Theorem caucvgprlemladdrl 7434

Description: Lemma for caucvgpr 7438. 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 3671	. . . . . . . . 9 |- ( j = a -> <. j , 1o >. = <. a , 1o >. )
2	1	eceq1d 6419	. . . . . . . 8 |- ( j = a -> [ <. j , 1o >. ] ~Q = [ <. a , 1o >. ] ~Q )
3	2	fveq2d 5379	. . . . . . 7 |- ( j = a -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. a , 1o >. ] ~Q ) )
4	3	oveq2d 5744	. . . . . 6 |- ( j = a -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
5		fveq2 5375	. . . . . . 7 |- ( j = a -> ( F ` j ) = ( F ` a ) )
6	5	oveq1d 5743	. . . . . 6 |- ( j = a -> ( ( F ` j ) +Q S ) = ( ( F ` a ) +Q S ) )
7	4, 6	breq12d 3908	. . . . 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 2629	. . . 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 2642	. 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 5735	. . . . . . 7 |- ( l = r -> ( l +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
12	11	breq1d 3905	. . . . . 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 2412	. . . . 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 2809	. . . 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 483	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 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 109	. . . . . . . . . . . . . 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 506	. . . . . . . . . . . . . 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 7423	. . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> F : N. --> Q. )
23	22	ffvelrnda 5509	. . . . . . . . . . . . . . . . 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 7178	. . . . . . . . . . . . . . . . . 18 |- ( b e. N. -> [ <. b , 1o >. ] ~Q e. Q. )
25		recclnq 7148	. . . . . . . . . . . . . . . . . 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 7131	. . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . 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 7178	. . . . . . . . . . . . . . . . 17 |- ( a e. N. -> [ <. a , 1o >. ] ~Q e. Q. )
30		recclnq 7148	. . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . 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 7138	. . . . . . . . . . . . . . . 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 1199	. . . . . . . . . . . . . . 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 3905	. . . . . . . . . . . . . 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 7156	. . . . . . . . . . . . . . . 16 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
38	37	adantl 273	. . . . . . . . . . . . . . 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 7131	. . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . 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	ffvelrnd 5510	. . . . . . . . . . . . . . 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 7137	. . . . . . . . . . . . . . . 16 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
43	42	adantl 273	. . . . . . . . . . . . . . 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 5894	. . . . . . . . . . . . . 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 7137	. . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . 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 5744	. . . . . . . . . . . . . . 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 3905	. . . . . . . . . . . . . 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 220	. . . . . . . . . . . . 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 645	. . . . . . . . . . . 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 2499	. . . . . . . . . . 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 899	. . . . . . . . . 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 481	. . . . . . . . . . . . 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 5375	. . . . . . . . . . . . . . . . 17 |- ( j = b -> ( F ` j ) = ( F ` b ) )
56	55	breq2d 3907	. . . . . . . . . . . . . . . 16 |- ( j = b -> ( A <Q ( F ` j ) <-> A <Q ( F ` b ) ) )
57	56	cbvralv 2628	. . . . . . . . . . . . . . 15 |- ( A. j e. N. A <Q ( F ` j ) <-> A. b e. N. A <Q ( F ` b ) )
58	54, 57	sylib 121	. . . . . . . . . . . . . 14 |- ( ph -> A. b e. N. A <Q ( F ` b ) )
59	58	ad3antrrr 481	. . . . . . . . . . . . 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 3671	. . . . . . . . . . . . . . . . . . . . . 22 |- ( j = b -> <. j , 1o >. = <. b , 1o >. )
62	61	eceq1d 6419	. . . . . . . . . . . . . . . . . . . . 21 |- ( j = b -> [ <. j , 1o >. ] ~Q = [ <. b , 1o >. ] ~Q )
63	62	fveq2d 5379	. . . . . . . . . . . . . . . . . . . 20 |- ( j = b -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. b , 1o >. ] ~Q ) )
64	63	oveq2d 5744	. . . . . . . . . . . . . . . . . . 19 |- ( j = b -> ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( l +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
65	64, 55	breq12d 3908	. . . . . . . . . . . . . . . . . 18 |- ( j = b -> ( ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) <-> ( l +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q ( F ` b ) ) )
66	65	cbvrexv 2629	. . . . . . . . . . . . . . . . 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 2642	. . . . . . . . . . . . . . 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 5746	. . . . . . . . . . . . . . . . . . 19 |- ( j = b -> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
70	69	breq1d 3905	. . . . . . . . . . . . . . . . . 18 |- ( j = b -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` b ) +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q u ) )
71	70	cbvrexv 2629	. . . . . . . . . . . . . . . . 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 2642	. . . . . . . . . . . . . . 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 3676	. . . . . . . . . . . . . 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 2135	. . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 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 478	. . . . . . . . . . . . . 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 7131	. . . . . . . . . . . . . 14 |- ( ( S e. Q. /\ ( *Q ` [ <. a , 1o >. ] ~Q ) e. Q. ) -> ( S +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) e. Q. )
80	76, 78, 79	syl2anc 406	. . . . . . . . . . . . 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 7433	. . . . . . . . . . . 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 3062	. . . . . . . . . . 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 3899	. . . . . . . . . . . . 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 2412	. . . . . . . . . . . 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 2809	. . . . . . . . . . 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	syl6ib 160	. . . . . . . . . 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 635	. . . . . . . . 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 7432	. . . . . . . . . . . 12 |- ( ph -> L e. P. )
89	88	ad3antrrr 481	. . . . . . . . . . 11 |- ( ( ( ( ph /\ r e. Q. ) /\ a e. N. ) /\ ( r +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q ( ( F ` a ) +Q S ) ) -> L e. P. )
90		nqprlu 7303	. . . . . . . . . . . 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 7293	. . . . . . . . . . 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 406	. . . . . . . . . 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 7231	. . . . . . . . . . 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 7247	. . . . . . . . . . 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 279	. . . . . . . . . 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 414	. . . . . . . . 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 1310	. . . . . . . 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 506	. . . . . . . . 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 7400	. . . . . . . 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 146	. . . . . . 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 114	. . . . . 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 2523	. . . . 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 358	. . . 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	syl5bi 151	. . 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 3069	. 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 3109	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 103   <-> wb 104   \/ wo 680   /\ w3a 945   = wceq 1314   e. wcel 1463   {cab 2101   A.wral 2390   E.wrex 2391   {crab 2394   C_ wss 3037   <.cop 3496   class class class wbr 3895   -->wf 5077   ` cfv 5081  (class class class)co 5728   1stc1st 5990   2ndc2nd 5991   1oc1o 6260   [cec 6381   N.cnpi 7028   <N clti 7031   ~Q ceq 7035   Q.cnq 7036   +Q cplq 7038   *Qcrq 7040   <Q cltq 7041   P.cnp 7047   +P. cpp 7049

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-eprel 4171  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-irdg 6221  df-1o 6267  df-2o 6268  df-oadd 6271  df-omul 6272  df-er 6383  df-ec 6385  df-qs 6389  df-ni 7060  df-pli 7061  df-mi 7062  df-lti 7063  df-plpq 7100  df-mpq 7101  df-enq 7103  df-nqqs 7104  df-plqqs 7105  df-mqqs 7106  df-1nqqs 7107  df-rq 7108  df-ltnqqs 7109  df-enq0 7180  df-nq0 7181  df-0nq0 7182  df-plq0 7183  df-mq0 7184  df-inp 7222  df-iplp 7224  df-iltp 7226

This theorem is referenced by:  caucvgprlem1  7435