Theorem caucvgprprlem2 7853

Description: Lemma for caucvgprpr 7855. 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 7826	. . . 4 |- ( ph -> <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. <P Q )
4		ltrelpi 7467	. . . . . . . . . 10 |- <N C_ ( N. X. N. )
5	4	brel 4740	. . . . . . . . 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 114	. . . . . . 7 |- ( ph -> K e. N. )
8		nnnq 7565	. . . . . . . 8 |- ( K e. N. -> [ <. K , 1o >. ] ~Q e. Q. )
9		recclnq 7535	. . . . . . . 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 7690	. . . . . 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	ffvelcdmd 5734	. . . . 5 |- ( ph -> ( F ` K ) e. P. )
17		ltaprg 7762	. . . . 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 1250	. . . 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 147	. . 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 7680	. . . . 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 411	. . . 4 |- ( ph -> ( ( F ` K ) +P. <. { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } >. ) e. P. )
22		addclpr 7680	. . . . 5 |- ( ( ( F ` K ) e. P. /\ Q e. P. ) -> ( ( F ` K ) +P. Q ) e. P. )
23	16, 14, 22	syl2anc 411	. . . 4 |- ( ph -> ( ( F ` K ) +P. Q ) e. P. )
24		ltdfpr 7649	. . . 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 411	. . 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 147	. 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 529	. . . 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 276	. . . . . 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 539	. . . . . . . 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 4057	. . . . . . . . . . . 12 |- ( l = p -> ( l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
31	30	cbvabv 2331	. . . . . . . . . . 11 |- { l | l <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) }
32		breq2 4058	. . . . . . . . . . . 12 |- ( u = q -> ( ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u <-> ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q ) )
33	32	cbvabv 2331	. . . . . . . . . . 11 |- { u | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q u } = { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q }
34	31, 33	opeq12i 3833	. . . . . . . . . 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 5973	. . . . . . . . 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 5597	. . . . . . . 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	eleqtrdi 2299	. . . . . . 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 7690	. . . . . . . . . . 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 7680	. . . . . . . . . 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 411	. . . . . . . . 9 |- ( ph -> ( ( F ` K ) +P. <. { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } >. ) e. P. )
42	41	adantr 276	. . . . . . . 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 7695	. . . . . . . 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 411	. . . . . . 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 147	. . . . . 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 5594	. . . . . . . . 9 |- ( r = K -> ( F ` r ) = ( F ` K ) )
47		opeq1 3828	. . . . . . . . . . . . . 14 |- ( r = K -> <. r , 1o >. = <. K , 1o >. )
48	47	eceq1d 6674	. . . . . . . . . . . . 13 |- ( r = K -> [ <. r , 1o >. ] ~Q = [ <. K , 1o >. ] ~Q )
49	48	fveq2d 5598	. . . . . . . . . . . 12 |- ( r = K -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. K , 1o >. ] ~Q ) )
50	49	breq2d 4066	. . . . . . . . . . 11 |- ( r = K -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
51	50	abbidv 2324	. . . . . . . . . 10 |- ( r = K -> { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. K , 1o >. ] ~Q ) } )
52	49	breq1d 4064	. . . . . . . . . . 11 |- ( r = K -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q ) )
53	52	abbidv 2324	. . . . . . . . . 10 |- ( r = K -> { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. K , 1o >. ] ~Q ) <Q q } )
54	51, 53	opeq12d 3836	. . . . . . . . 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 5980	. . . . . . . 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 4064	. . . . . . 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 2881	. . . . . 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 411	. . . . 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 4058	. . . . . . . . . 10 |- ( u = x -> ( p <Q u <-> p <Q x ) )
60	59	abbidv 2324	. . . . . . . . 9 |- ( u = x -> { p | p <Q u } = { p | p <Q x } )
61		breq1 4057	. . . . . . . . . 10 |- ( u = x -> ( u <Q q <-> x <Q q ) )
62	61	abbidv 2324	. . . . . . . . 9 |- ( u = x -> { q | u <Q q } = { q | x <Q q } )
63	60, 62	opeq12d 3836	. . . . . . . 8 |- ( u = x -> <. { p | p <Q u } , { q | u <Q q } >. = <. { p | p <Q x } , { q | x <Q q } >. )
64	63	breq2d 4066	. . . . . . 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 2508	. . . . . 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 5597	. . . . . . 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 7506	. . . . . . . . 9 |- Q. e. _V
69	68	rabex 4199	. . . . . . . 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 4199	. . . . . . . 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 6251	. . . . . . 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 2227	. . . . . 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 2936	. . . . 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 417	. . . 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 540	. . . 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 2556	. . . 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 1248	. . 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 7847	. . . . 5 |- ( ph -> L e. P. )
81	80	adantr 276	. . . 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 276	. . . 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 7649	. . . 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 411	. . 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 167	. 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 2629	1 |- ( ph -> L <P ( ( F ` K ) +P. Q ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2177   {cab 2192   A.wral 2485   E.wrex 2486   {crab 2489   <.cop 3641   class class class wbr 4054   -->wf 5281   ` cfv 5285  (class class class)co 5962   1stc1st 6242   2ndc2nd 6243   1oc1o 6513   [cec 6636   N.cnpi 7415   <N clti 7418   ~Q ceq 7422   Q.cnq 7423   +Q cplq 7425   *Qcrq 7427   <Q cltq 7428   P.cnp 7434   +P. cpp 7436   <P cltp 7438

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-eprel 4349  df-id 4353  df-po 4356  df-iso 4357  df-iord 4426  df-on 4428  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-recs 6409  df-irdg 6474  df-1o 6520  df-2o 6521  df-oadd 6524  df-omul 6525  df-er 6638  df-ec 6640  df-qs 6644  df-ni 7447  df-pli 7448  df-mi 7449  df-lti 7450  df-plpq 7487  df-mpq 7488  df-enq 7490  df-nqqs 7491  df-plqqs 7492  df-mqqs 7493  df-1nqqs 7494  df-rq 7495  df-ltnqqs 7496  df-enq0 7567  df-nq0 7568  df-0nq0 7569  df-plq0 7570  df-mq0 7571  df-inp 7609  df-iplp 7611  df-iltp 7613

This theorem is referenced by:  caucvgprprlemlim  7854