Theorem caucvgprprlemexb 8080

Description: Lemma for caucvgprpr 8085. 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 8078	. . . . 5 |- ( ph -> L e. P. )
6		caucvgprprlemexb.r	. . . . . 6 |- ( ph -> R e. N. )
7		recnnpr 7921	. . . . . 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 7910	. . . . 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 415	. . . 4 |- ( ph -> ( L +P. <. { p | p <Q ( *Q ` [ <. R , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. R , 1o >. ] ~Q ) <Q q } >. ) e. P. )
11	1, 6	ffvelcdmd 5847	. . . 4 |- ( ph -> ( F ` R ) e. P. )
12		caucvgprprlemexb.q	. . . 4 |- ( ph -> Q e. P. )
13		ltaprg 7992	. . . 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 1278	. . 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 7952	. . . . . 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 1278	. . . . 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 7951	. . . . . . 7 |- ( ( Q e. P. /\ L e. P. ) -> ( Q +P. L ) = ( L +P. Q ) )
18	12, 5, 17	syl2anc 415	. . . . . 6 |- ( ph -> ( Q +P. L ) = ( L +P. Q ) )
19	18	oveq1d 6103	. . . . 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 2273	. . . 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 7951	. . . . 5 |- ( ( Q e. P. /\ ( F ` R ) e. P. ) -> ( Q +P. ( F ` R ) ) = ( ( F ` R ) +P. Q ) )
22	12, 11, 21	syl2anc 415	. . . 4 |- ( ph -> ( Q +P. ( F ` R ) ) = ( ( F ` R ) +P. Q ) )
23	20, 22	breq12d 4144	. . 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 188	. 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 276	. . . . 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 276	. . . . 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 276	. . . . 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 7795	. . . . . . 7 |- ( R e. N. -> [ <. R , 1o >. ] ~Q e. Q. )
29		recclnq 7765	. . . . . . 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 276	. . . . 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 276	. . . . 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 110	. . . . 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 8079	. . . 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 7992	. . . . . . . 8 |- ( ( f e. P. /\ g e. P. /\ h e. P. ) -> ( f <P g <-> ( h +P. f ) <P ( h +P. g ) ) )
36	35	adantl 277	. . . . . . 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	ffvelcdmda 5846	. . . . . . . . 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 7921	. . . . . . . . . 10 |- ( b e. N. -> <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. e. P. )
39	38	adantl 277	. . . . . . . . 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 7910	. . . . . . . . 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 415	. . . . . . . 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 492	. . . . . . . . 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 7910	. . . . . . . 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 415	. . . . . . 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 492	. . . . . . 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 492	. . . . . . 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 7951	. . . . . . . 8 |- ( ( f e. P. /\ g e. P. ) -> ( f +P. g ) = ( g +P. f ) )
49	48	adantl 277	. . . . . . 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 6262	. . . . . 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 7952	. . . . . . . 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 1278	. . . . . . 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 4141	. . . . . 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 7951	. . . . . . . . 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 415	. . . . . . . 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 6104	. . . . . . 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 4141	. . . . . 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 214	. . . . 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 2547	. . . 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 147	. . 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 115	. 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 170	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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = 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   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-2o 6692  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-enq0 7797  df-nq0 7798  df-0nq0 7799  df-plq0 7800  df-mq0 7801  df-inp 7839  df-iplp 7841  df-iltp 7843

This theorem is used by:  caucvgprprlemaddq  8081