Theorem caucvgprlemnbj 7886

Description: Lemma for caucvgpr 7901. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 18-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 ) ) ) ) )
caucvgprlemnbj.b	|- ( ph -> B e. N. )
caucvgprlemnbj.j	|- ( ph -> J e. N. )

Assertion

Ref	Expression
caucvgprlemnbj	|- ( ph -> -. ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( F ` J ) )

Distinct variable groups:   B, k, n   k, F, n   k, J, n

Allowed substitution hints:   ph( k, n)

Proof of Theorem caucvgprlemnbj

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

Step	Hyp	Ref	Expression
1		caucvgpr.cau	. . . . . . 7 |- ( 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 ) ) ) ) )
2		caucvgprlemnbj.b	. . . . . . . 8 |- ( ph -> B e. N. )
3		caucvgprlemnbj.j	. . . . . . . 8 |- ( ph -> J e. N. )
4		breq1 4091	. . . . . . . . . 10 |- ( n = B -> ( n <N k <-> B <N k ) )
5		fveq2 5639	. . . . . . . . . . . 12 |- ( n = B -> ( F ` n ) = ( F ` B ) )
6		opeq1 3862	. . . . . . . . . . . . . . 15 |- ( n = B -> <. n , 1o >. = <. B , 1o >. )
7	6	eceq1d 6737	. . . . . . . . . . . . . 14 |- ( n = B -> [ <. n , 1o >. ] ~Q = [ <. B , 1o >. ] ~Q )
8	7	fveq2d 5643	. . . . . . . . . . . . 13 |- ( n = B -> ( *Q ` [ <. n , 1o >. ] ~Q ) = ( *Q ` [ <. B , 1o >. ] ~Q ) )
9	8	oveq2d 6033	. . . . . . . . . . . 12 |- ( n = B -> ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) = ( ( F ` k ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) )
10	5, 9	breq12d 4101	. . . . . . . . . . 11 |- ( n = B -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) <-> ( F ` B ) <Q ( ( F ` k ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) ) )
11	5, 8	oveq12d 6035	. . . . . . . . . . . 12 |- ( n = B -> ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) = ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) )
12	11	breq2d 4100	. . . . . . . . . . 11 |- ( n = B -> ( ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) <-> ( F ` k ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) ) )
13	10, 12	anbi12d 473	. . . . . . . . . 10 |- ( n = B -> ( ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) <-> ( ( F ` B ) <Q ( ( F ` k ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) ) ) )
14	4, 13	imbi12d 234	. . . . . . . . 9 |- ( n = B -> ( ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) <-> ( B <N k -> ( ( F ` B ) <Q ( ( F ` k ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) ) ) ) )
15		breq2 4092	. . . . . . . . . 10 |- ( k = J -> ( B <N k <-> B <N J ) )
16		fveq2 5639	. . . . . . . . . . . . 13 |- ( k = J -> ( F ` k ) = ( F ` J ) )
17	16	oveq1d 6032	. . . . . . . . . . . 12 |- ( k = J -> ( ( F ` k ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) = ( ( F ` J ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) )
18	17	breq2d 4100	. . . . . . . . . . 11 |- ( k = J -> ( ( F ` B ) <Q ( ( F ` k ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) <-> ( F ` B ) <Q ( ( F ` J ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) ) )
19	16	breq1d 4098	. . . . . . . . . . 11 |- ( k = J -> ( ( F ` k ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) <-> ( F ` J ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) ) )
20	18, 19	anbi12d 473	. . . . . . . . . 10 |- ( k = J -> ( ( ( F ` B ) <Q ( ( F ` k ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) ) <-> ( ( F ` B ) <Q ( ( F ` J ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) /\ ( F ` J ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) ) ) )
21	15, 20	imbi12d 234	. . . . . . . . 9 |- ( k = J -> ( ( B <N k -> ( ( F ` B ) <Q ( ( F ` k ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) ) ) <-> ( B <N J -> ( ( F ` B ) <Q ( ( F ` J ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) /\ ( F ` J ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) ) ) ) )
22	14, 21	rspc2v 2923	. . . . . . . 8 |- ( ( B e. N. /\ J 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 ) ) ) ) -> ( B <N J -> ( ( F ` B ) <Q ( ( F ` J ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) /\ ( F ` J ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) ) ) ) )
23	2, 3, 22	syl2anc 411	. . . . . . 7 |- ( 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 ) ) ) ) -> ( B <N J -> ( ( F ` B ) <Q ( ( F ` J ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) /\ ( F ` J ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) ) ) ) )
24	1, 23	mpd 13	. . . . . 6 |- ( ph -> ( B <N J -> ( ( F ` B ) <Q ( ( F ` J ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) /\ ( F ` J ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) ) ) )
25	24	imp 124	. . . . 5 |- ( ( ph /\ B <N J ) -> ( ( F ` B ) <Q ( ( F ` J ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) /\ ( F ` J ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) ) )
26	25	simprd 114	. . . 4 |- ( ( ph /\ B <N J ) -> ( F ` J ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) )
27		caucvgpr.f	. . . . . . . 8 |- ( ph -> F : N. --> Q. )
28	27, 2	ffvelcdmd 5783	. . . . . . 7 |- ( ph -> ( F ` B ) e. Q. )
29		nnnq 7641	. . . . . . . 8 |- ( B e. N. -> [ <. B , 1o >. ] ~Q e. Q. )
30		recclnq 7611	. . . . . . . 8 |- ( [ <. B , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. B , 1o >. ] ~Q ) e. Q. )
31	2, 29, 30	3syl 17	. . . . . . 7 |- ( ph -> ( *Q ` [ <. B , 1o >. ] ~Q ) e. Q. )
32		addclnq 7594	. . . . . . 7 |- ( ( ( F ` B ) e. Q. /\ ( *Q ` [ <. B , 1o >. ] ~Q ) e. Q. ) -> ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) e. Q. )
33	28, 31, 32	syl2anc 411	. . . . . 6 |- ( ph -> ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) e. Q. )
34		nnnq 7641	. . . . . . 7 |- ( J e. N. -> [ <. J , 1o >. ] ~Q e. Q. )
35		recclnq 7611	. . . . . . 7 |- ( [ <. J , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. J , 1o >. ] ~Q ) e. Q. )
36	3, 34, 35	3syl 17	. . . . . 6 |- ( ph -> ( *Q ` [ <. J , 1o >. ] ~Q ) e. Q. )
37		ltaddnq 7626	. . . . . 6 |- ( ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) e. Q. /\ ( *Q ` [ <. J , 1o >. ] ~Q ) e. Q. ) -> ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) <Q ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
38	33, 36, 37	syl2anc 411	. . . . 5 |- ( ph -> ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) <Q ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
39	38	adantr 276	. . . 4 |- ( ( ph /\ B <N J ) -> ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) <Q ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
40		ltsonq 7617	. . . . 5 |- <Q Or Q.
41		ltrelnq 7584	. . . . 5 |- <Q C_ ( Q. X. Q. )
42	40, 41	sotri 5132	. . . 4 |- ( ( ( F ` J ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) /\ ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) <Q ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) -> ( F ` J ) <Q ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
43	26, 39, 42	syl2anc 411	. . 3 |- ( ( ph /\ B <N J ) -> ( F ` J ) <Q ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
44		ltaddnq 7626	. . . . . . 7 |- ( ( ( F ` B ) e. Q. /\ ( *Q ` [ <. B , 1o >. ] ~Q ) e. Q. ) -> ( F ` B ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) )
45	28, 31, 44	syl2anc 411	. . . . . 6 |- ( ph -> ( F ` B ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) )
46	45	adantr 276	. . . . 5 |- ( ( ph /\ B = J ) -> ( F ` B ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) )
47		fveq2 5639	. . . . . . 7 |- ( B = J -> ( F ` B ) = ( F ` J ) )
48	47	breq1d 4098	. . . . . 6 |- ( B = J -> ( ( F ` B ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) <-> ( F ` J ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) ) )
49	48	adantl 277	. . . . 5 |- ( ( ph /\ B = J ) -> ( ( F ` B ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) <-> ( F ` J ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) ) )
50	46, 49	mpbid 147	. . . 4 |- ( ( ph /\ B = J ) -> ( F ` J ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) )
51	38	adantr 276	. . . 4 |- ( ( ph /\ B = J ) -> ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) <Q ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
52	50, 51, 42	syl2anc 411	. . 3 |- ( ( ph /\ B = J ) -> ( F ` J ) <Q ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
53		breq1 4091	. . . . . . . . . 10 |- ( n = J -> ( n <N k <-> J <N k ) )
54		fveq2 5639	. . . . . . . . . . . 12 |- ( n = J -> ( F ` n ) = ( F ` J ) )
55		opeq1 3862	. . . . . . . . . . . . . . 15 |- ( n = J -> <. n , 1o >. = <. J , 1o >. )
56	55	eceq1d 6737	. . . . . . . . . . . . . 14 |- ( n = J -> [ <. n , 1o >. ] ~Q = [ <. J , 1o >. ] ~Q )
57	56	fveq2d 5643	. . . . . . . . . . . . 13 |- ( n = J -> ( *Q ` [ <. n , 1o >. ] ~Q ) = ( *Q ` [ <. J , 1o >. ] ~Q ) )
58	57	oveq2d 6033	. . . . . . . . . . . 12 |- ( n = J -> ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) = ( ( F ` k ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
59	54, 58	breq12d 4101	. . . . . . . . . . 11 |- ( n = J -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) <-> ( F ` J ) <Q ( ( F ` k ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) )
60	54, 57	oveq12d 6035	. . . . . . . . . . . 12 |- ( n = J -> ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) = ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
61	60	breq2d 4100	. . . . . . . . . . 11 |- ( n = J -> ( ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) <-> ( F ` k ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) )
62	59, 61	anbi12d 473	. . . . . . . . . 10 |- ( n = J -> ( ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) <-> ( ( F ` J ) <Q ( ( F ` k ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) ) )
63	53, 62	imbi12d 234	. . . . . . . . 9 |- ( n = J -> ( ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) <-> ( J <N k -> ( ( F ` J ) <Q ( ( F ` k ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) ) ) )
64		breq2 4092	. . . . . . . . . 10 |- ( k = B -> ( J <N k <-> J <N B ) )
65		fveq2 5639	. . . . . . . . . . . . 13 |- ( k = B -> ( F ` k ) = ( F ` B ) )
66	65	oveq1d 6032	. . . . . . . . . . . 12 |- ( k = B -> ( ( F ` k ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) = ( ( F ` B ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
67	66	breq2d 4100	. . . . . . . . . . 11 |- ( k = B -> ( ( F ` J ) <Q ( ( F ` k ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <-> ( F ` J ) <Q ( ( F ` B ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) )
68	65	breq1d 4098	. . . . . . . . . . 11 |- ( k = B -> ( ( F ` k ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <-> ( F ` B ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) )
69	67, 68	anbi12d 473	. . . . . . . . . 10 |- ( k = B -> ( ( ( F ` J ) <Q ( ( F ` k ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) <-> ( ( F ` J ) <Q ( ( F ` B ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( F ` B ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) ) )
70	64, 69	imbi12d 234	. . . . . . . . 9 |- ( k = B -> ( ( J <N k -> ( ( F ` J ) <Q ( ( F ` k ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) ) <-> ( J <N B -> ( ( F ` J ) <Q ( ( F ` B ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( F ` B ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) ) ) )
71	63, 70	rspc2v 2923	. . . . . . . 8 |- ( ( J e. N. /\ 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 ) ) ) ) -> ( J <N B -> ( ( F ` J ) <Q ( ( F ` B ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( F ` B ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) ) ) )
72	3, 2, 71	syl2anc 411	. . . . . . 7 |- ( 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 ) ) ) ) -> ( J <N B -> ( ( F ` J ) <Q ( ( F ` B ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( F ` B ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) ) ) )
73	1, 72	mpd 13	. . . . . 6 |- ( ph -> ( J <N B -> ( ( F ` J ) <Q ( ( F ` B ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( F ` B ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) ) )
74	73	imp 124	. . . . 5 |- ( ( ph /\ J <N B ) -> ( ( F ` J ) <Q ( ( F ` B ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( F ` B ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) )
75	74	simpld 112	. . . 4 |- ( ( ph /\ J <N B ) -> ( F ` J ) <Q ( ( F ` B ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
76		ltanqg 7619	. . . . . . . 8 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
77	76	adantl 277	. . . . . . 7 |- ( ( ph /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
78		addcomnqg 7600	. . . . . . . 8 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
79	78	adantl 277	. . . . . . 7 |- ( ( ph /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
80	77, 28, 33, 36, 79	caovord2d 6191	. . . . . 6 |- ( ph -> ( ( F ` B ) <Q ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) <-> ( ( F ` B ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) )
81	45, 80	mpbid 147	. . . . 5 |- ( ph -> ( ( F ` B ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
82	81	adantr 276	. . . 4 |- ( ( ph /\ J <N B ) -> ( ( F ` B ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
83	40, 41	sotri 5132	. . . 4 |- ( ( ( F ` J ) <Q ( ( F ` B ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( ( F ` B ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) -> ( F ` J ) <Q ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
84	75, 82, 83	syl2anc 411	. . 3 |- ( ( ph /\ J <N B ) -> ( F ` J ) <Q ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
85		pitri3or 7541	. . . 4 |- ( ( B e. N. /\ J e. N. ) -> ( B <N J \/ B = J \/ J <N B ) )
86	2, 3, 85	syl2anc 411	. . 3 |- ( ph -> ( B <N J \/ B = J \/ J <N B ) )
87	43, 52, 84, 86	mpjao3dan 1343	. 2 |- ( ph -> ( F ` J ) <Q ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
88	27, 3	ffvelcdmd 5783	. . . 4 |- ( ph -> ( F ` J ) e. Q. )
89		addclnq 7594	. . . . 5 |- ( ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) e. Q. /\ ( *Q ` [ <. J , 1o >. ] ~Q ) e. Q. ) -> ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) e. Q. )
90	33, 36, 89	syl2anc 411	. . . 4 |- ( ph -> ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) e. Q. )
91		so2nr 4418	. . . . 5 |- ( ( <Q Or Q. /\ ( ( F ` J ) e. Q. /\ ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) e. Q. ) ) -> -. ( ( F ` J ) <Q ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( F ` J ) ) )
92	40, 91	mpan 424	. . . 4 |- ( ( ( F ` J ) e. Q. /\ ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) e. Q. ) -> -. ( ( F ` J ) <Q ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( F ` J ) ) )
93	88, 90, 92	syl2anc 411	. . 3 |- ( ph -> -. ( ( F ` J ) <Q ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( F ` J ) ) )
94		imnan 696	. . 3 |- ( ( ( F ` J ) <Q ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) -> -. ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( F ` J ) ) <-> -. ( ( F ` J ) <Q ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( F ` J ) ) )
95	93, 94	sylibr 134	. 2 |- ( ph -> ( ( F ` J ) <Q ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) -> -. ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( F ` J ) ) )
96	87, 95	mpd 13	1 |- ( ph -> -. ( ( ( F ` B ) +Q ( *Q ` [ <. B , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( F ` J ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 1003   /\ w3a 1004   = wceq 1397   e. wcel 2202   A.wral 2510   <.cop 3672   class class class wbr 4088   Or wor 4392   -->wf 5322   ` cfv 5326  (class class class)co 6017   1oc1o 6574   [cec 6699   N.cnpi 7491   <N clti 7494   ~Q ceq 7498   Q.cnq 7499   +Q cplq 7501   *Qcrq 7503   <Q cltq 7504

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-1o 6581  df-oadd 6585  df-omul 6586  df-er 6701  df-ec 6703  df-qs 6707  df-ni 7523  df-pli 7524  df-mi 7525  df-lti 7526  df-plpq 7563  df-mpq 7564  df-enq 7566  df-nqqs 7567  df-plqqs 7568  df-mqqs 7569  df-1nqqs 7570  df-rq 7571  df-ltnqqs 7572

This theorem is referenced by:  caucvgprlemladdrl  7897