Theorem caucvgprlemladdrl 7740

Description: Lemma for caucvgpr 7744. Adding S after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 8-Oct-2020.)

Hypotheses

Ref	Expression
caucvgpr.f	|- ( ph -> F : N. --> Q. )
caucvgpr.cau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )
caucvgpr.bnd	|- ( ph -> A. j e. N. A <Q ( F ` j ) )
caucvgpr.lim	|- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.
caucvgprlemladd.s	|- ( ph -> S e. Q. )

Assertion

Ref	Expression
caucvgprlemladdrl	|- ( ph -> { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q S ) } C_ ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) )

Distinct variable groups:   A, j   j, F, u, l   n, F, k   k, L, j   S, l, u, j   j, k, S

Allowed substitution hints:   ph( u, j, k, n, l)   A( u, k, n, l)   S( n)   L( u, n, l)

Proof of Theorem caucvgprlemladdrl

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

Step	Hyp	Ref	Expression
1		opeq1 3805	. . . . . . . . 9 |- ( j = a -> <. j , 1o >. = <. a , 1o >. )
2	1	eceq1d 6625	. . . . . . . 8 |- ( j = a -> [ <. j , 1o >. ] ~Q = [ <. a , 1o >. ] ~Q )
3	2	fveq2d 5559	. . . . . . 7 |- ( j = a -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. a , 1o >. ] ~Q ) )
4	3	oveq2d 5935	. . . . . 6 |- ( j = a -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
5		fveq2 5555	. . . . . . 7 |- ( j = a -> ( F ` j ) = ( F ` a ) )
6	5	oveq1d 5934	. . . . . 6 |- ( j = a -> ( ( F ` j ) +Q S ) = ( ( F ` a ) +Q S ) )
7	4, 6	breq12d 4043	. . . . 5 |- ( j = a -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q S ) <-> ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) )
8	7	cbvrexv 2727	. . . 4 |- ( E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q S ) <-> E. a e. N. ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) )
9	8	a1i 9	. . 3 |- ( l e. Q. -> ( E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q S ) <-> E. a e. N. ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) )
10	9	rabbiia 2745	. 2 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q S ) } = { l e. Q. | E. a e. N. ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) }
11		oveq1 5926	. . . . . . 7 |- ( l = r -> ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
12	11	breq1d 4040	. . . . . 6 |- ( l = r -> ( ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) <-> ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) )
13	12	rexbidv 2495	. . . . 5 |- ( l = r -> ( E. a e. N. ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) <-> E. a e. N. ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) )
14	13	elrab 2917	. . . 4 |- ( r e. { l e. Q. | E. a e. N. ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) } <-> ( r e. Q. /\ E. a e. N. ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) )
15		caucvgpr.f	. . . . . . . . . . . . . . 15 |- ( ph -> F : N. --> Q. )
16	15	ad4antr 494	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) -> F : N. --> Q. )
17		caucvgpr.cau	. . . . . . . . . . . . . . 15 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )
18	17	ad4antr 494	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )
19		simpr 110	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) -> b e. N. )
20		simpllr 534	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) -> a e. N. )
21	16, 18, 19, 20	caucvgprlemnbj 7729	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) -> -. ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( F ` a ) )
22	15	ad3antrrr 492	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> F : N. --> Q. )
23	22	ffvelcdmda 5694	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) -> ( F ` b ) e. Q. )
24		nnnq 7484	. . . . . . . . . . . . . . . . . 18 |- ( b e. N. -> [ <. b , 1o >. ] ~Q e. Q. )
25		recclnq 7454	. . . . . . . . . . . . . . . . . 18 |- ( [ <. b , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. b , 1o >. ] ~Q ) e. Q. )
26	19, 24, 25	3syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) -> ( *Q ` [ <. b , 1o >. ] ~Q ) e. Q. )
27		addclnq 7437	. . . . . . . . . . . . . . . . 17 |- ( ( ( F ` b ) e. Q. /\ ( *Q ` [ <. b , 1o >. ] ~Q ) e. Q. ) -> ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) e. Q. )
28	23, 26, 27	syl2anc 411	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) -> ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) e. Q. )
29		nnnq 7484	. . . . . . . . . . . . . . . . 17 |- ( a e. N. -> [ <. a , 1o >. ] ~Q e. Q. )
30		recclnq 7454	. . . . . . . . . . . . . . . . 17 |- ( [ <. a , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. a , 1o >. ] ~Q ) e. Q. )
31	20, 29, 30	3syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) -> ( *Q ` [ <. a , 1o >. ] ~Q ) e. Q. )
32		caucvgprlemladd.s	. . . . . . . . . . . . . . . . 17 |- ( ph -> S e. Q. )
33	32	ad4antr 494	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) -> S e. Q. )
34		addassnqg 7444	. . . . . . . . . . . . . . . 16 |- ( ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) e. Q. /\ ( *Q ` [ <. a , 1o >. ] ~Q ) e. Q. /\ S e. Q. ) -> ( ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) +Q S ) = ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( ( *Q ` [ <. a , 1o >. ] ~Q ) +Q S ) ) )
35	28, 31, 33, 34	syl3anc 1249	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) -> ( ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) +Q S ) = ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( ( *Q ` [ <. a , 1o >. ] ~Q ) +Q S ) ) )
36	35	breq1d 4040	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) -> ( ( ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) +Q S ) <Q ( ( F ` a ) +Q S ) <-> ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( ( *Q ` [ <. a , 1o >. ] ~Q ) +Q S ) ) <Q ( ( F ` a ) +Q S ) ) )
37		ltanqg 7462	. . . . . . . . . . . . . . . 16 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
38	37	adantl 277	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
39		addclnq 7437	. . . . . . . . . . . . . . . 16 |- ( ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) e. Q. /\ ( *Q ` [ <. a , 1o >. ] ~Q ) e. Q. ) -> ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) e. Q. )
40	28, 31, 39	syl2anc 411	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) -> ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) e. Q. )
41	16, 20	ffvelcdmd 5695	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) -> ( F ` a ) e. Q. )
42		addcomnqg 7443	. . . . . . . . . . . . . . . 16 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
43	42	adantl 277	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
44	38, 40, 41, 33, 43	caovord2d 6090	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) -> ( ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( F ` a ) <-> ( ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) +Q S ) <Q ( ( F ` a ) +Q S ) ) )
45		addcomnqg 7443	. . . . . . . . . . . . . . . . 17 |- ( ( S e. Q. /\ ( *Q ` [ <. a , 1o >. ] ~Q ) e. Q. ) -> ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( ( *Q ` [ <. a , 1o >. ] ~Q ) +Q S ) )
46	33, 31, 45	syl2anc 411	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) -> ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( ( *Q ` [ <. a , 1o >. ] ~Q ) +Q S ) )
47	46	oveq2d 5935	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) -> ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) = ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( ( *Q ` [ <. a , 1o >. ] ~Q ) +Q S ) ) )
48	47	breq1d 4040	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) -> ( ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) <Q ( ( F ` a ) +Q S ) <-> ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( ( *Q ` [ <. a , 1o >. ] ~Q ) +Q S ) ) <Q ( ( F ` a ) +Q S ) ) )
49	36, 44, 48	3bitr4rd 221	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) -> ( ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) <Q ( ( F ` a ) +Q S ) <-> ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( F ` a ) ) )
50	21, 49	mtbird 674	. . . . . . . . . . . 12 |- ( ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) /\ b e. N. ) -> -. ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) <Q ( ( F ` a ) +Q S ) )
51	50	nrexdv 2587	. . . . . . . . . . 11 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> -. E. b e. N. ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) <Q ( ( F ` a ) +Q S ) )
52	51	intnand 932	. . . . . . . . . 10 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> -. ( ( ( F ` a ) +Q S ) e. Q. /\ E. b e. N. ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) <Q ( ( F ` a ) +Q S ) ) )
53	17	ad3antrrr 492	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) )
54		caucvgpr.bnd	. . . . . . . . . . . . . . 15 |- ( ph -> A. j e. N. A <Q ( F ` j ) )
55		fveq2 5555	. . . . . . . . . . . . . . . . 17 |- ( j = b -> ( F ` j ) = ( F ` b ) )
56	55	breq2d 4042	. . . . . . . . . . . . . . . 16 |- ( j = b -> ( A <Q ( F ` j ) <-> A <Q ( F ` b ) ) )
57	56	cbvralv 2726	. . . . . . . . . . . . . . 15 |- ( A. j e. N. A <Q ( F ` j ) <-> A. b e. N. A <Q ( F ` b ) )
58	54, 57	sylib 122	. . . . . . . . . . . . . 14 |- ( ph -> A. b e. N. A <Q ( F ` b ) )
59	58	ad3antrrr 492	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> A. b e. N. A <Q ( F ` b ) )
60		caucvgpr.lim	. . . . . . . . . . . . . 14 |- L = <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >.
61		opeq1 3805	. . . . . . . . . . . . . . . . . . . . . 22 |- ( j = b -> <. j , 1o >. = <. b , 1o >. )
62	61	eceq1d 6625	. . . . . . . . . . . . . . . . . . . . 21 |- ( j = b -> [ <. j , 1o >. ] ~Q = [ <. b , 1o >. ] ~Q )
63	62	fveq2d 5559	. . . . . . . . . . . . . . . . . . . 20 |- ( j = b -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. b , 1o >. ] ~Q ) )
64	63	oveq2d 5935	. . . . . . . . . . . . . . . . . . 19 |- ( j = b -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( l +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
65	64, 55	breq12d 4043	. . . . . . . . . . . . . . . . . 18 |- ( j = b -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( l +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q ( F ` b ) ) )
66	65	cbvrexv 2727	. . . . . . . . . . . . . . . . 17 |- ( E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> E. b e. N. ( l +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q ( F ` b ) )
67	66	a1i 9	. . . . . . . . . . . . . . . 16 |- ( l e. Q. -> ( E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> E. b e. N. ( l +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q ( F ` b ) ) )
68	67	rabbiia 2745	. . . . . . . . . . . . . . 15 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } = { l e. Q. | E. b e. N. ( l +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q ( F ` b ) }
69	55, 63	oveq12d 5937	. . . . . . . . . . . . . . . . . . 19 |- ( j = b -> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
70	69	breq1d 4040	. . . . . . . . . . . . . . . . . 18 |- ( j = b -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q u ) )
71	70	cbvrexv 2727	. . . . . . . . . . . . . . . . 17 |- ( E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> E. b e. N. ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q u )
72	71	a1i 9	. . . . . . . . . . . . . . . 16 |- ( u e. Q. -> ( E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> E. b e. N. ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q u ) )
73	72	rabbiia 2745	. . . . . . . . . . . . . . 15 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } = { u e. Q. | E. b e. N. ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q u }
74	68, 73	opeq12i 3810	. . . . . . . . . . . . . 14 |- <. { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } , { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } >. = <. { l e. Q. | E. b e. N. ( l +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q ( F ` b ) } , { u e. Q. | E. b e. N. ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q u } >.
75	60, 74	eqtri 2214	. . . . . . . . . . . . 13 |- L = <. { l e. Q. | E. b e. N. ( l +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q ( F ` b ) } , { u e. Q. | E. b e. N. ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q u } >.
76	32	ad3antrrr 492	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> S e. Q. )
77	29, 30	syl 14	. . . . . . . . . . . . . . 15 |- ( a e. N. -> ( *Q ` [ <. a , 1o >. ] ~Q ) e. Q. )
78	77	ad2antlr 489	. . . . . . . . . . . . . 14 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> ( *Q ` [ <. a , 1o >. ] ~Q ) e. Q. )
79		addclnq 7437	. . . . . . . . . . . . . 14 |- ( ( S e. Q. /\ ( *Q ` [ <. a , 1o >. ] ~Q ) e. Q. ) -> ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) e. Q. )
80	76, 78, 79	syl2anc 411	. . . . . . . . . . . . 13 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) e. Q. )
81	22, 53, 59, 75, 80	caucvgprlemladdfu 7739	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> ( 2nd ` ( L +P. <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. ) ) C_ { u e. Q. | E. b e. N. ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) <Q u } )
82	81	sseld 3179	. . . . . . . . . . 11 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> ( ( ( F ` a ) +Q S ) e. ( 2nd ` ( L +P. <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. ) ) -> ( ( F ` a ) +Q S ) e. { u e. Q. | E. b e. N. ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) <Q u } ) )
83		breq2 4034	. . . . . . . . . . . . 13 |- ( u = ( ( F ` a ) +Q S ) -> ( ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) <Q u <-> ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) <Q ( ( F ` a ) +Q S ) ) )
84	83	rexbidv 2495	. . . . . . . . . . . 12 |- ( u = ( ( F ` a ) +Q S ) -> ( E. b e. N. ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) <Q u <-> E. b e. N. ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) <Q ( ( F ` a ) +Q S ) ) )
85	84	elrab 2917	. . . . . . . . . . 11 |- ( ( ( F ` a ) +Q S ) e. { u e. Q. | E. b e. N. ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) <Q u } <-> ( ( ( F ` a ) +Q S ) e. Q. /\ E. b e. N. ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) <Q ( ( F ` a ) +Q S ) ) )
86	82, 85	imbitrdi 161	. . . . . . . . . 10 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> ( ( ( F ` a ) +Q S ) e. ( 2nd ` ( L +P. <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. ) ) -> ( ( ( F ` a ) +Q S ) e. Q. /\ E. b e. N. ( ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) +Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) <Q ( ( F ` a ) +Q S ) ) ) )
87	52, 86	mtod 664	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> -. ( ( F ` a ) +Q S ) e. ( 2nd ` ( L +P. <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. ) ) )
88	15, 17, 54, 60	caucvgprlemcl 7738	. . . . . . . . . . . 12 |- ( ph -> L e. P. )
89	88	ad3antrrr 492	. . . . . . . . . . 11 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> L e. P. )
90		nqprlu 7609	. . . . . . . . . . . 12 |- ( ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) e. Q. -> <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. e. P. )
91	80, 90	syl 14	. . . . . . . . . . 11 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. e. P. )
92		addclpr 7599	. . . . . . . . . . 11 |- ( ( L e. P. /\ <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. e. P. ) -> ( L +P. <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. ) e. P. )
93	89, 91, 92	syl2anc 411	. . . . . . . . . 10 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> ( L +P. <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. ) e. P. )
94		prop 7537	. . . . . . . . . . 11 |- ( ( L +P. <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. ) e. P. -> <. ( 1st ` ( L +P. <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. ) ) , ( 2nd ` ( L +P. <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. ) ) >. e. P. )
95		prloc 7553	. . . . . . . . . . 11 |- ( ( <. ( 1st ` ( L +P. <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. ) ) , ( 2nd ` ( L +P. <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. ) ) >. e. P. /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> ( ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) e. ( 1st ` ( L +P. <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. ) ) \/ ( ( F ` a ) +Q S ) e. ( 2nd ` ( L +P. <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. ) ) ) )
96	94, 95	sylan 283	. . . . . . . . . 10 |- ( ( ( L +P. <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. ) e. P. /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> ( ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) e. ( 1st ` ( L +P. <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. ) ) \/ ( ( F ` a ) +Q S ) e. ( 2nd ` ( L +P. <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. ) ) ) )
97	93, 96	sylancom 420	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> ( ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) e. ( 1st ` ( L +P. <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. ) ) \/ ( ( F ` a ) +Q S ) e. ( 2nd ` ( L +P. <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. ) ) ) )
98	87, 97	ecased 1360	. . . . . . . 8 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) e. ( 1st ` ( L +P. <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. ) ) )
99		simpllr 534	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> r e. Q. )
100	89, 76, 99, 78	caucvgprlemcanl 7706	. . . . . . . 8 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> ( ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) e. ( 1st ` ( L +P. <. { l | l <Q ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { u | ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q u } >. ) ) <-> r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) )
101	98, 100	mpbid 147	. . . . . . 7 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) )
102	101	ex 115	. . . . . 6 |- ( ( ( ph /\ r e. Q. ) /\ a e. N. ) -> ( ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) -> r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) )
103	102	rexlimdva 2611	. . . . 5 |- ( ( ph /\ r e. Q. ) -> ( E. a e. N. ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) -> r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) )
104	103	expimpd 363	. . . 4 |- ( ph -> ( ( r e. Q. /\ E. a e. N. ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) )
105	14, 104	biimtrid 152	. . 3 |- ( ph -> ( r e. { l e. Q. | E. a e. N. ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) } -> r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) )
106	105	ssrdv 3186	. 2 |- ( ph -> { l e. Q. | E. a e. N. ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) } C_ ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) )
107	10, 106	eqsstrid 3226	1 |- ( ph -> { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( ( F ` j ) +Q S ) } C_ ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2164   {cab 2179   A.wral 2472   E.wrex 2473   {crab 2476   C_ wss 3154   <.cop 3622   class class class wbr 4030   -->wf 5251   ` cfv 5255  (class class class)co 5919   1stc1st 6193   2ndc2nd 6194   1oc1o 6464   [cec 6587   N.cnpi 7334   <N clti 7337   ~Q ceq 7341   Q.cnq 7342   +Q cplq 7344   *Qcrq 7346   <Q cltq 7347   P.cnp 7353   +P. cpp 7355

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-eprel 4321  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-1o 6471  df-2o 6472  df-oadd 6475  df-omul 6476  df-er 6589  df-ec 6591  df-qs 6595  df-ni 7366  df-pli 7367  df-mi 7368  df-lti 7369  df-plpq 7406  df-mpq 7407  df-enq 7409  df-nqqs 7410  df-plqqs 7411  df-mqqs 7412  df-1nqqs 7413  df-rq 7414  df-ltnqqs 7415  df-enq0 7486  df-nq0 7487  df-0nq0 7488  df-plq0 7489  df-mq0 7490  df-inp 7528  df-iplp 7530  df-iltp 7532

This theorem is referenced by:  caucvgprlem1  7741