . . . . . . . . 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 406	. . . . . . . . . . . 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 7533	. . . . . . . . . . . . . . . 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 466	. . . . . . . . . . . . . . 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 466	. . . . . . . . . . . . . 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 466	. . . . . . . . . . . . 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 7528	. . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . 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 7530	. . . . . . . . . . . . . . . . 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 5744	. . . . . . . . . . . . . . . 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 465	. . . . . . . . . . . . . . 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 2147	. . . . . . . . . . . . . 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 272	. . . . . . . . . . . . 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 3908	. . . . . . . . . . . 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 187	. . . . . . . . . . 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 272	. . . . . . . . . . . . 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 272	. . . . . . . . . . . . . 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 7293	. . . . . . . . . . . . . 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 406	. . . . . . . . . . . . 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 7529	. . . . . . . . . . . . 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 406	. . . . . . . . . . . 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 7528	. . . . . . . . . . . . . 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 406	. . . . . . . . . . . . 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 3907	. . . . . . . . . . . 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 465	. . . . . . . . . . . . . . 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 272	. . . . . . . . . . . . . 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 5746	. . . . . . . . . . . . 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 3907	. . . . . . . . . . . 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 213	. . . . . . . . . . 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 462	. . . . . . . . . 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 229	. . . . . . . . 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 2405	. . . . . . . 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 2410	. . . . . . 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 146	. . . . . 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 3899	. . . . . . . . 9 |- ( k = i -> ( j <N k <-> j <N i ) )
92		fveq2 5375	. . . . . . . . . . 11 |- ( k = i -> ( F ` k ) = ( F ` i ) )
93	92	breq1d 3905	. . . . . . . . . 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 5743	. . . . . . . . . . 11 |- ( k = i -> ( ( F ` k ) +R x ) = ( ( F ` i ) +R x ) )
95	94	breq2d 3907	. . . . . . . . . 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 462	. . . . . . . . 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 233	. . . . . . . 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 2628	. . . . . . 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 2416	. . . . . 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 121	. . . . 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 114	. . . 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 2479	. . 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 5735	. . . . . . . . . 10 |- ( y = [ <. ( a +P. 1P ) , 1P >. ] ~R -> ( y +R x ) = ( [ <. ( a +P. 1P ) , 1P >. ] ~R +R x ) )
104	103	breq2d 3907	. . . . . . . . 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 3898	. . . . . . . . 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 462	. . . . . . . 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 229	. . . . . . 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 2435	. . . . . 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 229	. . . . 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 2411	. . . 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 2760	. . 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 406	. 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 2528	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 103   <-> wb 104   = wceq 1314   e. wcel 1463   {cab 2101   A.wral 2390   E.wrex 2391   E!wreu 2392   <.cop 3496   class class class wbr 3895   |-> cmpt 3949   -->wf 5077   ` cfv 5081   iota_crio 5683  (class class class)co 5728   1oc1o 6260   [cec 6381   N.cnpi 7028   <N clti 7031   ~Q ceq 7035   *Qcrq 7040   <Q cltq 7041   P.cnp 7047   1Pc1p 7048   +P. cpp 7049   <P cltp 7051   ~R cer 7052   R.cnr 7053   0Rc0r 7054   1Rc1r 7055   +R cplr 7057   <R cltr 7059

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-eprel 4171  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-irdg 6221  df-1o 6267  df-2o 6268  df-oadd 6271  df-omul 6272  df-er 6383  df-ec 6385  df-qs 6389  df-ni 7060  df-pli 7061  df-mi 7062  df-lti 7063  df-plpq 7100  df-mpq 7101  df-enq 7103  df-nqqs 7104  df-plqqs 7105  df-mqqs 7106  df-1nqqs 7107  df-rq 7108  df-ltnqqs 7109  df-enq0 7180  df-nq0 7181  df-0nq0 7182  df-plq0 7183  df-mq0 7184  df-inp 7222  df-i1p 7223  df-iplp 7224  df-iltp 7226  df-enr 7469  df-nr 7470  df-plr 7471  df-ltr 7473  df-0r 7474  df-1r 7475

This theorem is referenced by:  caucvgsrlemoffres  7542