Theorem caucvgprprlemopu 7500

Description: Lemma for caucvgprpr 7513. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 21-Dec-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 } >. } >.

Assertion

Ref	Expression
caucvgprprlemopu	|- ( ( ph /\ t e. ( 2nd ` L ) ) -> E. s e. Q. ( s <Q t /\ s e. ( 2nd ` L ) ) )

Distinct variable groups:   A, m   m, F   F, l, r, s   u, F, r, s   L, s   p, l, q, t, r, s   u, p, q, t   ph, r, s

Allowed substitution hints:   ph( u, t, k, m, n, q, p, l)   A( u, t, k, n, s, r, q, p, l)   F( t, k, n, q, p)   L( u, t, k, m, n, r, q, p, l)

Proof of Theorem caucvgprprlemopu

Dummy variable b is distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgprpr.lim	. . . . 5 |- 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 } >. } >.
2	1	caucvgprprlemelu 7487	. . . 4 |- ( t e. ( 2nd ` L ) <-> ( t e. Q. /\ E. b e. N. ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) )
3	2	simprbi 273	. . 3 |- ( t e. ( 2nd ` L ) -> E. b e. N. ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. )
4	3	adantl 275	. 2 |- ( ( ph /\ t e. ( 2nd ` L ) ) -> E. b e. N. ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. )
5		simprr 521	. . . . 5 |- ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) -> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. )
6		caucvgprpr.f	. . . . . . . . 9 |- ( ph -> F : N. --> P. )
7	6	ffvelrnda 5548	. . . . . . . 8 |- ( ( ph /\ b e. N. ) -> ( F ` b ) e. P. )
8		recnnpr 7349	. . . . . . . . 9 |- ( b e. N. -> <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. e. P. )
9	8	adantl 275	. . . . . . . 8 |- ( ( ph /\ b e. N. ) -> <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. e. P. )
10		addclpr 7338	. . . . . . . 8 |- ( ( ( F ` b ) e. P. /\ <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. e. P. ) -> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) e. P. )
11	7, 9, 10	syl2anc 408	. . . . . . 7 |- ( ( ph /\ b e. N. ) -> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) e. P. )
12	11	ad2ant2r 500	. . . . . 6 |- ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) -> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) e. P. )
13	2	simplbi 272	. . . . . . . 8 |- ( t e. ( 2nd ` L ) -> t e. Q. )
14	13	ad2antlr 480	. . . . . . 7 |- ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) -> t e. Q. )
15		nqprlu 7348	. . . . . . 7 |- ( t e. Q. -> <. { p | p <Q t } , { q | t <Q q } >. e. P. )
16	14, 15	syl 14	. . . . . 6 |- ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) -> <. { p | p <Q t } , { q | t <Q q } >. e. P. )
17		ltdfpr 7307	. . . . . 6 |- ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) e. P. /\ <. { p | p <Q t } , { q | t <Q q } >. e. P. ) -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. <-> E. s e. Q. ( s e. ( 2nd ` ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) ) /\ s e. ( 1st ` <. { p | p <Q t } , { q | t <Q q } >. ) ) ) )
18	12, 16, 17	syl2anc 408	. . . . 5 |- ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. <-> E. s e. Q. ( s e. ( 2nd ` ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) ) /\ s e. ( 1st ` <. { p | p <Q t } , { q | t <Q q } >. ) ) ) )
19	5, 18	mpbid 146	. . . 4 |- ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) -> E. s e. Q. ( s e. ( 2nd ` ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) ) /\ s e. ( 1st ` <. { p | p <Q t } , { q | t <Q q } >. ) ) )
20		simpr 109	. . . . . . . 8 |- ( ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) /\ s e. Q. ) -> s e. Q. )
21	12	adantr 274	. . . . . . . 8 |- ( ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) /\ s e. Q. ) -> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) e. P. )
22		nqpru 7353	. . . . . . . 8 |- ( ( s e. Q. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) e. P. ) -> ( s e. ( 2nd ` ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) ) <-> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q s } , { q | s <Q q } >. ) )
23	20, 21, 22	syl2anc 408	. . . . . . 7 |- ( ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) /\ s e. Q. ) -> ( s e. ( 2nd ` ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) ) <-> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q s } , { q | s <Q q } >. ) )
24		vex 2684	. . . . . . . . 9 |- s e. _V
25		breq1 3927	. . . . . . . . 9 |- ( p = s -> ( p <Q t <-> s <Q t ) )
26		ltnqex 7350	. . . . . . . . . 10 |- { p | p <Q t } e. _V
27		gtnqex 7351	. . . . . . . . . 10 |- { q | t <Q q } e. _V
28	26, 27	op1st 6037	. . . . . . . . 9 |- ( 1st ` <. { p | p <Q t } , { q | t <Q q } >. ) = { p | p <Q t }
29	24, 25, 28	elab2 2827	. . . . . . . 8 |- ( s e. ( 1st ` <. { p | p <Q t } , { q | t <Q q } >. ) <-> s <Q t )
30	29	a1i 9	. . . . . . 7 |- ( ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) /\ s e. Q. ) -> ( s e. ( 1st ` <. { p | p <Q t } , { q | t <Q q } >. ) <-> s <Q t ) )
31	23, 30	anbi12d 464	. . . . . 6 |- ( ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) /\ s e. Q. ) -> ( ( s e. ( 2nd ` ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) ) /\ s e. ( 1st ` <. { p | p <Q t } , { q | t <Q q } >. ) ) <-> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q s } , { q | s <Q q } >. /\ s <Q t ) ) )
32	31	biimpd 143	. . . . 5 |- ( ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) /\ s e. Q. ) -> ( ( s e. ( 2nd ` ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) ) /\ s e. ( 1st ` <. { p | p <Q t } , { q | t <Q q } >. ) ) -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q s } , { q | s <Q q } >. /\ s <Q t ) ) )
33	32	reximdva 2532	. . . 4 |- ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) -> ( E. s e. Q. ( s e. ( 2nd ` ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) ) /\ s e. ( 1st ` <. { p | p <Q t } , { q | t <Q q } >. ) ) -> E. s e. Q. ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q s } , { q | s <Q q } >. /\ s <Q t ) ) )
34	19, 33	mpd 13	. . 3 |- ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) -> E. s e. Q. ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q s } , { q | s <Q q } >. /\ s <Q t ) )
35		simprr 521	. . . . . 6 |- ( ( ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) /\ s e. Q. ) /\ ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q s } , { q | s <Q q } >. /\ s <Q t ) ) -> s <Q t )
36		simplr 519	. . . . . . 7 |- ( ( ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) /\ s e. Q. ) /\ ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q s } , { q | s <Q q } >. /\ s <Q t ) ) -> s e. Q. )
37		simplrl 524	. . . . . . . . 9 |- ( ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) /\ s e. Q. ) -> b e. N. )
38	37	adantr 274	. . . . . . . 8 |- ( ( ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) /\ s e. Q. ) /\ ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q s } , { q | s <Q q } >. /\ s <Q t ) ) -> b e. N. )
39		simprl 520	. . . . . . . 8 |- ( ( ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) /\ s e. Q. ) /\ ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q s } , { q | s <Q q } >. /\ s <Q t ) ) -> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q s } , { q | s <Q q } >. )
40		fveq2 5414	. . . . . . . . . . 11 |- ( r = b -> ( F ` r ) = ( F ` b ) )
41		opeq1 3700	. . . . . . . . . . . . . . . 16 |- ( r = b -> <. r , 1o >. = <. b , 1o >. )
42	41	eceq1d 6458	. . . . . . . . . . . . . . 15 |- ( r = b -> [ <. r , 1o >. ] ~Q = [ <. b , 1o >. ] ~Q )
43	42	fveq2d 5418	. . . . . . . . . . . . . 14 |- ( r = b -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. b , 1o >. ] ~Q ) )
44	43	breq2d 3936	. . . . . . . . . . . . 13 |- ( r = b -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
45	44	abbidv 2255	. . . . . . . . . . . 12 |- ( r = b -> { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } )
46	43	breq1d 3934	. . . . . . . . . . . . 13 |- ( r = b -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q ) )
47	46	abbidv 2255	. . . . . . . . . . . 12 |- ( r = b -> { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } )
48	45, 47	opeq12d 3708	. . . . . . . . . . 11 |- ( r = b -> <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. = <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. )
49	40, 48	oveq12d 5785	. . . . . . . . . 10 |- ( r = b -> ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) = ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) )
50	49	breq1d 3934	. . . . . . . . 9 |- ( r = b -> ( ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q s } , { q | s <Q q } >. <-> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q s } , { q | s <Q q } >. ) )
51	50	rspcev 2784	. . . . . . . 8 |- ( ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q s } , { q | s <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 s } , { q | s <Q q } >. )
52	38, 39, 51	syl2anc 408	. . . . . . 7 |- ( ( ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) /\ s e. Q. ) /\ ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q s } , { q | s <Q q } >. /\ s <Q t ) ) -> 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 s } , { q | s <Q q } >. )
53	1	caucvgprprlemelu 7487	. . . . . . 7 |- ( s e. ( 2nd ` L ) <-> ( s 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 s } , { q | s <Q q } >. ) )
54	36, 52, 53	sylanbrc 413	. . . . . 6 |- ( ( ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) /\ s e. Q. ) /\ ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q s } , { q | s <Q q } >. /\ s <Q t ) ) -> s e. ( 2nd ` L ) )
55	35, 54	jca 304	. . . . 5 |- ( ( ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) /\ s e. Q. ) /\ ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q s } , { q | s <Q q } >. /\ s <Q t ) ) -> ( s <Q t /\ s e. ( 2nd ` L ) ) )
56	55	ex 114	. . . 4 |- ( ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) /\ s e. Q. ) -> ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q s } , { q | s <Q q } >. /\ s <Q t ) -> ( s <Q t /\ s e. ( 2nd ` L ) ) ) )
57	56	reximdva 2532	. . 3 |- ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) -> ( E. s e. Q. ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q s } , { q | s <Q q } >. /\ s <Q t ) -> E. s e. Q. ( s <Q t /\ s e. ( 2nd ` L ) ) ) )
58	34, 57	mpd 13	. 2 |- ( ( ( ph /\ t e. ( 2nd ` L ) ) /\ ( b e. N. /\ ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q t } , { q | t <Q q } >. ) ) -> E. s e. Q. ( s <Q t /\ s e. ( 2nd ` L ) ) )
59	4, 58	rexlimddv 2552	1 |- ( ( ph /\ t e. ( 2nd ` L ) ) -> E. s e. Q. ( s <Q t /\ s e. ( 2nd ` L ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   {cab 2123   A.wral 2414   E.wrex 2415   {crab 2418   <.cop 3525   class class class wbr 3924   -->wf 5114   ` cfv 5118  (class class class)co 5767   1stc1st 6029   2ndc2nd 6030   1oc1o 6299   [cec 6420   N.cnpi 7073   <N clti 7076   ~Q ceq 7080   Q.cnq 7081   +Q cplq 7083   *Qcrq 7085   <Q cltq 7086   P.cnp 7092   +P. cpp 7094   <P cltp 7096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-iplp 7269  df-iltp 7271

This theorem is referenced by:  caucvgprprlemrnd  7502