Theorem caucvgprprlemexb 6897

Description: Lemma for caucvgprpr 6902. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 15-Jun-2021.)

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 } >. } >.
caucvgprprlemexb.q	|- ( ph -> Q e. P. )
caucvgprprlemexb.r	|- ( ph -> R e. N. )

Assertion

Ref

Expression

caucvgprprlemexb

|- ( ph -> ( ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` R ) +P. Q ) -> E. b e. N. ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. ( Q +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) ) <P ( ( F ` R ) +P. Q ) ) )

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

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

Proof of Theorem caucvgprprlemexb

Dummy variables f g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgprpr.f	. . . . . 6 |- ( ph -> F : N. --> P. )
2		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 } >. ) ) ) )
3		caucvgprpr.bnd	. . . . . 6 |- ( ph -> A. m e. N. A <P ( F ` m ) )
4		caucvgprpr.lim	. . . . . 6 |- 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 } >. } >.
5	1, 2, 3, 4	caucvgprprlemclphr 6895	. . . . 5 |- ( ph -> L e. P. )
6		caucvgprprlemexb.r	. . . . . 6 |- ( ph -> R e. N. )
7		recnnpr 6738	. . . . . 6 |- ( R e. N. -> <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. e. P. )
8	6, 7	syl 14	. . . . 5 |- ( ph -> <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. e. P. )
9		addclpr 6727	. . . . 5 |- ( ( L e. P. /\ <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. e. P. ) -> ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) e. P. )
10	5, 8, 9	syl2anc 403	. . . 4 |- ( ph -> ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) e. P. )
11	1, 6	ffvelrnd 5324	. . . 4 |- ( ph -> ( F ` R ) e. P. )
12		caucvgprprlemexb.q	. . . 4 |- ( ph -> Q e. P. )
13		ltaprg 6809	. . . 4 |- ( ( ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) e. P. /\ ( F ` R ) e. P. /\ Q e. P. ) -> ( ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) <-> ( Q +P. ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) ) <P ( Q +P. ( F ` R ) ) ) )
14	10, 11, 12, 13	syl3anc 1169	. . 3 |- ( ph -> ( ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) <-> ( Q +P. ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) ) <P ( Q +P. ( F ` R ) ) ) )
15		addassprg 6769	. . . . . 6 |- ( ( Q e. P. /\ L e. P. /\ <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. e. P. ) -> ( ( Q +P. L ) +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) = ( Q +P. ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) ) )
16	12, 5, 8, 15	syl3anc 1169	. . . . 5 |- ( ph -> ( ( Q +P. L ) +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) = ( Q +P. ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) ) )
17		addcomprg 6768	. . . . . . 7 |- ( ( Q e. P. /\ L e. P. ) -> ( Q +P. L ) = ( L +P. Q ) )
18	12, 5, 17	syl2anc 403	. . . . . 6 |- ( ph -> ( Q +P. L ) = ( L +P. Q ) )
19	18	oveq1d 5547	. . . . 5 |- ( ph -> ( ( Q +P. L ) +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) = ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) )
20	16, 19	eqtr3d 2115	. . . 4 |- ( ph -> ( Q +P. ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) ) = ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) )
21		addcomprg 6768	. . . . 5 |- ( ( Q e. P. /\ ( F ` R ) e. P. ) -> ( Q +P. ( F ` R ) ) = ( ( F ` R ) +P. Q ) )
22	12, 11, 21	syl2anc 403	. . . 4 |- ( ph -> ( Q +P. ( F ` R ) ) = ( ( F ` R ) +P. Q ) )
23	20, 22	breq12d 3798	. . 3 |- ( ph -> ( ( Q +P. ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) ) <P ( Q +P. ( F ` R ) ) <-> ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` R ) +P. Q ) ) )
24	14, 23	bitrd 186	. 2 |- ( ph -> ( ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) <-> ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` R ) +P. Q ) ) )
25	1	adantr 270	. . . . 5 |- ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) -> F : N. --> P. )
26	2	adantr 270	. . . . 5 |- ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) -> 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 } >. ) ) ) )
27	3	adantr 270	. . . . 5 |- ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) -> A. m e. N. A <P ( F ` m ) )
28		nnnq 6612	. . . . . . 7 |- ( R e. N. -> [ <. R , 1o >. ] ~Q e. Q. )
29		recclnq 6582	. . . . . . 7 |- ( [ <. R , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. R , 1o >. ] ~Q ) e. Q. )
30	6, 28, 29	3syl 17	. . . . . 6 |- ( ph -> ( *Q ` [ <. R , 1o >. ] ~Q ) e. Q. )
31	30	adantr 270	. . . . 5 |- ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) -> ( *Q ` [ <. R , 1o >. ] ~Q ) e. Q. )
32	11	adantr 270	. . . . 5 |- ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) -> ( F ` R ) e. P. )
33		simpr 108	. . . . 5 |- ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) -> ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) )
34	25, 26, 27, 4, 31, 32, 33	caucvgprprlemexbt 6896	. . . 4 |- ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) -> 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 ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) )
35		ltaprg 6809	. . . . . . . 8 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
36	35	adantl 271	. . . . . . 7 |- ( ( ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) /\ b e. N. ) /\ ( f e. P. /\ g e. P. /\ h e. P. ) ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
37	25	ffvelrnda 5323	. . . . . . . . 9 |- ( ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) /\ b e. N. ) -> ( F ` b ) e. P. )
38		recnnpr 6738	. . . . . . . . . 10 |- ( b e. N. -> <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. e. P. )
39	38	adantl 271	. . . . . . . . 9 |- ( ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) /\ b e. N. ) -> <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. e. P. )
40		addclpr 6727	. . . . . . . . 9 |- ( ( ( 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. )
41	37, 39, 40	syl2anc 403	. . . . . . . 8 |- ( ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) /\ b e. N. ) -> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) e. P. )
42	6	ad2antrr 471	. . . . . . . . 9 |- ( ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) /\ b e. N. ) -> R e. N. )
43	42, 7	syl 14	. . . . . . . 8 |- ( ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) /\ b e. N. ) -> <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. e. P. )
44		addclpr 6727	. . . . . . . 8 |- ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) e. P. /\ <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <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 ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) e. P. )
45	41, 43, 44	syl2anc 403	. . . . . . 7 |- ( ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) /\ b e. N. ) -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) e. P. )
46	11	ad2antrr 471	. . . . . . 7 |- ( ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) /\ b e. N. ) -> ( F ` R ) e. P. )
47	12	ad2antrr 471	. . . . . . 7 |- ( ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) /\ b e. N. ) -> Q e. P. )
48		addcomprg 6768	. . . . . . . 8 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
49	48	adantl 271	. . . . . . 7 |- ( ( ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) /\ b e. N. ) /\ ( f e. P. /\ g e. P. ) ) -> ( f +P. g ) = ( g +P. f ) )
50	36, 45, 46, 47, 49	caovord2d 5690	. . . . . 6 |- ( ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) /\ b e. N. ) -> ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) <-> ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) +P. Q ) <P ( ( F ` R ) +P. Q ) ) )
51		addassprg 6769	. . . . . . . 8 |- ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) e. P. /\ <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. e. P. /\ Q e. P. ) -> ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) +P. Q ) = ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. ( <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. +P. Q ) ) )
52	41, 43, 47, 51	syl3anc 1169	. . . . . . 7 |- ( ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) /\ b e. N. ) -> ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) +P. Q ) = ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. ( <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. +P. Q ) ) )
53	52	breq1d 3795	. . . . . 6 |- ( ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) /\ b e. N. ) -> ( ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) +P. Q ) <P ( ( F ` R ) +P. Q ) <-> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. ( <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. +P. Q ) ) <P ( ( F ` R ) +P. Q ) ) )
54		addcomprg 6768	. . . . . . . . 9 |- ( ( <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. e. P. /\ Q e. P. ) -> ( <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. +P. Q ) = ( Q +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) )
55	43, 47, 54	syl2anc 403	. . . . . . . 8 |- ( ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) /\ b e. N. ) -> ( <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. +P. Q ) = ( Q +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) )
56	55	oveq2d 5548	. . . . . . 7 |- ( ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) /\ b e. N. ) -> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. ( <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. +P. Q ) ) = ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. ( Q +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) ) )
57	56	breq1d 3795	. . . . . 6 |- ( ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) /\ b e. N. ) -> ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. ( <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. +P. Q ) ) <P ( ( F ` R ) +P. Q ) <-> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. ( Q +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) ) <P ( ( F ` R ) +P. Q ) ) )
58	50, 53, 57	3bitrd 212	. . . . 5 |- ( ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) /\ b e. N. ) -> ( ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) <-> ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. ( Q +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) ) <P ( ( F ` R ) +P. Q ) ) )
59	58	rexbidva 2365	. . . 4 |- ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) -> ( 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 ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) <-> E. b e. N. ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. ( Q +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) ) <P ( ( F ` R ) +P. Q ) ) )
60	34, 59	mpbid 145	. . 3 |- ( ( ph /\ ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) ) -> E. b e. N. ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. ( Q +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) ) <P ( ( F ` R ) +P. Q ) )
61	60	ex 113	. 2 |- ( ph -> ( ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( F ` R ) -> E. b e. N. ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. ( Q +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) ) <P ( ( F ` R ) +P. Q ) ) )
62	24, 61	sylbird 168	1 |- ( ph -> ( ( ( L +P. Q ) +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) <P ( ( F ` R ) +P. Q ) -> E. b e. N. ( ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) +P. ( Q +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) ) <P ( ( F ` R ) +P. Q ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = 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   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:  caucvgprprlemaddq  6898