Theorem caucvgprlemnkj 8023

Description: Lemma for caucvgpr 8039. 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 7755	. . . 4 |- <Q Or Q.
2		ltrelnq 7722	. . . 4 |- <Q C_ ( Q. X. Q. )
3	1, 2	son2lpi 5179	. . 3 |- -. ( S <Q ( F ` J ) /\ ( F ` J ) <Q S )
4		simprl 535	. . . . . . 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 4128	. . . . . . . . . . . . . 14 |- ( n = a -> ( n <N k <-> a <N k ) )
7		fveq2 5690	. . . . . . . . . . . . . . . 16 |- ( n = a -> ( F ` n ) = ( F ` a ) )
8		opeq1 3899	. . . . . . . . . . . . . . . . . . 19 |- ( n = a -> <. n , 1o >. = <. a , 1o >. )
9	8	eceq1d 6833	. . . . . . . . . . . . . . . . . 18 |- ( n = a -> [ <. n , 1o >. ] ~Q = [ <. a , 1o >. ] ~Q )
10	9	fveq2d 5694	. . . . . . . . . . . . . . . . 17 |- ( n = a -> ( *Q ` [ <. n , 1o >. ] ~Q ) = ( *Q ` [ <. a , 1o >. ] ~Q ) )
11	10	oveq2d 6091	. . . . . . . . . . . . . . . 16 |- ( n = a -> ( ( F ` k ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) = ( ( F ` k ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
12	7, 11	breq12d 4138	. . . . . . . . . . . . . . 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 6093	. . . . . . . . . . . . . . . 16 |- ( n = a -> ( ( F ` n ) +Q ( *Q ` [ <. n , 1o >. ] ~Q ) ) = ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
14	13	breq2d 4137	. . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . 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 4129	. . . . . . . . . . . . . 14 |- ( k = b -> ( a <N k <-> a <N b ) )
18		fveq2 5690	. . . . . . . . . . . . . . . . 17 |- ( k = b -> ( F ` k ) = ( F ` b ) )
19	18	oveq1d 6090	. . . . . . . . . . . . . . . 16 |- ( k = b -> ( ( F ` k ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( ( F ` b ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
20	19	breq2d 4137	. . . . . . . . . . . . . . 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 4135	. . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . 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 2799	. . . . . . . . . . . 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 4128	. . . . . . . . . . . . . 14 |- ( a = K -> ( a <N b <-> K <N b ) )
29		fveq2 5690	. . . . . . . . . . . . . . . 16 |- ( a = K -> ( F ` a ) = ( F ` K ) )
30		opeq1 3899	. . . . . . . . . . . . . . . . . . 19 |- ( a = K -> <. a , 1o >. = <. K , 1o >. )
31	30	eceq1d 6833	. . . . . . . . . . . . . . . . . 18 |- ( a = K -> [ <. a , 1o >. ] ~Q = [ <. K , 1o >. ] ~Q )
32	31	fveq2d 5694	. . . . . . . . . . . . . . . . 17 |- ( a = K -> ( *Q ` [ <. a , 1o >. ] ~Q ) = ( *Q ` [ <. K , 1o >. ] ~Q ) )
33	32	oveq2d 6091	. . . . . . . . . . . . . . . 16 |- ( a = K -> ( ( F ` b ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( ( F ` b ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
34	29, 33	breq12d 4138	. . . . . . . . . . . . . . 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 6093	. . . . . . . . . . . . . . . 16 |- ( a = K -> ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( ( F ` K ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
36	35	breq2d 4137	. . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . 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 4129	. . . . . . . . . . . . . 14 |- ( b = J -> ( K <N b <-> K <N J ) )
40		fveq2 5690	. . . . . . . . . . . . . . . . 17 |- ( b = J -> ( F ` b ) = ( F ` J ) )
41	40	oveq1d 6090	. . . . . . . . . . . . . . . 16 |- ( b = J -> ( ( F ` b ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) = ( ( F ` J ) +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
42	41	breq2d 4137	. . . . . . . . . . . . . . 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 4135	. . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . 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 2943	. . . . . . . . . . . 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 415	. . . . . . . . . . 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 5178	. . . . . . 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 415	. . . . . 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 7757	. . . . . . . 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 492	. . . . . . 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 5835	. . . . . . . 8 |- ( ph -> ( F ` J ) e. Q. )
60	59	ad2antrr 492	. . . . . . 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 7779	. . . . . . . . 9 |- ( K e. N. -> [ <. K , 1o >. ] ~Q e. Q. )
62		recclnq 7749	. . . . . . . . 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 492	. . . . . . 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 7738	. . . . . . . 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 6249	. . . . . 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 7779	. . . . . . . . 9 |- ( J e. N. -> [ <. J , 1o >. ] ~Q e. Q. )
70		recclnq 7749	. . . . . . . . 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 492	. . . . . . 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 7764	. . . . . . 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 415	. . . . . 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 537	. . . . . 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 5178	. . . . . 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 415	. . . . 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 674	. 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 5179	. . 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 3899	. . . . . . . . . . . 12 |- ( K = J -> <. K , 1o >. = <. J , 1o >. )
83	82	eceq1d 6833	. . . . . . . . . . 11 |- ( K = J -> [ <. K , 1o >. ] ~Q = [ <. J , 1o >. ] ~Q )
84	83	fveq2d 5694	. . . . . . . . . 10 |- ( K = J -> ( *Q ` [ <. K , 1o >. ] ~Q ) = ( *Q ` [ <. J , 1o >. ] ~Q ) )
85	84	oveq2d 6091	. . . . . . . . 9 |- ( K = J -> ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) = ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
86		fveq2 5690	. . . . . . . . 9 |- ( K = J -> ( F ` K ) = ( F ` J ) )
87	85, 86	breq12d 4138	. . . . . . . 8 |- ( K = J -> ( ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) <Q ( F ` K ) <-> ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( F ` J ) ) )
88	87	anbi1d 469	. . . . . . 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 7732	. . . . . . . . . 10 |- ( ( S e. Q. /\ ( *Q ` [ <. J , 1o >. ] ~Q ) e. Q. ) -> ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) e. Q. )
92	56, 71, 91	syl2anc 415	. . . . . . . . 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 6249	. . . . . . . 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 469	. . . . . 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 5178	. . . . 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 7764	. . . . . . 7 |- ( ( S e. Q. /\ ( *Q ` [ <. J , 1o >. ] ~Q ) e. Q. ) -> S <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
101	56, 71, 100	syl2anc 415	. . . . . 6 |- ( ph -> S <Q ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
102		ltaddnq 7764	. . . . . . 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 415	. . . . . 6 |- ( ph -> ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q ( ( S +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
104	1, 2	sotri 5178	. . . . . 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 415	. . . . 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 674	. 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 5179	. . 3 |- -. ( S <Q ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) /\ ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) <Q S )
110	56	ad2antrr 492	. . . . . . 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 492	. . . . . . 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 7764	. . . . . . 7 |- ( ( S e. Q. /\ ( *Q ` [ <. K , 1o >. ] ~Q ) e. Q. ) -> S <Q ( S +Q ( *Q ` [ <. K , 1o >. ] ~Q ) ) )
113	110, 111, 112	syl2anc 415	. . . . . 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 535	. . . . . . 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 4128	. . . . . . . . . . . . . 14 |- ( a = J -> ( a <N b <-> J <N b ) )
116		fveq2 5690	. . . . . . . . . . . . . . . 16 |- ( a = J -> ( F ` a ) = ( F ` J ) )
117		opeq1 3899	. . . . . . . . . . . . . . . . . . 19 |- ( a = J -> <. a , 1o >. = <. J , 1o >. )
118	117	eceq1d 6833	. . . . . . . . . . . . . . . . . 18 |- ( a = J -> [ <. a , 1o >. ] ~Q = [ <. J , 1o >. ] ~Q )
119	118	fveq2d 5694	. . . . . . . . . . . . . . . . 17 |- ( a = J -> ( *Q ` [ <. a , 1o >. ] ~Q ) = ( *Q ` [ <. J , 1o >. ] ~Q ) )
120	119	oveq2d 6091	. . . . . . . . . . . . . . . 16 |- ( a = J -> ( ( F ` b ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( ( F ` b ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
121	116, 120	breq12d 4138	. . . . . . . . . . . . . . 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 6093	. . . . . . . . . . . . . . . 16 |- ( a = J -> ( ( F ` a ) +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( ( F ` J ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
123	122	breq2d 4137	. . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . 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 4129	. . . . . . . . . . . . . 14 |- ( b = K -> ( J <N b <-> J <N K ) )
127		fveq2 5690	. . . . . . . . . . . . . . . . 17 |- ( b = K -> ( F ` b ) = ( F ` K ) )
128	127	oveq1d 6090	. . . . . . . . . . . . . . . 16 |- ( b = K -> ( ( F ` b ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) = ( ( F ` K ) +Q ( *Q ` [ <. J , 1o >. ] ~Q ) ) )
129	128	breq2d 4137	. . . . . . . . . . . . . . 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 4135	. . . . . . . . . . . . . . 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 477	. . . . . . . . . . . . . 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 2943	. . . . . . . . . . . 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 415	. . . . . . . . . . 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 5178	. . . . . . 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 415	. . . . . 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 5178	. . . . . 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 415	. . . . 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 537	. . . . 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 674	. 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 7679	. . 3 |- ( ( K e. N. /\ J e. N. ) -> ( K <N J \/ K = J \/ J <N K ) )
148	26, 27, 147	syl2anc 415	. 2 |- ( ph -> ( K <N J \/ K = J \/ J <N K ) )
149	80, 108, 146, 148	mpjao3dan 1348	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 1008   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   <.cop 3708   class class class wbr 4125   -->wf 5368   ` cfv 5372  (class class class)co 6075   1oc1o 6670   [cec 6795   N.cnpi 7629   <N clti 7632   ~Q ceq 7636   Q.cnq 7637   +Q cplq 7639   *Qcrq 7641   <Q cltq 7642

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-eprel 4429  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-1o 6677  df-oadd 6681  df-omul 6682  df-er 6797  df-ec 6799  df-qs 6803  df-ni 7661  df-pli 7662  df-mi 7663  df-lti 7664  df-plpq 7701  df-mpq 7702  df-enq 7704  df-nqqs 7705  df-plqqs 7706  df-mqqs 7707  df-1nqqs 7708  df-rq 7709  df-ltnqqs 7710

This theorem is referenced by:  caucvgprlemdisj  8031