Theorem caucvgprprlem2 6900

Description: Lemma for caucvgprpr 6902. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 25-Nov-2020.)

Hypotheses

Ref	Expression
caucvgprpr.f	|- ( ph -> F : N. --> P. )
caucvgprpr.cau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )
caucvgprpr.bnd	|- ( ph -> A. m e. N. A <P ( F ` m ) )
caucvgprpr.lim	|- L = <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.
caucvgprprlemlim.q	|- ( ph -> Q e. P. )
caucvgprprlemlim.jk	|- ( ph -> J <N K )
caucvgprprlemlim.jkq	|- ( ph -> <. { l | l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u } >. <P Q )

Assertion

Ref	Expression
caucvgprprlem2	|- ( ph -> L <P ( ( F ` K ) +P. Q ) )

Distinct variable groups:   A, m   m, F   A, r   F, r, u, l, k   n, F   K, l, p, u, q, r   J, l, u   k, L   ph, r   k, n   k, r   q, l, r   m, r   k, p, q   u, n, l, k

Allowed substitution hints:   ph( u, k, m, n, q, p, l)   A( u, k, n, q, p, l)   Q( u, k, m, n, r, q, p, l)   F( q, p)   J( k, m, n, r, q, p)   K( k, m, n)   L( u, m, n, r, q, p, l)

Proof of Theorem caucvgprprlem2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgprprlemlim.jk	. . . . 5 |- ( ph -> J <N K )
2		caucvgprprlemlim.jkq	. . . . 5 |- ( ph -> <. { l | l <Q ( *Q ` [ <. J , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. J , 1o >. ] ~Q ) <Q u } >. <P Q )
3	1, 2	caucvgprprlemk 6873	. . . 4 |- ( ph -> <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. <P Q )
4		ltrelpi 6514	. . . . . . . . . 10 |- <N C_ ( N. X. N. )
5	4	brel 4410	. . . . . . . . 9 |- ( J <N K -> ( J e. N. /\ K e. N. ) )
6	1, 5	syl 14	. . . . . . . 8 |- ( ph -> ( J e. N. /\ K e. N. ) )
7	6	simprd 112	. . . . . . 7 |- ( ph -> K e. N. )
8		nnnq 6612	. . . . . . . 8 |- ( K e. N. -> [ <. K , 1o >. ] ~Q e. Q. )
9		recclnq 6582	. . . . . . . 8 |- ( [ <. K , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. )
10	8, 9	syl 14	. . . . . . 7 |- ( K e. N. -> ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. )
11	7, 10	syl 14	. . . . . 6 |- ( ph -> ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. )
12		nqprlu 6737	. . . . . 6 |- ( ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. -> <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. e. P. )
13	11, 12	syl 14	. . . . 5 |- ( ph -> <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. e. P. )
14		caucvgprprlemlim.q	. . . . 5 |- ( ph -> Q e. P. )
15		caucvgprpr.f	. . . . . 6 |- ( ph -> F : N. --> P. )
16	15, 7	ffvelrnd 5324	. . . . 5 |- ( ph -> ( F ` K ) e. P. )
17		ltaprg 6809	. . . . 5 |- ( ( <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. e. P. /\ Q e. P. /\ ( F ` K ) e. P. ) -> ( <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. <P Q <-> ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) <P ( ( F ` K ) +P. Q ) ) )
18	13, 14, 16, 17	syl3anc 1169	. . . 4 |- ( ph -> ( <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. <P Q <-> ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) <P ( ( F ` K ) +P. Q ) ) )
19	3, 18	mpbid 145	. . 3 |- ( ph -> ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) <P ( ( F ` K ) +P. Q ) )
20		addclpr 6727	. . . . 5 |- ( ( ( F ` K ) e. P. /\ <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. e. P. ) -> ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) e. P. )
21	16, 13, 20	syl2anc 403	. . . 4 |- ( ph -> ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) e. P. )
22		addclpr 6727	. . . . 5 |- ( ( ( F ` K ) e. P. /\ Q e. P. ) -> ( ( F ` K ) +P. Q ) e. P. )
23	16, 14, 22	syl2anc 403	. . . 4 |- ( ph -> ( ( F ` K ) +P. Q ) e. P. )
24		ltdfpr 6696	. . . 4 |- ( ( ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) e. P. /\ ( ( F ` K ) +P. Q ) e. P. ) -> ( ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) <P ( ( F ` K ) +P. Q ) <-> E. x e. Q. ( x e. ( 2nd ` ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) ) )
25	21, 23, 24	syl2anc 403	. . 3 |- ( ph -> ( ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) <P ( ( F ` K ) +P. Q ) <-> E. x e. Q. ( x e. ( 2nd ` ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) ) )
26	19, 25	mpbid 145	. 2 |- ( ph -> E. x e. Q. ( x e. ( 2nd ` ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) )
27		simprl 497	. . . 4 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) ) ) -> x e. Q. )
28	7	adantr 270	. . . . . 6 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) ) ) -> K e. N. )
29		simprrl 505	. . . . . . . 8 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) ) ) -> x e. ( 2nd ` ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) ) )
30		breq1 3788	. . . . . . . . . . . 12 |- ( l = p -> ( l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
31	30	cbvabv 2202	. . . . . . . . . . 11 |- { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) }
32		breq2 3789	. . . . . . . . . . . 12 |- ( u = q -> ( ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u <-> ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q ) )
33	32	cbvabv 2202	. . . . . . . . . . 11 |- { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } = { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q }
34	31, 33	opeq12i 3575	. . . . . . . . . 10 |- <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. = <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >.
35	34	oveq2i 5543	. . . . . . . . 9 |- ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) = ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. )
36	35	fveq2i 5201	. . . . . . . 8 |- ( 2nd ` ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) ) = ( 2nd ` ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) )
37	29, 36	syl6eleq 2171	. . . . . . 7 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) ) ) -> x e. ( 2nd ` ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) ) )
38		nqprlu 6737	. . . . . . . . . . 11 |- ( ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. -> <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. e. P. )
39	11, 38	syl 14	. . . . . . . . . 10 |- ( ph -> <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. e. P. )
40		addclpr 6727	. . . . . . . . . 10 |- ( ( ( F ` K ) e. P. /\ <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. e. P. ) -> ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) e. P. )
41	16, 39, 40	syl2anc 403	. . . . . . . . 9 |- ( ph -> ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) e. P. )
42	41	adantr 270	. . . . . . . 8 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) ) ) -> ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) e. P. )
43		nqpru 6742	. . . . . . . 8 |- ( ( x e. Q. /\ ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) e. P. ) -> ( x e. ( 2nd ` ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) ) <-> ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q x } , { q | x <Q q } >. ) )
44	27, 42, 43	syl2anc 403	. . . . . . 7 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) ) ) -> ( x e. ( 2nd ` ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) ) <-> ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q x } , { q | x <Q q } >. ) )
45	37, 44	mpbid 145	. . . . . 6 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) ) ) -> ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q x } , { q | x <Q q } >. )
46		fveq2 5198	. . . . . . . . 9 |- ( r = K -> ( F ` r ) = ( F ` K ) )
47		opeq1 3570	. . . . . . . . . . . . . 14 |- ( r = K -> <. r , 1o >. = <. K , 1o >. )
48	47	eceq1d 6165	. . . . . . . . . . . . 13 |- ( r = K -> [ <. r , 1o >. ] ~Q = [ <. K , 1o >. ] ~Q )
49	48	fveq2d 5202	. . . . . . . . . . . 12 |- ( r = K -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. K , 1o >. ] ~Q ) )
50	49	breq2d 3797	. . . . . . . . . . 11 |- ( r = K -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
51	50	abbidv 2196	. . . . . . . . . 10 |- ( r = K -> { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } )
52	49	breq1d 3795	. . . . . . . . . . 11 |- ( r = K -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q ) )
53	52	abbidv 2196	. . . . . . . . . 10 |- ( r = K -> { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } )
54	51, 53	opeq12d 3578	. . . . . . . . 9 |- ( r = K -> <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. = <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. )
55	46, 54	oveq12d 5550	. . . . . . . 8 |- ( r = K -> ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) = ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) )
56	55	breq1d 3795	. . . . . . 7 |- ( r = K -> ( ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q x } , { q | x <Q q } >. <-> ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q x } , { q | x <Q q } >. ) )
57	56	rspcev 2701	. . . . . 6 |- ( ( K e. N. /\ ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q x } , { q | x <Q q } >. ) -> E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q x } , { q | x <Q q } >. )
58	28, 45, 57	syl2anc 403	. . . . 5 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) ) ) -> E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q x } , { q | x <Q q } >. )
59		breq2 3789	. . . . . . . . . 10 |- ( u = x -> ( p <Q u <-> p <Q x ) )
60	59	abbidv 2196	. . . . . . . . 9 |- ( u = x -> { p | p <Q u } = { p | p <Q x } )
61		breq1 3788	. . . . . . . . . 10 |- ( u = x -> ( u <Q q <-> x <Q q ) )
62	61	abbidv 2196	. . . . . . . . 9 |- ( u = x -> { q | u <Q q } = { q | x <Q q } )
63	60, 62	opeq12d 3578	. . . . . . . 8 |- ( u = x -> <. { p | p <Q u } , { q | u <Q q } >. = <. { p | p <Q x } , { q | x <Q q } >. )
64	63	breq2d 3797	. . . . . . 7 |- ( u = x -> ( ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. <-> ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q x } , { q | x <Q q } >. ) )
65	64	rexbidv 2369	. . . . . 6 |- ( u = x -> ( E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. <-> E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q x } , { q | x <Q q } >. ) )
66		caucvgprpr.lim	. . . . . . . 8 |- L = <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.
67	66	fveq2i 5201	. . . . . . 7 |- ( 2nd ` L ) = ( 2nd ` <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >. )
68		nqex 6553	. . . . . . . . 9 |- Q. e. _V
69	68	rabex 3922	. . . . . . . 8 |- { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } e. _V
70	68	rabex 3922	. . . . . . . 8 |- { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } e. _V
71	69, 70	op2nd 5794	. . . . . . 7 |- ( 2nd ` <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >. ) = { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. }
72	67, 71	eqtri 2101	. . . . . 6 |- ( 2nd ` L ) = { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. }
73	65, 72	elrab2 2751	. . . . 5 |- ( x e. ( 2nd ` L ) <-> ( x e. Q. /\ E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q x } , { q | x <Q q } >. ) )
74	27, 58, 73	sylanbrc 408	. . . 4 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) ) ) -> x e. ( 2nd ` L ) )
75		simprrr 506	. . . 4 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) ) ) -> x e. ( 1st ` ( ( F ` K ) +P. Q ) ) )
76		rspe 2412	. . . 4 |- ( ( x e. Q. /\ ( x e. ( 2nd ` L ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) ) -> E. x e. Q. ( x e. ( 2nd ` L ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) )
77	27, 74, 75, 76	syl12anc 1167	. . 3 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) ) ) -> E. x e. Q. ( x e. ( 2nd ` L ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) )
78		caucvgprpr.cau	. . . . . 6 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )
79		caucvgprpr.bnd	. . . . . 6 |- ( ph -> A. m e. N. A <P ( F ` m ) )
80	15, 78, 79, 66	caucvgprprlemcl 6894	. . . . 5 |- ( ph -> L e. P. )
81	80	adantr 270	. . . 4 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) ) ) -> L e. P. )
82	23	adantr 270	. . . 4 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) ) ) -> ( ( F ` K ) +P. Q ) e. P. )
83		ltdfpr 6696	. . . 4 |- ( ( L e. P. /\ ( ( F ` K ) +P. Q ) e. P. ) -> ( L <P ( ( F ` K ) +P. Q ) <-> E. x e. Q. ( x e. ( 2nd ` L ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) ) )
84	81, 82, 83	syl2anc 403	. . 3 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) ) ) -> ( L <P ( ( F ` K ) +P. Q ) <-> E. x e. Q. ( x e. ( 2nd ` L ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) ) )
85	77, 84	mpbird 165	. 2 |- ( ( ph /\ ( x e. Q. /\ ( x e. ( 2nd ` ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) ) /\ x e. ( 1st ` ( ( F ` K ) +P. Q ) ) ) ) ) -> L <P ( ( F ` K ) +P. Q ) )
86	26, 85	rexlimddv 2481	1 |- ( ph -> L <P ( ( F ` K ) +P. Q ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   {cab 2067   A.wral 2348   E.wrex 2349   {crab 2352   <.cop 3401   class class class wbr 3785   -->wf 4918   ` cfv 4922  (class class class)co 5532   1stc1st 5785   2ndc2nd 5786   1oc1o 6017   [cec 6127   N.cnpi 6462   <N clti 6465   ~Q ceq 6469   Q.cnq 6470   +Q cplq 6472   *Qcrq 6474   <Q cltq 6475   P.cnp 6481   +P. cpp 6483   <P cltp 6485

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

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

This theorem is referenced by:  caucvgprprlemlim  6901