Theorem caucvgprlemnkj 7667

Description: Lemma for caucvgpr 7683. Part of disjointness. (Contributed by Jim Kingdon, 23-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 ) ) ) ) )
caucvgprlemnkj.k	|- ( ph -> K e. N. )
caucvgprlemnkj.j	|- ( ph -> J e. N. )
caucvgprlemnkj.s	|- ( ph -> S e. Q. )

Assertion

Ref	Expression
caucvgprlemnkj	|- ( ph -> -. ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) )

Distinct variable group:   k, F, n

Allowed substitution hints:   ph( k, n)   S( k, n)   J( k, n)   K( k, n)

Proof of Theorem caucvgprlemnkj

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

Step	Hyp	Ref	Expression
1		ltsonq 7399	. . . 4 |- <Q Or Q.
2		ltrelnq 7366	. . . 4 |- <Q C_ ( Q. X. Q. )
3	1, 2	son2lpi 5027	. . 3 |- -. ( S <Q ( F ` J ) /\ ( F ` J ) <Q S )
4		simprl 529	. . . . . . 7 |- ( ( ( ph /\ K <N J ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) )
5		caucvgpr.cau	. . . . . . . . . . . 12 |- ( 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 ) ) ) ) )
6		breq1 4008	. . . . . . . . . . . . . 14 |- ( n = a -> ( n <N k <-> a <N k ) )
7		fveq2 5517	. . . . . . . . . . . . . . . 16 |- ( n = a -> ( F ` n ) = ( F ` a ) )
8		opeq1 3780	. . . . . . . . . . . . . . . . . . 19 |- ( n = a -> <. n , 1o >. = <. a , 1o >. )
9	8	eceq1d 6573	. . . . . . . . . . . . . . . . . 18 |- ( n = a -> [ <. n , 1o >. ] ~Q = [ <. a , 1o >. ] ~Q )
10	9	fveq2d 5521	. . . . . . . . . . . . . . . . 17 |- ( n = a -> ( *Q ` [ <. n , 1o >. ] ~Q ) = ( *Q ` [ <. a , 1o >. ] ~Q ) )
11	10	oveq2d 5893	. . . . . . . . . . . . . . . 16 |- ( n = a -> ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) = ( ( F ` k ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
12	7, 11	breq12d 4018	. . . . . . . . . . . . . . 15 |- ( n = a -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) <-> ( F ` a ) <Q ( ( F ` k ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) )
13	7, 10	oveq12d 5895	. . . . . . . . . . . . . . . 16 |- ( n = a -> ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) = ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
14	13	breq2d 4017	. . . . . . . . . . . . . . 15 |- ( n = a -> ( ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) <-> ( F ` k ) <Q ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) )
15	12, 14	anbi12d 473	. . . . . . . . . . . . . 14 |- ( n = a -> ( ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) <-> ( ( F ` a ) <Q ( ( F ` k ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) ) )
16	6, 15	imbi12d 234	. . . . . . . . . . . . 13 |- ( n = a -> ( ( n <N k -> ( ( F ` n ) <Q ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) ) ) <-> ( a <N k -> ( ( F ` a ) <Q ( ( F ` k ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) ) ) )
17		breq2 4009	. . . . . . . . . . . . . 14 |- ( k = b -> ( a <N k <-> a <N b ) )
18		fveq2 5517	. . . . . . . . . . . . . . . . 17 |- ( k = b -> ( F ` k ) = ( F ` b ) )
19	18	oveq1d 5892	. . . . . . . . . . . . . . . 16 |- ( k = b -> ( ( F ` k ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( ( F ` b ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
20	19	breq2d 4017	. . . . . . . . . . . . . . 15 |- ( k = b -> ( ( F ` a ) <Q ( ( F ` k ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <-> ( F ` a ) <Q ( ( F ` b ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) )
21	18	breq1d 4015	. . . . . . . . . . . . . . 15 |- ( k = b -> ( ( F ` k ) <Q ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <-> ( F ` b ) <Q ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) )
22	20, 21	anbi12d 473	. . . . . . . . . . . . . 14 |- ( k = b -> ( ( ( F ` a ) <Q ( ( F ` k ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) <-> ( ( F ` a ) <Q ( ( F ` b ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) /\ ( F ` b ) <Q ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) ) )
23	17, 22	imbi12d 234	. . . . . . . . . . . . 13 |- ( k = b -> ( ( a <N k -> ( ( F ` a ) <Q ( ( F ` k ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) /\ ( F ` k ) <Q ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) ) <-> ( a <N b -> ( ( F ` a ) <Q ( ( F ` b ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) /\ ( F ` b ) <Q ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) ) ) )
24	16, 23	cbvral2v 2718	. . . . . . . . . . . 12 |- ( 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 ) ) ) ) <-> A. a e. N. A. b e. N. ( a <N b -> ( ( F ` a ) <Q ( ( F ` b ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) /\ ( F ` b ) <Q ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) ) )
25	5, 24	sylib 122	. . . . . . . . . . 11 |- ( ph -> A. a e. N. A. b e. N. ( a <N b -> ( ( F ` a ) <Q ( ( F ` b ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) /\ ( F ` b ) <Q ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) ) )
26		caucvgprlemnkj.k	. . . . . . . . . . . 12 |- ( ph -> K e. N. )
27		caucvgprlemnkj.j	. . . . . . . . . . . 12 |- ( ph -> J e. N. )
28		breq1 4008	. . . . . . . . . . . . . 14 |- ( a = K -> ( a <N b <-> K <N b ) )
29		fveq2 5517	. . . . . . . . . . . . . . . 16 |- ( a = K -> ( F ` a ) = ( F ` K ) )
30		opeq1 3780	. . . . . . . . . . . . . . . . . . 19 |- ( a = K -> <. a , 1o >. = <. K , 1o >. )
31	30	eceq1d 6573	. . . . . . . . . . . . . . . . . 18 |- ( a = K -> [ <. a , 1o >. ] ~Q = [ <. K , 1o >. ] ~Q )
32	31	fveq2d 5521	. . . . . . . . . . . . . . . . 17 |- ( a = K -> ( *Q ` [ <. a , 1o >. ] ~Q ) = ( *Q ` [ <. K , 1o >. ] ~Q ) )
33	32	oveq2d 5893	. . . . . . . . . . . . . . . 16 |- ( a = K -> ( ( F ` b ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( ( F ` b ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
34	29, 33	breq12d 4018	. . . . . . . . . . . . . . 15 |- ( a = K -> ( ( F ` a ) <Q ( ( F ` b ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <-> ( F ` K ) <Q ( ( F ` b ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) ) )
35	29, 32	oveq12d 5895	. . . . . . . . . . . . . . . 16 |- ( a = K -> ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
36	35	breq2d 4017	. . . . . . . . . . . . . . 15 |- ( a = K -> ( ( F ` b ) <Q ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <-> ( F ` b ) <Q ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) ) )
37	34, 36	anbi12d 473	. . . . . . . . . . . . . 14 |- ( a = K -> ( ( ( F ` a ) <Q ( ( F ` b ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) /\ ( F ` b ) <Q ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) <-> ( ( F ` K ) <Q ( ( F ` b ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) /\ ( F ` b ) <Q ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) ) ) )
38	28, 37	imbi12d 234	. . . . . . . . . . . . 13 |- ( a = K -> ( ( a <N b -> ( ( F ` a ) <Q ( ( F ` b ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) /\ ( F ` b ) <Q ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) ) <-> ( K <N b -> ( ( F ` K ) <Q ( ( F ` b ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) /\ ( F ` b ) <Q ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) ) ) ) )
39		breq2 4009	. . . . . . . . . . . . . 14 |- ( b = J -> ( K <N b <-> K <N J ) )
40		fveq2 5517	. . . . . . . . . . . . . . . . 17 |- ( b = J -> ( F ` b ) = ( F ` J ) )
41	40	oveq1d 5892	. . . . . . . . . . . . . . . 16 |- ( b = J -> ( ( F ` b ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) = ( ( F ` J ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
42	41	breq2d 4017	. . . . . . . . . . . . . . 15 |- ( b = J -> ( ( F ` K ) <Q ( ( F ` b ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <-> ( F ` K ) <Q ( ( F ` J ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) ) )
43	40	breq1d 4015	. . . . . . . . . . . . . . 15 |- ( b = J -> ( ( F ` b ) <Q ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <-> ( F ` J ) <Q ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) ) )
44	42, 43	anbi12d 473	. . . . . . . . . . . . . 14 |- ( b = J -> ( ( ( F ` K ) <Q ( ( F ` b ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) /\ ( F ` b ) <Q ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) ) <-> ( ( F ` K ) <Q ( ( F ` J ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) /\ ( F ` J ) <Q ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) ) ) )
45	39, 44	imbi12d 234	. . . . . . . . . . . . 13 |- ( b = J -> ( ( K <N b -> ( ( F ` K ) <Q ( ( F ` b ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) /\ ( F ` b ) <Q ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) ) ) <-> ( K <N J -> ( ( F ` K ) <Q ( ( F ` J ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) /\ ( F ` J ) <Q ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) ) ) ) )
46	38, 45	rspc2v 2856	. . . . . . . . . . . 12 |- ( ( K e. N. /\ J e. N. ) -> ( A. a e. N. A. b e. N. ( a <N b -> ( ( F ` a ) <Q ( ( F ` b ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) /\ ( F ` b ) <Q ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) ) -> ( K <N J -> ( ( F ` K ) <Q ( ( F ` J ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) /\ ( F ` J ) <Q ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) ) ) ) )
47	26, 27, 46	syl2anc 411	. . . . . . . . . . 11 |- ( ph -> ( A. a e. N. A. b e. N. ( a <N b -> ( ( F ` a ) <Q ( ( F ` b ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) /\ ( F ` b ) <Q ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) ) -> ( K <N J -> ( ( F ` K ) <Q ( ( F ` J ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) /\ ( F ` J ) <Q ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) ) ) ) )
48	25, 47	mpd 13	. . . . . . . . . 10 |- ( ph -> ( K <N J -> ( ( F ` K ) <Q ( ( F ` J ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) /\ ( F ` J ) <Q ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) ) ) )
49	48	imp 124	. . . . . . . . 9 |- ( ( ph /\ K <N J ) -> ( ( F ` K ) <Q ( ( F ` J ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) /\ ( F ` J ) <Q ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) ) )
50	49	simpld 112	. . . . . . . 8 |- ( ( ph /\ K <N J ) -> ( F ` K ) <Q ( ( F ` J ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
51	50	adantr 276	. . . . . . 7 |- ( ( ( ph /\ K <N J ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> ( F ` K ) <Q ( ( F ` J ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
52	1, 2	sotri 5026	. . . . . . 7 |- ( ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( F ` K ) <Q ( ( F ` J ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) ) -> ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( ( F ` J ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
53	4, 51, 52	syl2anc 411	. . . . . 6 |- ( ( ( ph /\ K <N J ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( ( F ` J ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
54		ltanqg 7401	. . . . . . . 8 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
55	54	adantl 277	. . . . . . 7 |- ( ( ( ( ph /\ K <N J ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
56		caucvgprlemnkj.s	. . . . . . . 8 |- ( ph -> S e. Q. )
57	56	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ K <N J ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> S e. Q. )
58		caucvgpr.f	. . . . . . . . 9 |- ( ph -> F : N. --> Q. )
59	58, 27	ffvelcdmd 5654	. . . . . . . 8 |- ( ph -> ( F ` J ) e. Q. )
60	59	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ K <N J ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> ( F ` J ) e. Q. )
61		nnnq 7423	. . . . . . . . 9 |- ( K e. N. -> [ <. K , 1o >. ] ~Q e. Q. )
62		recclnq 7393	. . . . . . . . 9 |- ( [ <. K , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. )
63	26, 61, 62	3syl 17	. . . . . . . 8 |- ( ph -> ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. )
64	63	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ K <N J ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. )
65		addcomnqg 7382	. . . . . . . 8 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
66	65	adantl 277	. . . . . . 7 |- ( ( ( ( ph /\ K <N J ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
67	55, 57, 60, 64, 66	caovord2d 6046	. . . . . 6 |- ( ( ( ph /\ K <N J ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> ( S <Q ( F ` J ) <-> ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( ( F ` J ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) ) )
68	53, 67	mpbird 167	. . . . 5 |- ( ( ( ph /\ K <N J ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> S <Q ( F ` J ) )
69		nnnq 7423	. . . . . . . . 9 |- ( J e. N. -> [ <. J , 1o >. ] ~Q e. Q. )
70		recclnq 7393	. . . . . . . . 9 |- ( [ <. J , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. J , 1o >. ] ~Q ) e. Q. )
71	27, 69, 70	3syl 17	. . . . . . . 8 |- ( ph -> ( *Q ` [ <. J , 1o >. ] ~Q ) e. Q. )
72	71	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ K <N J ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> ( *Q ` [ <. J , 1o >. ] ~Q ) e. Q. )
73		ltaddnq 7408	. . . . . . 7 |- ( ( ( F ` J ) e. Q. /\ ( *Q ` [ <. J , 1o >. ] ~Q ) e. Q. ) -> ( F ` J ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
74	60, 72, 73	syl2anc 411	. . . . . 6 |- ( ( ( ph /\ K <N J ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> ( F ` J ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
75		simprr 531	. . . . . 6 |- ( ( ( ph /\ K <N J ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S )
76	1, 2	sotri 5026	. . . . . 6 |- ( ( ( F ` J ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) -> ( F ` J ) <Q S )
77	74, 75, 76	syl2anc 411	. . . . 5 |- ( ( ( ph /\ K <N J ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> ( F ` J ) <Q S )
78	68, 77	jca 306	. . . 4 |- ( ( ( ph /\ K <N J ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> ( S <Q ( F ` J ) /\ ( F ` J ) <Q S ) )
79	78	ex 115	. . 3 |- ( ( ph /\ K <N J ) -> ( ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) -> ( S <Q ( F ` J ) /\ ( F ` J ) <Q S ) ) )
80	3, 79	mtoi 664	. 2 |- ( ( ph /\ K <N J ) -> -. ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) )
81	1, 2	son2lpi 5027	. . 3 |- -. ( ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S /\ S <Q ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
82		opeq1 3780	. . . . . . . . . . . 12 |- ( K = J -> <. K , 1o >. = <. J , 1o >. )
83	82	eceq1d 6573	. . . . . . . . . . 11 |- ( K = J -> [ <. K , 1o >. ] ~Q = [ <. J , 1o >. ] ~Q )
84	83	fveq2d 5521	. . . . . . . . . 10 |- ( K = J -> ( *Q ` [ <. K , 1o >. ] ~Q ) = ( *Q ` [ <. J , 1o >. ] ~Q ) )
85	84	oveq2d 5893	. . . . . . . . 9 |- ( K = J -> ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) = ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
86		fveq2 5517	. . . . . . . . 9 |- ( K = J -> ( F ` K ) = ( F ` J ) )
87	85, 86	breq12d 4018	. . . . . . . 8 |- ( K = J -> ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) <-> ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( F ` J ) ) )
88	87	anbi1d 465	. . . . . . 7 |- ( K = J -> ( ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) <-> ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( F ` J ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) )
89	88	adantl 277	. . . . . 6 |- ( ( ph /\ K = J ) -> ( ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) <-> ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( F ` J ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) )
90	54	adantl 277	. . . . . . . . 9 |- ( ( ph /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
91		addclnq 7376	. . . . . . . . . 10 |- ( ( S e. Q. /\ ( *Q ` [ <. J , 1o >. ] ~Q ) e. Q. ) -> ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) e. Q. )
92	56, 71, 91	syl2anc 411	. . . . . . . . 9 |- ( ph -> ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) e. Q. )
93	65	adantl 277	. . . . . . . . 9 |- ( ( ph /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
94	90, 92, 59, 71, 93	caovord2d 6046	. . . . . . . 8 |- ( ph -> ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( F ` J ) <-> ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) )
95	94	adantr 276	. . . . . . 7 |- ( ( ph /\ K = J ) -> ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( F ` J ) <-> ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) )
96	95	anbi1d 465	. . . . . 6 |- ( ( ph /\ K = J ) -> ( ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( F ` J ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) <-> ( ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) )
97	89, 96	bitrd 188	. . . . 5 |- ( ( ph /\ K = J ) -> ( ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) <-> ( ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) )
98	1, 2	sotri 5026	. . . . 5 |- ( ( ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) -> ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S )
99	97, 98	biimtrdi 163	. . . 4 |- ( ( ph /\ K = J ) -> ( ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) -> ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) )
100		ltaddnq 7408	. . . . . . 7 |- ( ( S e. Q. /\ ( *Q ` [ <. J , 1o >. ] ~Q ) e. Q. ) -> S <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
101	56, 71, 100	syl2anc 411	. . . . . 6 |- ( ph -> S <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
102		ltaddnq 7408	. . . . . . 7 |- ( ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) e. Q. /\ ( *Q ` [ <. J , 1o >. ] ~Q ) e. Q. ) -> ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
103	92, 71, 102	syl2anc 411	. . . . . 6 |- ( ph -> ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
104	1, 2	sotri 5026	. . . . . 6 |- ( ( S <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) -> S <Q ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
105	101, 103, 104	syl2anc 411	. . . . 5 |- ( ph -> S <Q ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
106	105	adantr 276	. . . 4 |- ( ( ph /\ K = J ) -> S <Q ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
107	99, 106	jctird 317	. . 3 |- ( ( ph /\ K = J ) -> ( ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) -> ( ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S /\ S <Q ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) ) )
108	81, 107	mtoi 664	. 2 |- ( ( ph /\ K = J ) -> -. ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) )
109	1, 2	son2lpi 5027	. . 3 |- -. ( S <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S )
110	56	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ J <N K ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> S e. Q. )
111	63	ad2antrr 488	. . . . . . 7 |- ( ( ( ph /\ J <N K ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. )
112		ltaddnq 7408	. . . . . . 7 |- ( ( S e. Q. /\ ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. ) -> S <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
113	110, 111, 112	syl2anc 411	. . . . . 6 |- ( ( ( ph /\ J <N K ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> S <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
114		simprl 529	. . . . . . 7 |- ( ( ( ph /\ J <N K ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) )
115		breq1 4008	. . . . . . . . . . . . . 14 |- ( a = J -> ( a <N b <-> J <N b ) )
116		fveq2 5517	. . . . . . . . . . . . . . . 16 |- ( a = J -> ( F ` a ) = ( F ` J ) )
117		opeq1 3780	. . . . . . . . . . . . . . . . . . 19 |- ( a = J -> <. a , 1o >. = <. J , 1o >. )
118	117	eceq1d 6573	. . . . . . . . . . . . . . . . . 18 |- ( a = J -> [ <. a , 1o >. ] ~Q = [ <. J , 1o >. ] ~Q )
119	118	fveq2d 5521	. . . . . . . . . . . . . . . . 17 |- ( a = J -> ( *Q ` [ <. a , 1o >. ] ~Q ) = ( *Q ` [ <. J , 1o >. ] ~Q ) )
120	119	oveq2d 5893	. . . . . . . . . . . . . . . 16 |- ( a = J -> ( ( F ` b ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( ( F ` b ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
121	116, 120	breq12d 4018	. . . . . . . . . . . . . . 15 |- ( a = J -> ( ( F ` a ) <Q ( ( F ` b ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <-> ( F ` J ) <Q ( ( F ` b ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) )
122	116, 119	oveq12d 5895	. . . . . . . . . . . . . . . 16 |- ( a = J -> ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
123	122	breq2d 4017	. . . . . . . . . . . . . . 15 |- ( a = J -> ( ( F ` b ) <Q ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <-> ( F ` b ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) )
124	121, 123	anbi12d 473	. . . . . . . . . . . . . 14 |- ( a = J -> ( ( ( F ` a ) <Q ( ( F ` b ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) /\ ( F ` b ) <Q ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) <-> ( ( F ` J ) <Q ( ( F ` b ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( F ` b ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) ) )
125	115, 124	imbi12d 234	. . . . . . . . . . . . 13 |- ( a = J -> ( ( a <N b -> ( ( F ` a ) <Q ( ( F ` b ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) /\ ( F ` b ) <Q ( ( F ` a ) +Q ( *Q ` [ <. a , 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 ) ) ) ) ) )
126		breq2 4009	. . . . . . . . . . . . . 14 |- ( b = K -> ( J <N b <-> J <N K ) )
127		fveq2 5517	. . . . . . . . . . . . . . . . 17 |- ( b = K -> ( F ` b ) = ( F ` K ) )
128	127	oveq1d 5892	. . . . . . . . . . . . . . . 16 |- ( b = K -> ( ( F ` b ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) = ( ( F ` K ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
129	128	breq2d 4017	. . . . . . . . . . . . . . 15 |- ( b = K -> ( ( F ` J ) <Q ( ( F ` b ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <-> ( F ` J ) <Q ( ( F ` K ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) )
130	127	breq1d 4015	. . . . . . . . . . . . . . 15 |- ( b = K -> ( ( F ` b ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <-> ( F ` K ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) )
131	129, 130	anbi12d 473	. . . . . . . . . . . . . 14 |- ( b = K -> ( ( ( F ` J ) <Q ( ( F ` b ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( F ` b ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) <-> ( ( F ` J ) <Q ( ( F ` K ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( F ` K ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) ) )
132	126, 131	imbi12d 234	. . . . . . . . . . . . 13 |- ( b = K -> ( ( J <N b -> ( ( F ` J ) <Q ( ( F ` b ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( F ` b ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 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 ) ) ) ) ) )
133	125, 132	rspc2v 2856	. . . . . . . . . . . 12 |- ( ( J e. N. /\ K e. N. ) -> ( A. a e. N. A. b e. N. ( a <N b -> ( ( F ` a ) <Q ( ( F ` b ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) /\ ( F ` b ) <Q ( ( F ` a ) +Q ( *Q ` [ <. a , 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 ) ) ) ) ) )
134	27, 26, 133	syl2anc 411	. . . . . . . . . . 11 |- ( ph -> ( A. a e. N. A. b e. N. ( a <N b -> ( ( F ` a ) <Q ( ( F ` b ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) /\ ( F ` b ) <Q ( ( F ` a ) +Q ( *Q ` [ <. a , 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 ) ) ) ) ) )
135	25, 134	mpd 13	. . . . . . . . . 10 |- ( ph -> ( J <N K -> ( ( F ` J ) <Q ( ( F ` K ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( F ` K ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) ) )
136	135	imp 124	. . . . . . . . 9 |- ( ( ph /\ J <N K ) -> ( ( F ` J ) <Q ( ( F ` K ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( F ` K ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) )
137	136	simprd 114	. . . . . . . 8 |- ( ( ph /\ J <N K ) -> ( F ` K ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
138	137	adantr 276	. . . . . . 7 |- ( ( ( ph /\ J <N K ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> ( F ` K ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
139	1, 2	sotri 5026	. . . . . . 7 |- ( ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( F ` K ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) -> ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
140	114, 138, 139	syl2anc 411	. . . . . 6 |- ( ( ( ph /\ J <N K ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
141	1, 2	sotri 5026	. . . . . 6 |- ( ( S <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) /\ ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) ) -> S <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
142	113, 140, 141	syl2anc 411	. . . . 5 |- ( ( ( ph /\ J <N K ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> S <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
143		simprr 531	. . . . 5 |- ( ( ( ph /\ J <N K ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S )
144	142, 143	jca 306	. . . 4 |- ( ( ( ph /\ J <N K ) /\ ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) -> ( S <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) )
145	144	ex 115	. . 3 |- ( ( ph /\ J <N K ) -> ( ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) -> ( S <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) ) )
146	109, 145	mtoi 664	. 2 |- ( ( ph /\ J <N K ) -> -. ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) )
147		pitri3or 7323	. . 3 |- ( ( K e. N. /\ J e. N. ) -> ( K <N J \/ K = J \/ J <N K ) )
148	26, 27, 147	syl2anc 411	. 2 |- ( ph -> ( K <N J \/ K = J \/ J <N K ) )
149	80, 108, 146, 148	mpjao3dan 1307	1 |- ( ph -> -. ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 977   /\ w3a 978   = wceq 1353   e. wcel 2148   A.wral 2455   <.cop 3597   class class class wbr 4005   -->wf 5214   ` cfv 5218  (class class class)co 5877   1oc1o 6412   [cec 6535   N.cnpi 7273   <N clti 7276   ~Q ceq 7280   Q.cnq 7281   +Q cplq 7283   *Qcrq 7285   <Q cltq 7286

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354

This theorem is referenced by:  caucvgprlemdisj  7675