Theorem caucvgprprlemopu 7812

Description: Lemma for caucvgprpr 7825. 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 7799	. . . 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 275	. . 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 277	. 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 531	. . . . 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	ffvelcdmda 5715	. . . . . . . 8 |- ( ( ph /\ b e. N. ) -> ( F ` b ) e. P. )
8		recnnpr 7661	. . . . . . . . 9 |- ( b e. N. -> <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. e. P. )
9	8	adantl 277	. . . . . . . 8 |- ( ( ph /\ b e. N. ) -> <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. e. P. )
10		addclpr 7650	. . . . . . . 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 411	. . . . . . 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 509	. . . . . 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 274	. . . . . . . 8 |- ( t e. ( 2nd ` L ) -> t e. Q. )
14	13	ad2antlr 489	. . . . . . 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 7660	. . . . . . 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 7619	. . . . . 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 411	. . . . 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 147	. . . 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 110	. . . . . . . 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 276	. . . . . . . 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 7665	. . . . . . . 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 411	. . . . . . 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 2775	. . . . . . . . 9 |- s e. _V
25		breq1 4047	. . . . . . . . 9 |- ( p = s -> ( p <Q t <-> s <Q t ) )
26		ltnqex 7662	. . . . . . . . . 10 |- { p | p <Q t } e. _V
27		gtnqex 7663	. . . . . . . . . 10 |- { q | t <Q q } e. _V
28	26, 27	op1st 6232	. . . . . . . . 9 |- ( 1st ` <. { p | p <Q t } , { q | t <Q q } >. ) = { p | p <Q t }
29	24, 25, 28	elab2 2921	. . . . . . . 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 473	. . . . . 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 144	. . . . 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 2608	. . . 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 531	. . . . . 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 528	. . . . . . 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 535	. . . . . . . . 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 276	. . . . . . . 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 529	. . . . . . . 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 5576	. . . . . . . . . . 11 |- ( r = b -> ( F ` r ) = ( F ` b ) )
41		opeq1 3819	. . . . . . . . . . . . . . . 16 |- ( r = b -> <. r , 1o >. = <. b , 1o >. )
42	41	eceq1d 6656	. . . . . . . . . . . . . . 15 |- ( r = b -> [ <. r , 1o >. ] ~Q = [ <. b , 1o >. ] ~Q )
43	42	fveq2d 5580	. . . . . . . . . . . . . 14 |- ( r = b -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. b , 1o >. ] ~Q ) )
44	43	breq2d 4056	. . . . . . . . . . . . 13 |- ( r = b -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
45	44	abbidv 2323	. . . . . . . . . . . 12 |- ( r = b -> { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } )
46	43	breq1d 4054	. . . . . . . . . . . . 13 |- ( r = b -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q ) )
47	46	abbidv 2323	. . . . . . . . . . . 12 |- ( r = b -> { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } )
48	45, 47	opeq12d 3827	. . . . . . . . . . 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 5962	. . . . . . . . . 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 4054	. . . . . . . . 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 2877	. . . . . . . 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 411	. . . . . . 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 7799	. . . . . . 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 417	. . . . . 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 306	. . . . 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 115	. . . 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 2608	. . 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 2628	1 |- ( ( ph /\ t e. ( 2nd ` L ) ) -> E. s e. Q. ( s <Q t /\ s e. ( 2nd ` L ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   {cab 2191   A.wral 2484   E.wrex 2485   {crab 2488   <.cop 3636   class class class wbr 4044   -->wf 5267   ` cfv 5271  (class class class)co 5944   1stc1st 6224   2ndc2nd 6225   1oc1o 6495   [cec 6618   N.cnpi 7385   <N clti 7388   ~Q ceq 7392   Q.cnq 7393   +Q cplq 7395   *Qcrq 7397   <Q cltq 7398   P.cnp 7404   +P. cpp 7406   <P cltp 7408

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-eprel 4336  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-irdg 6456  df-1o 6502  df-2o 6503  df-oadd 6506  df-omul 6507  df-er 6620  df-ec 6622  df-qs 6626  df-ni 7417  df-pli 7418  df-mi 7419  df-lti 7420  df-plpq 7457  df-mpq 7458  df-enq 7460  df-nqqs 7461  df-plqqs 7462  df-mqqs 7463  df-1nqqs 7464  df-rq 7465  df-ltnqqs 7466  df-enq0 7537  df-nq0 7538  df-0nq0 7539  df-plq0 7540  df-mq0 7541  df-inp 7579  df-iplp 7581  df-iltp 7583

This theorem is referenced by:  caucvgprprlemrnd  7814