Theorem caucvgprlemladdfu 6648

Description: Lemma for caucvgpr 6653. 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 6647	. . . . . 6 |- (ph -> L e. P.)
6		caucvgprlemladd.s	. . . . . . 7 |- (ph -> S e. Q.)
7		nqprlu 6530	. . . . . . 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 6451	. . . . . . 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 6359	. . . . . . 7 |- ((g e. Q. /\ h e. Q.) -> (g +Q h) e. Q.)
11	9, 10	genpelvu 6496	. . . . . 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 3759	. . . . . . . . . . . . . . . 16 |- (u = s -> ((( F ` j) +Q ( *Q ` [ <. j , 1o >.] ~Q )) <Q u <-> (( F ` j) +Q ( *Q ` [ <. j , 1o >.] ~Q )) <Q s))
15	14	rexbidv 2321	. . . . . . . . . . . . . . 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 5124	. . . . . . . . . . . . . . . 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 6347	. . . . . . . . . . . . . . . . . 18 |- Q. e. _V
18	17	rabex 3892	. . . . . . . . . . . . . . . . 17 |- { l e. Q. | E. j e. N. ( l +Q ( *Q ` [ <. j , 1o >.] ~Q )) <Q ( F ` j) } e. _V
19	17	rabex 3892	. . . . . . . . . . . . . . . . 17 |- {u e. Q. | E. j e. N. (( F ` j) +Q ( *Q ` [ <. j , 1o >.] ~Q )) <Q u } e. _V
20	18, 19	op2nd 5716	. . . . . . . . . . . . . . . 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 2057	. . . . . . . . . . . . . . 15 |- ( 2nd ` L) = {u e. Q. | E. j e. N. (( F ` j) +Q ( *Q ` [ <. j , 1o >.] ~Q )) <Q u }
22	15, 21	elrab2 2694	. . . . . . . . . . . . . 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 2554	. . . . . . . . . . . . . 14 |- t e. _V
29		breq2 3759	. . . . . . . . . . . . . 14 |- (u = t -> ( S <Q u <-> S <Q t))
30		ltnqex 6531	. . . . . . . . . . . . . . 15 |- { l | l <Q S } e. _V
31		gtnqex 6532	. . . . . . . . . . . . . . 15 |- {u | S <Q u } e. _V
32	30, 31	op2nd 5716	. . . . . . . . . . . . . 14 |- ( 2nd ` <. { l | l <Q S } , {u | S <Q u } >.) = {u | S <Q u }
33	28, 29, 32	elab2 2684	. . . . . . . . . . . . 13 |- ( t e. ( 2nd ` <. { l | l <Q S } , {u | S <Q u } >.) <-> S <Q t)
34		ltrelnq 6349	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q.)
35	34	brel 4335	. . . . . . . . . . . . 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 6359	. . . . . . . . 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 2097	. . . . . . . . 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 5121	. . . . . . . . . . . . 13 |- ( j = m -> ( F ` j) = ( F ` m))
47		opeq1 3540	. . . . . . . . . . . . . . 15 |- ( j = m -> <. j , 1o >. = <. m , 1o >.)
48	47	eceq1d 6078	. . . . . . . . . . . . . 14 |- ( j = m -> [ <. j , 1o >.] ~Q = [ <. m , 1o >.] ~Q )
49	48	fveq2d 5125	. . . . . . . . . . . . 13 |- ( j = m -> ( *Q ` [ <. j , 1o >.] ~Q ) = ( *Q ` [ <. m , 1o >.] ~Q ))
50	46, 49	oveq12d 5473	. . . . . . . . . . . 12 |- ( j = m -> (( F ` j) +Q ( *Q ` [ <. j , 1o >.] ~Q )) = (( F ` m) +Q ( *Q ` [ <. m , 1o >.] ~Q )))
51	50	breq1d 3765	. . . . . . . . . . 11 |- ( j = m -> ((( F ` j) +Q ( *Q ` [ <. j , 1o >.] ~Q )) <Q s <-> (( F ` m) +Q ( *Q ` [ <. m , 1o >.] ~Q )) <Q s))
52	51	cbvrexv 2528	. . . . . . . . . 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 5246	. . . . . . . . . . . . . . . 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 6405	. . . . . . . . . . . . . . . . 17 |- ( m e. N. -> [ <. m , 1o >.] ~Q e. Q.)
64		recclnq 6376	. . . . . . . . . . . . . . . . 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 6359	. . . . . . . . . . . . . . . 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 6384	. . . . . . . . . . . . . . 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 1134	. . . . . . . . . . . . . 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 6384	. . . . . . . . . . . . . . . 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 6365	. . . . . . . . . . . . . . . 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 5612	. . . . . . . . . . . . . 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 6382	. . . . . . . . . . . . . 14 |- <Q Or Q.
80	79, 34	sotri 4663	. . . . . . . . . . . . 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 3781	. . . . . . . . . . 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 2415	. . . . . . . . 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 5470	. . . . . . . . . 10 |- ( j = m -> ((( F ` j) +Q ( *Q ` [ <. j , 1o >.] ~Q )) +Q S) = ((( F ` m) +Q ( *Q ` [ <. m , 1o >.] ~Q )) +Q S))
88	87	breq1d 3765	. . . . . . . . 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 2528	. . . . . . . 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 3759	. . . . . . . . 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 2321	. . . . . . . 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 2692	. . . . . . 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 2434	. . . 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 2945	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 884   = wceq 1242   e. wcel 1390   {cab 2023  A.wral 2300  E.wrex 2301   {crab 2304   C_ wss 2911   <.cop 3370   class class class wbr 3755   -->wf 4841   ` cfv 4845  (class class class)co 5455   2ndc2nd 5708   1oc1o 5933  [cec 6040   N.cnpi 6256   <N clti 6259   ~Q ceq 6263   Q.cnq 6264   +Q cplq 6266   *Qcrq 6268   <Q cltq 6269   P.cnp 6275   +P. cpp 6277

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-inp 6449  df-iplp 6451

This theorem is referenced by:  caucvgprlemladdrl  6649