Theorem caucvgprlemladdfu 6747

Description: Lemma for caucvgpr 6752. Adding S after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 9-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
caucvgprlemladdfu	|- ( ph -> ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) C_ { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } )

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

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

Proof of Theorem caucvgprlemladdfu

Dummy variables m r s t v w z f g h x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		caucvgpr.f	. . . . . . 7 |- ( ph -> F : N. --> Q. )
2		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 ) ) ) ) )
3		caucvgpr.bnd	. . . . . . 7 |- ( ph -> A. j e. N. A <Q ( F ` j ) )
4		caucvgpr.lim	. . . . . . 7 |- 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 } >.
5	1, 2, 3, 4	caucvgprlemcl 6746	. . . . . 6 |- ( ph -> L e. P. )
6		caucvgprlemladd.s	. . . . . . 7 |- ( ph -> S e. Q. )
7		nqprlu 6617	. . . . . . 7 |- ( S e. Q. -> <. { l | l <Q S } , { u | S <Q u } >. e. P. )
8	6, 7	syl 14	. . . . . 6 |- ( ph -> <. { l | l <Q S } , { u | S <Q u } >. e. P. )
9		df-iplp 6538	. . . . . . 7 |- +P. = ( x e. P. , y e. P. |-> <. { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 1st ` x ) /\ h e. ( 1st ` y ) /\ f = ( g +Q h ) ) } , { f e. Q. | E. g e. Q. E. h e. Q. ( g e. ( 2nd ` x ) /\ h e. ( 2nd ` y ) /\ f = ( g +Q h ) ) } >. )
10		addclnq 6445	. . . . . . 7 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
11	9, 10	genpelvu 6583	. . . . . 6 |- ( ( L e. P. /\ <. { l | l <Q S } , { u | S <Q u } >. e. P. ) -> ( r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) <-> E. s e. ( 2nd ` L ) E. t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) r = ( s +Q t ) ) )
12	5, 8, 11	syl2anc 391	. . . . 5 |- ( ph -> ( r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) <-> E. s e. ( 2nd ` L ) E. t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) r = ( s +Q t ) ) )
13	12	biimpa 280	. . . 4 |- ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> E. s e. ( 2nd ` L ) E. t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) r = ( s +Q t ) )
14		breq2 3765	. . . . . . . . . . . . . . . 16 |- ( u = s -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
15	14	rexbidv 2324	. . . . . . . . . . . . . . 15 |- ( u = s -> ( E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u <-> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
16	4	fveq2i 5159	. . . . . . . . . . . . . . . 16 |- ( 2nd ` L ) = ( 2nd ` <. { 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 } >. )
17		nqex 6433	. . . . . . . . . . . . . . . . . 18 |- Q. e. _V
18	17	rabex 3898	. . . . . . . . . . . . . . . . 17 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q ( F ` j ) } e. _V
19	17	rabex 3898	. . . . . . . . . . . . . . . . 17 |- { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u } e. _V
20	18, 19	op2nd 5752	. . . . . . . . . . . . . . . 16 |- ( 2nd ` <. { 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 } >. ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
21	16, 20	eqtri 2060	. . . . . . . . . . . . . . 15 |- ( 2nd ` L ) = { u e. Q. | E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q u }
22	15, 21	elrab2 2697	. . . . . . . . . . . . . 14 |- ( s e. ( 2nd ` L ) <-> ( s e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
23	22	biimpi 113	. . . . . . . . . . . . 13 |- ( s e. ( 2nd ` L ) -> ( s e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
24	23	adantr 261	. . . . . . . . . . . 12 |- ( ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) -> ( s e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
25	24	adantl 262	. . . . . . . . . . 11 |- ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> ( s e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
26	25	adantr 261	. . . . . . . . . 10 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( s e. Q. /\ E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s ) )
27	26	simpld 105	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> s e. Q. )
28		vex 2557	. . . . . . . . . . . . . 14 |- t e. _V
29		breq2 3765	. . . . . . . . . . . . . 14 |- ( u = t -> ( S <Q u <-> S <Q t ) )
30		ltnqex 6619	. . . . . . . . . . . . . . 15 |- { l | l <Q S } e. _V
31		gtnqex 6620	. . . . . . . . . . . . . . 15 |- { u | S <Q u } e. _V
32	30, 31	op2nd 5752	. . . . . . . . . . . . . 14 |- ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) = { u | S <Q u }
33	28, 29, 32	elab2 2687	. . . . . . . . . . . . 13 |- ( t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) <-> S <Q t )
34		ltrelnq 6435	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
35	34	brel 4370	. . . . . . . . . . . . 13 |- ( S <Q t -> ( S e. Q. /\ t e. Q. ) )
36	33, 35	sylbi 114	. . . . . . . . . . . 12 |- ( t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) -> ( S e. Q. /\ t e. Q. ) )
37	36	simprd 107	. . . . . . . . . . 11 |- ( t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) -> t e. Q. )
38	37	ad2antll 460	. . . . . . . . . 10 |- ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> t e. Q. )
39	38	adantr 261	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> t e. Q. )
40		addclnq 6445	. . . . . . . . 9 |- ( ( s e. Q. /\ t e. Q. ) -> ( s +Q t ) e. Q. )
41	27, 39, 40	syl2anc 391	. . . . . . . 8 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( s +Q t ) e. Q. )
42		eleq1 2100	. . . . . . . . 9 |- ( r = ( s +Q t ) -> ( r e. Q. <-> ( s +Q t ) e. Q. ) )
43	42	adantl 262	. . . . . . . 8 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( r e. Q. <-> ( s +Q t ) e. Q. ) )
44	41, 43	mpbird 156	. . . . . . 7 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> r e. Q. )
45	26	simprd 107	. . . . . . . . . 10 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s )
46		fveq2 5156	. . . . . . . . . . . . 13 |- ( j = m -> ( F ` j ) = ( F ` m ) )
47		opeq1 3546	. . . . . . . . . . . . . . 15 |- ( j = m -> <. j , 1o >. = <. m , 1o >. )
48	47	eceq1d 6120	. . . . . . . . . . . . . 14 |- ( j = m -> [ <. j , 1o >. ] ~Q = [ <. m , 1o >. ] ~Q )
49	48	fveq2d 5160	. . . . . . . . . . . . 13 |- ( j = m -> ( *Q ` [ <. j , 1o >. ] ~Q ) = ( *Q ` [ <. m , 1o >. ] ~Q ) )
50	46, 49	oveq12d 5508	. . . . . . . . . . . 12 |- ( j = m -> ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) = ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) )
51	50	breq1d 3771	. . . . . . . . . . 11 |- ( j = m -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s <-> ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) )
52	51	cbvrexv 2531	. . . . . . . . . 10 |- ( E. j e. N. ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) <Q s <-> E. m e. N. ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s )
53	45, 52	sylib 127	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> E. m e. N. ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s )
54	33	biimpi 113	. . . . . . . . . . . . . . . . 17 |- ( t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) -> S <Q t )
55	54	ad2antll 460	. . . . . . . . . . . . . . . 16 |- ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> S <Q t )
56	55	adantr 261	. . . . . . . . . . . . . . 15 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> S <Q t )
57	56	ad2antrr 457	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> S <Q t )
58	6	ad5antr 465	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> S e. Q. )
59	39	ad2antrr 457	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> t e. Q. )
60	1	ad5antr 465	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> F : N. --> Q. )
61		simplr 482	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> m e. N. )
62	60, 61	ffvelrnd 5281	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( F ` m ) e. Q. )
63		nnnq 6492	. . . . . . . . . . . . . . . . 17 |- ( m e. N. -> [ <. m , 1o >. ] ~Q e. Q. )
64		recclnq 6462	. . . . . . . . . . . . . . . . 17 |- ( [ <. m , 1o >. ] ~Q e. Q. -> ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. )
65	61, 63, 64	3syl 17	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. )
66		addclnq 6445	. . . . . . . . . . . . . . . 16 |- ( ( ( F ` m ) e. Q. /\ ( *Q ` [ <. m , 1o >. ] ~Q ) e. Q. ) -> ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) e. Q. )
67	62, 65, 66	syl2anc 391	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) e. Q. )
68		ltanqg 6470	. . . . . . . . . . . . . . 15 |- ( ( S e. Q. /\ t e. Q. /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) e. Q. ) -> ( S <Q t <-> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q t ) ) )
69	58, 59, 67, 68	syl3anc 1135	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( S <Q t <-> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q t ) ) )
70	57, 69	mpbid 135	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q t ) )
71		simpr 103	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s )
72		ltanqg 6470	. . . . . . . . . . . . . . . 16 |- ( ( z e. Q. /\ w e. Q. /\ v e. Q. ) -> ( z <Q w <-> ( v +Q z ) <Q ( v +Q w ) ) )
73	72	adantl 262	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) /\ ( z e. Q. /\ w e. Q. /\ v e. Q. ) ) -> ( z <Q w <-> ( v +Q z ) <Q ( v +Q w ) ) )
74	27	ad2antrr 457	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> s e. Q. )
75		addcomnqg 6451	. . . . . . . . . . . . . . . 16 |- ( ( z e. Q. /\ w e. Q. ) -> ( z +Q w ) = ( w +Q z ) )
76	75	adantl 262	. . . . . . . . . . . . . . 15 |- ( ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) /\ ( z e. Q. /\ w e. Q. ) ) -> ( z +Q w ) = ( w +Q z ) )
77	73, 67, 74, 59, 76	caovord2d 5648	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s <-> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q t ) <Q ( s +Q t ) ) )
78	71, 77	mpbid 135	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q t ) <Q ( s +Q t ) )
79		ltsonq 6468	. . . . . . . . . . . . . 14 |- <Q Or Q.
80	79, 34	sotri 4698	. . . . . . . . . . . . 13 |- ( ( ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q t ) /\ ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q t ) <Q ( s +Q t ) ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q ( s +Q t ) )
81	70, 78, 80	syl2anc 391	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q ( s +Q t ) )
82		simpllr 486	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> r = ( s +Q t ) )
83	81, 82	breqtrrd 3787	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) /\ ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q r )
84	83	ex 108	. . . . . . . . . 10 |- ( ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ m e. N. ) -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s -> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q r ) )
85	84	reximdva 2418	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( E. m e. N. ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) <Q s -> E. m e. N. ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q r ) )
86	53, 85	mpd 13	. . . . . . . 8 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> E. m e. N. ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q r )
87	50	oveq1d 5505	. . . . . . . . . 10 |- ( j = m -> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) = ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) )
88	87	breq1d 3771	. . . . . . . . 9 |- ( j = m -> ( ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q r <-> ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q r ) )
89	88	cbvrexv 2531	. . . . . . . 8 |- ( E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q r <-> E. m e. N. ( ( ( F ` m ) +Q ( *Q ` [ <. m , 1o >. ] ~Q ) ) +Q S ) <Q r )
90	86, 89	sylibr 137	. . . . . . 7 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q r )
91		breq2 3765	. . . . . . . . 9 |- ( u = r -> ( ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u <-> ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q r ) )
92	91	rexbidv 2324	. . . . . . . 8 |- ( u = r -> ( E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u <-> E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q r ) )
93	92	elrab 2695	. . . . . . 7 |- ( r e. { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } <-> ( r e. Q. /\ E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q r ) )
94	44, 90, 93	sylanbrc 394	. . . . . 6 |- ( ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> r e. { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } )
95	94	ex 108	. . . . 5 |- ( ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> ( r = ( s +Q t ) -> r e. { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } ) )
96	95	rexlimdvva 2437	. . . 4 |- ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> ( E. s e. ( 2nd ` L ) E. t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) r = ( s +Q t ) -> r e. { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } ) )
97	13, 96	mpd 13	. . 3 |- ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> r e. { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } )
98	97	ex 108	. 2 |- ( ph -> ( r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) -> r e. { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } ) )
99	98	ssrdv 2948	1 |- ( ph -> ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) C_ { u e. Q. | E. j e. N. ( ( ( F ` j ) +Q ( *Q ` [ <. j , 1o >. ] ~Q ) ) +Q S ) <Q u } )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   /\ w3a 885   = wceq 1243   e. wcel 1393   {cab 2026   A.wral 2303   E.wrex 2304   {crab 2307   C_ wss 2914   <.cop 3375   class class class wbr 3761   -->wf 4876   ` cfv 4880  (class class class)co 5490   2ndc2nd 5744   1oc1o 5972   [cec 6082   N.cnpi 6342   <N clti 6345   ~Q ceq 6349   Q.cnq 6350   +Q cplq 6352   *Qcrq 6354   <Q cltq 6355   P.cnp 6361   +P. cpp 6363

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3869  ax-sep 3872  ax-nul 3880  ax-pow 3924  ax-pr 3941  ax-un 4157  ax-setind 4247  ax-iinf 4289

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2308  df-rex 2309  df-reu 2310  df-rab 2312  df-v 2556  df-sbc 2762  df-csb 2850  df-dif 2917  df-un 2919  df-in 2921  df-ss 2928  df-nul 3222  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-uni 3578  df-int 3613  df-iun 3656  df-br 3762  df-opab 3816  df-mpt 3817  df-tr 3852  df-eprel 4023  df-id 4027  df-po 4030  df-iso 4031  df-iord 4090  df-on 4092  df-suc 4095  df-iom 4292  df-xp 4329  df-rel 4330  df-cnv 4331  df-co 4332  df-dm 4333  df-rn 4334  df-res 4335  df-ima 4336  df-iota 4845  df-fun 4882  df-fn 4883  df-f 4884  df-f1 4885  df-fo 4886  df-f1o 4887  df-fv 4888  df-ov 5493  df-oprab 5494  df-mpt2 5495  df-1st 5745  df-2nd 5746  df-recs 5898  df-irdg 5935  df-1o 5979  df-oadd 5983  df-omul 5984  df-er 6084  df-ec 6086  df-qs 6090  df-ni 6374  df-pli 6375  df-mi 6376  df-lti 6377  df-plpq 6414  df-mpq 6415  df-enq 6417  df-nqqs 6418  df-plqqs 6419  df-mqqs 6420  df-1nqqs 6421  df-rq 6422  df-ltnqqs 6423  df-inp 6536  df-iplp 6538

This theorem is referenced by:  caucvgprlemladdrl  6748