Theorem caucvgprprlemml 8067

Description: Lemma for caucvgprpr 8085. The lower cut of the putative limit is inhabited. (Contributed by Jim Kingdon, 29-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
caucvgprprlemml	|- ( ph -> E. s e. Q. s e. ( 1st ` L ) )

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

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

Proof of Theorem caucvgprprlemml

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5698	. . . . . 6 |- ( m = 1o -> ( F ` m ) = ( F ` 1o ) )
2	1	breq2d 4143	. . . . 5 |- ( m = 1o -> ( A <P ( F ` m ) <-> A <P ( F ` 1o ) ) )
3		caucvgprpr.bnd	. . . . 5 |- ( ph -> A. m e. N. A <P ( F ` m ) )
4		1pi 7688	. . . . . 6 |- 1o e. N.
5	4	a1i 9	. . . . 5 |- ( ph -> 1o e. N. )
6	2, 3, 5	rspcdva 2934	. . . 4 |- ( ph -> A <P ( F ` 1o ) )
7		ltrelpr 7878	. . . . . 6 |- <P C_ ( P. X. P. )
8	7	brel 4829	. . . . 5 |- ( A <P ( F ` 1o ) -> ( A e. P. /\ ( F ` 1o ) e. P. ) )
9	8	simpld 112	. . . 4 |- ( A <P ( F ` 1o ) -> A e. P. )
10	6, 9	syl 14	. . 3 |- ( ph -> A e. P. )
11		prop 7848	. . . 4 |- ( A e. P. -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. )
12		prml 7850	. . . 4 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. -> E. x e. Q. x e. ( 1st ` A ) )
13	11, 12	syl 14	. . 3 |- ( A e. P. -> E. x e. Q. x e. ( 1st ` A ) )
14	10, 13	syl 14	. 2 |- ( ph -> E. x e. Q. x e. ( 1st ` A ) )
15		subhalfnqq 7787	. . . 4 |- ( x e. Q. -> E. s e. Q. ( s +Q s ) <Q x )
16	15	ad2antrl 494	. . 3 |- ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) -> E. s e. Q. ( s +Q s ) <Q x )
17		simplr 533	. . . . . 6 |- ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) -> s e. Q. )
18		archrecnq 8036	. . . . . . . 8 |- ( s e. Q. -> E. r e. N. ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s )
19	17, 18	syl 14	. . . . . . 7 |- ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) -> E. r e. N. ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s )
20		simpr 110	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) /\ r e. N. ) /\ ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s ) -> ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s )
21		simplr 533	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) /\ r e. N. ) /\ ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s ) -> r e. N. )
22		nnnq 7795	. . . . . . . . . . . . . . . 16 |- ( r e. N. -> [ <. r , 1o >. ] ~Q e. Q. )
23		recclnq 7765	. . . . . . . . . . . . . . . 16 |- ( [ <. r , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. r , 1o >. ] ~Q ) e. Q. )
24	21, 22, 23	3syl 17	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) /\ r e. N. ) /\ ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s ) -> ( *Q ` [ <. r , 1o >. ] ~Q ) e. Q. )
25	17	ad2antrr 492	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) /\ r e. N. ) /\ ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s ) -> s e. Q. )
26		ltanqg 7773	. . . . . . . . . . . . . . 15 |- ( ( ( *Q ` [ <. r , 1o >. ] ~Q ) e. Q. /\ s e. Q. /\ s e. Q. ) -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s <-> ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q ( s +Q s ) ) )
27	24, 25, 25, 26	syl3anc 1278	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) /\ r e. N. ) /\ ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s ) -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s <-> ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q ( s +Q s ) ) )
28	20, 27	mpbid 147	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) /\ r e. N. ) /\ ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s ) -> ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q ( s +Q s ) )
29		simpllr 540	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) /\ r e. N. ) /\ ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s ) -> ( s +Q s ) <Q x )
30		ltsonq 7771	. . . . . . . . . . . . . 14 |- <Q Or Q.
31		ltrelnq 7738	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
32	30, 31	sotri 5186	. . . . . . . . . . . . 13 |- ( ( ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q ( s +Q s ) /\ ( s +Q s ) <Q x ) -> ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q x )
33	28, 29, 32	syl2anc 415	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) /\ r e. N. ) /\ ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s ) -> ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q x )
34	10	ad5antr 500	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) /\ r e. N. ) /\ ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s ) -> A e. P. )
35		simprr 537	. . . . . . . . . . . . . 14 |- ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) -> x e. ( 1st ` A ) )
36	35	ad4antr 498	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) /\ r e. N. ) /\ ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s ) -> x e. ( 1st ` A ) )
37		prcdnql 7857	. . . . . . . . . . . . . 14 |- ( ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. P. /\ x e. ( 1st ` A ) ) -> ( ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q x -> ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) e. ( 1st ` A ) ) )
38	11, 37	sylan 283	. . . . . . . . . . . . 13 |- ( ( A e. P. /\ x e. ( 1st ` A ) ) -> ( ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q x -> ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) e. ( 1st ` A ) ) )
39	34, 36, 38	syl2anc 415	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) /\ r e. N. ) /\ ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s ) -> ( ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q x -> ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) e. ( 1st ` A ) ) )
40	33, 39	mpd 13	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) /\ r e. N. ) /\ ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s ) -> ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) e. ( 1st ` A ) )
41		addclnq 7748	. . . . . . . . . . . . 13 |- ( ( s e. Q. /\ ( *Q ` [ <. r , 1o >. ] ~Q ) e. Q. ) -> ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) e. Q. )
42	25, 24, 41	syl2anc 415	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) /\ r e. N. ) /\ ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s ) -> ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) e. Q. )
43		nqprl 7924	. . . . . . . . . . . 12 |- ( ( ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) e. Q. /\ A e. P. ) -> ( ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) e. ( 1st ` A ) <-> <. { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P A ) )
44	42, 34, 43	syl2anc 415	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) /\ r e. N. ) /\ ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s ) -> ( ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) e. ( 1st ` A ) <-> <. { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P A ) )
45	40, 44	mpbid 147	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) /\ r e. N. ) /\ ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s ) -> <. { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P A )
46		fveq2 5698	. . . . . . . . . . . 12 |- ( m = r -> ( F ` m ) = ( F ` r ) )
47	46	breq2d 4143	. . . . . . . . . . 11 |- ( m = r -> ( A <P ( F ` m ) <-> A <P ( F ` r ) ) )
48	3	ad5antr 500	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) /\ r e. N. ) /\ ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s ) -> A. m e. N. A <P ( F ` m ) )
49	47, 48, 21	rspcdva 2934	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) /\ r e. N. ) /\ ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s ) -> A <P ( F ` r ) )
50		ltsopr 7969	. . . . . . . . . . 11 |- <P Or P.
51	50, 7	sotri 5186	. . . . . . . . . 10 |- ( ( <. { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P A /\ A <P ( F ` r ) ) -> <. { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) )
52	45, 49, 51	syl2anc 415	. . . . . . . . 9 |- ( ( ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) /\ r e. N. ) /\ ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s ) -> <. { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) )
53	52	ex 115	. . . . . . . 8 |- ( ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) /\ r e. N. ) -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s -> <. { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) ) )
54	53	reximdva 2652	. . . . . . 7 |- ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) -> ( E. r e. N. ( *Q ` [ <. r , 1o >. ] ~Q ) <Q s -> E. r e. N. <. { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) ) )
55	19, 54	mpd 13	. . . . . 6 |- ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) -> E. r e. N. <. { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) )
56		oveq1 6095	. . . . . . . . . . . 12 |- ( l = s -> ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) = ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) )
57	56	breq2d 4143	. . . . . . . . . . 11 |- ( l = s -> ( p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <-> p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) ) )
58	57	abbidv 2358	. . . . . . . . . 10 |- ( l = s -> { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } = { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } )
59	56	breq1d 4141	. . . . . . . . . . 11 |- ( l = s -> ( ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q <-> ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q ) )
60	59	abbidv 2358	. . . . . . . . . 10 |- ( l = s -> { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } = { q | ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } )
61	58, 60	opeq12d 3912	. . . . . . . . 9 |- ( l = s -> <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. = <. { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. )
62	61	breq1d 4141	. . . . . . . 8 |- ( l = s -> ( <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) <-> <. { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) ) )
63	62	rexbidv 2551	. . . . . . 7 |- ( l = s -> ( 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. r e. N. <. { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) ) )
64		caucvgprpr.lim	. . . . . . . . 9 |- 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 } >. } >.
65	64	fveq2i 5701	. . . . . . . 8 |- ( 1st ` L ) = ( 1st ` <. { 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 } >. } >. )
66		nqex 7736	. . . . . . . . . 10 |- Q. e. _V
67	66	rabex 4282	. . . . . . . . 9 |- { 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
68	66	rabex 4282	. . . . . . . . 9 |- { 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
69	67, 68	op1st 6384	. . . . . . . 8 |- ( 1st ` <. { 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 } >. } >. ) = { 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 ) }
70	65, 69	eqtri 2259	. . . . . . 7 |- ( 1st ` 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 ) }
71	63, 70	elrab2 2985	. . . . . 6 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. r e. N. <. { p | p <Q ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( s +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) ) )
72	17, 55, 71	sylanbrc 421	. . . . 5 |- ( ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) /\ ( s +Q s ) <Q x ) -> s e. ( 1st ` L ) )
73	72	ex 115	. . . 4 |- ( ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) /\ s e. Q. ) -> ( ( s +Q s ) <Q x -> s e. ( 1st ` L ) ) )
74	73	reximdva 2652	. . 3 |- ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) -> ( E. s e. Q. ( s +Q s ) <Q x -> E. s e. Q. s e. ( 1st ` L ) ) )
75	16, 74	mpd 13	. 2 |- ( ( ph /\ ( x e. Q. /\ x e. ( 1st ` A ) ) ) -> E. s e. Q. s e. ( 1st ` L ) )
76	14, 75	rexlimddv 2673	1 |- ( ph -> E. s e. Q. s e. ( 1st ` L ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   E.wrex 2529   {crab 2532   <.cop 3712   class class class wbr 4131   -->wf 5376   ` cfv 5380  (class class class)co 6088   1stc1st 6376   2ndc2nd 6377   1oc1o 6684   [cec 6809   N.cnpi 7645   <N clti 7648   ~Q ceq 7652   Q.cnq 7653   +Q cplq 7655   *Qcrq 7657   <Q cltq 7658   P.cnp 7664   +P. cpp 7666   <P cltp 7668

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-pli 7678  df-mi 7679  df-lti 7680  df-plpq 7717  df-mpq 7718  df-enq 7720  df-nqqs 7721  df-plqqs 7722  df-mqqs 7723  df-1nqqs 7724  df-rq 7725  df-ltnqqs 7726  df-inp 7839  df-iltp 7843

This theorem is used by:  caucvgprprlemm  8069