Theorem cauappcvgprlemladdfu 7874

Description: Lemma for cauappcvgprlemladd 7878. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.)

Hypotheses

Ref	Expression
cauappcvgpr.f	|- ( ph -> F : Q. --> Q. )
cauappcvgpr.app	|- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )
cauappcvgpr.bnd	|- ( ph -> A. p e. Q. A <Q ( F ` p ) )
cauappcvgpr.lim	|- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.
cauappcvgprlemladd.s	|- ( ph -> S e. Q. )

Assertion

Ref	Expression
cauappcvgprlemladdfu	|- ( ph -> ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) C_ ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) )

Distinct variable groups:   A, p   L, p, q   ph, p, q   F, l, u, p, q   S, l, q, u

Allowed substitution hints:   ph( u, l)   A( u, q, l)   S( p)   L( u, l)

Proof of Theorem cauappcvgprlemladdfu

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

Step	Hyp	Ref	Expression
1		cauappcvgpr.f	. . . . . . 7 |- ( ph -> F : Q. --> Q. )
2		cauappcvgpr.app	. . . . . . 7 |- ( ph -> A. p e. Q. A. q e. Q. ( ( F ` p ) <Q ( ( F ` q ) +Q ( p +Q q ) ) /\ ( F ` q ) <Q ( ( F ` p ) +Q ( p +Q q ) ) ) )
3		cauappcvgpr.bnd	. . . . . . 7 |- ( ph -> A. p e. Q. A <Q ( F ` p ) )
4		cauappcvgpr.lim	. . . . . . 7 |- L = <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >.
5	1, 2, 3, 4	cauappcvgprlemcl 7873	. . . . . 6 |- ( ph -> L e. P. )
6		cauappcvgprlemladd.s	. . . . . . 7 |- ( ph -> S e. Q. )
7		nqprlu 7767	. . . . . . 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 7688	. . . . . . 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 7595	. . . . . . 7 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
11	9, 10	genpelvu 7733	. . . . . 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 411	. . . . 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 296	. . . 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 4092	. . . . . . . . . . . . . . . 16 |- ( u = s -> ( ( ( F ` q ) +Q q ) <Q u <-> ( ( F ` q ) +Q q ) <Q s ) )
15	14	rexbidv 2533	. . . . . . . . . . . . . . 15 |- ( u = s -> ( E. q e. Q. ( ( F ` q ) +Q q ) <Q u <-> E. q e. Q. ( ( F ` q ) +Q q ) <Q s ) )
16	4	fveq2i 5642	. . . . . . . . . . . . . . . 16 |- ( 2nd ` L ) = ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >. )
17		nqex 7583	. . . . . . . . . . . . . . . . . 18 |- Q. e. _V
18	17	rabex 4234	. . . . . . . . . . . . . . . . 17 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
19	17	rabex 4234	. . . . . . . . . . . . . . . . 17 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
20	18, 19	op2nd 6310	. . . . . . . . . . . . . . . 16 |- ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } , { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } >. ) = { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u }
21	16, 20	eqtri 2252	. . . . . . . . . . . . . . 15 |- ( 2nd ` L ) = { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u }
22	15, 21	elrab2 2965	. . . . . . . . . . . . . 14 |- ( s e. ( 2nd ` L ) <-> ( s e. Q. /\ E. q e. Q. ( ( F ` q ) +Q q ) <Q s ) )
23	22	biimpi 120	. . . . . . . . . . . . 13 |- ( s e. ( 2nd ` L ) -> ( s e. Q. /\ E. q e. Q. ( ( F ` q ) +Q q ) <Q s ) )
24	23	adantr 276	. . . . . . . . . . . 12 |- ( ( s e. ( 2nd ` L ) /\ t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) ) -> ( s e. Q. /\ E. q e. Q. ( ( F ` q ) +Q q ) <Q s ) )
25	24	adantl 277	. . . . . . . . . . 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. q e. Q. ( ( F ` q ) +Q q ) <Q s ) )
26	25	adantr 276	. . . . . . . . . 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. q e. Q. ( ( F ` q ) +Q q ) <Q s ) )
27	26	simpld 112	. . . . . . . . 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 2805	. . . . . . . . . . . . . 14 |- t e. _V
29		breq2 4092	. . . . . . . . . . . . . 14 |- ( u = t -> ( S <Q u <-> S <Q t ) )
30		ltnqex 7769	. . . . . . . . . . . . . . 15 |- { l | l <Q S } e. _V
31		gtnqex 7770	. . . . . . . . . . . . . . 15 |- { u | S <Q u } e. _V
32	30, 31	op2nd 6310	. . . . . . . . . . . . . 14 |- ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) = { u | S <Q u }
33	28, 29, 32	elab2 2954	. . . . . . . . . . . . 13 |- ( t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) <-> S <Q t )
34		ltrelnq 7585	. . . . . . . . . . . . . 14 |- <Q C_ ( Q. X. Q. )
35	34	brel 4778	. . . . . . . . . . . . 13 |- ( S <Q t -> ( S e. Q. /\ t e. Q. ) )
36	33, 35	sylbi 121	. . . . . . . . . . . 12 |- ( t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) -> ( S e. Q. /\ t e. Q. ) )
37	36	simprd 114	. . . . . . . . . . 11 |- ( t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) -> t e. Q. )
38	37	ad2antll 491	. . . . . . . . . 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 276	. . . . . . . . 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 7595	. . . . . . . . 9 |- ( ( s e. Q. /\ t e. Q. ) -> ( s +Q t ) e. Q. )
41	27, 39, 40	syl2anc 411	. . . . . . . 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 2294	. . . . . . . . 9 |- ( r = ( s +Q t ) -> ( r e. Q. <-> ( s +Q t ) e. Q. ) )
43	42	adantl 277	. . . . . . . 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 167	. . . . . . 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 114	. . . . . . . 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. q e. Q. ( ( F ` q ) +Q q ) <Q s )
46	33	biimpi 120	. . . . . . . . . . . . . . . 16 |- ( t e. ( 2nd ` <. { l | l <Q S } , { u | S <Q u } >. ) -> S <Q t )
47	46	ad2antll 491	. . . . . . . . . . . . . . 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 } >. ) ) ) -> S <Q t )
48	47	adantr 276	. . . . . . . . . . . . . 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 ) ) -> S <Q t )
49	48	ad2antrr 488	. . . . . . . . . . . . 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 ) ) /\ q e. Q. ) /\ ( ( F ` q ) +Q q ) <Q s ) -> S <Q t )
50	6	ad5antr 496	. . . . . . . . . . . . . 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 ) ) /\ q e. Q. ) /\ ( ( F ` q ) +Q q ) <Q s ) -> S e. Q. )
51	39	ad2antrr 488	. . . . . . . . . . . . . 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 ) ) /\ q e. Q. ) /\ ( ( F ` q ) +Q q ) <Q s ) -> t e. Q. )
52	1	ad5antr 496	. . . . . . . . . . . . . . . 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 ) ) /\ q e. Q. ) /\ ( ( F ` q ) +Q q ) <Q s ) -> F : Q. --> Q. )
53		simplr 529	. . . . . . . . . . . . . . . 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 ) ) /\ q e. Q. ) /\ ( ( F ` q ) +Q q ) <Q s ) -> q e. Q. )
54	52, 53	ffvelcdmd 5783	. . . . . . . . . . . . . . 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 ) ) /\ q e. Q. ) /\ ( ( F ` q ) +Q q ) <Q s ) -> ( F ` q ) e. Q. )
55		addclnq 7595	. . . . . . . . . . . . . . 15 |- ( ( ( F ` q ) e. Q. /\ q e. Q. ) -> ( ( F ` q ) +Q q ) e. Q. )
56	54, 53, 55	syl2anc 411	. . . . . . . . . . . . . 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 ) ) /\ q e. Q. ) /\ ( ( F ` q ) +Q q ) <Q s ) -> ( ( F ` q ) +Q q ) e. Q. )
57		ltanqg 7620	. . . . . . . . . . . . . 14 |- ( ( S e. Q. /\ t e. Q. /\ ( ( F ` q ) +Q q ) e. Q. ) -> ( S <Q t <-> ( ( ( F ` q ) +Q q ) +Q S ) <Q ( ( ( F ` q ) +Q q ) +Q t ) ) )
58	50, 51, 56, 57	syl3anc 1273	. . . . . . . . . . . . 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 ) ) /\ q e. Q. ) /\ ( ( F ` q ) +Q q ) <Q s ) -> ( S <Q t <-> ( ( ( F ` q ) +Q q ) +Q S ) <Q ( ( ( F ` q ) +Q q ) +Q t ) ) )
59	49, 58	mpbid 147	. . . . . . . . . . . 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 ) ) /\ q e. Q. ) /\ ( ( F ` q ) +Q q ) <Q s ) -> ( ( ( F ` q ) +Q q ) +Q S ) <Q ( ( ( F ` q ) +Q q ) +Q t ) )
60		simpr 110	. . . . . . . . . . . . 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 ) ) /\ q e. Q. ) /\ ( ( F ` q ) +Q q ) <Q s ) -> ( ( F ` q ) +Q q ) <Q s )
61		ltanqg 7620	. . . . . . . . . . . . . . 15 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
62	61	adantl 277	. . . . . . . . . . . . . 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 ) ) /\ q e. Q. ) /\ ( ( F ` q ) +Q q ) <Q s ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( f <Q g <-> ( h +Q f ) <Q ( h +Q g ) ) )
63	27	ad2antrr 488	. . . . . . . . . . . . . 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 ) ) /\ q e. Q. ) /\ ( ( F ` q ) +Q q ) <Q s ) -> s e. Q. )
64		addcomnqg 7601	. . . . . . . . . . . . . . 15 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
65	64	adantl 277	. . . . . . . . . . . . . 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 ) ) /\ q e. Q. ) /\ ( ( F ` q ) +Q q ) <Q s ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
66	62, 56, 63, 51, 65	caovord2d 6192	. . . . . . . . . . . . 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 ) ) /\ q e. Q. ) /\ ( ( F ` q ) +Q q ) <Q s ) -> ( ( ( F ` q ) +Q q ) <Q s <-> ( ( ( F ` q ) +Q q ) +Q t ) <Q ( s +Q t ) ) )
67	60, 66	mpbid 147	. . . . . . . . . . . 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 ) ) /\ q e. Q. ) /\ ( ( F ` q ) +Q q ) <Q s ) -> ( ( ( F ` q ) +Q q ) +Q t ) <Q ( s +Q t ) )
68		ltsonq 7618	. . . . . . . . . . . . 13 |- <Q Or Q.
69	68, 34	sotri 5132	. . . . . . . . . . . 12 |- ( ( ( ( ( F ` q ) +Q q ) +Q S ) <Q ( ( ( F ` q ) +Q q ) +Q t ) /\ ( ( ( F ` q ) +Q q ) +Q t ) <Q ( s +Q t ) ) -> ( ( ( F ` q ) +Q q ) +Q S ) <Q ( s +Q t ) )
70	59, 67, 69	syl2anc 411	. . . . . . . . . . 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 ) ) /\ q e. Q. ) /\ ( ( F ` q ) +Q q ) <Q s ) -> ( ( ( F ` q ) +Q q ) +Q S ) <Q ( s +Q t ) )
71		simpllr 536	. . . . . . . . . . 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 ) ) /\ q e. Q. ) /\ ( ( F ` q ) +Q q ) <Q s ) -> r = ( s +Q t ) )
72	70, 71	breqtrrd 4116	. . . . . . . . . 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 ) ) /\ q e. Q. ) /\ ( ( F ` q ) +Q q ) <Q s ) -> ( ( ( F ` q ) +Q q ) +Q S ) <Q r )
73	72	ex 115	. . . . . . . . 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 ) ) /\ q e. Q. ) -> ( ( ( F ` q ) +Q q ) <Q s -> ( ( ( F ` q ) +Q q ) +Q S ) <Q r ) )
74	73	reximdva 2634	. . . . . . . 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. q e. Q. ( ( F ` q ) +Q q ) <Q s -> E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q r ) )
75	45, 74	mpd 13	. . . . . . 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. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q r )
76		breq2 4092	. . . . . . . . 9 |- ( u = r -> ( ( ( ( F ` q ) +Q q ) +Q S ) <Q u <-> ( ( ( F ` q ) +Q q ) +Q S ) <Q r ) )
77	76	rexbidv 2533	. . . . . . . 8 |- ( u = r -> ( E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u <-> E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q r ) )
78	17	rabex 4234	. . . . . . . . 9 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } e. _V
79	17	rabex 4234	. . . . . . . . 9 |- { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } e. _V
80	78, 79	op2nd 6310	. . . . . . . 8 |- ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) = { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u }
81	77, 80	elrab2 2965	. . . . . . 7 |- ( r e. ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) <-> ( r e. Q. /\ E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q r ) )
82	44, 75, 81	sylanbrc 417	. . . . . 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. ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) )
83	82	ex 115	. . . . 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. ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) ) )
84	83	rexlimdvva 2658	. . . 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. ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) ) )
85	13, 84	mpd 13	. . 3 |- ( ( ph /\ r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> r e. ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) )
86	85	ex 115	. 2 |- ( ph -> ( r e. ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) -> r e. ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) ) )
87	86	ssrdv 3233	1 |- ( ph -> ( 2nd ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) C_ ( 2nd ` <. { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } , { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } >. ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   {cab 2217   A.wral 2510   E.wrex 2511   {crab 2514   C_ wss 3200   <.cop 3672   class class class wbr 4088   -->wf 5322   ` cfv 5326  (class class class)co 6018   2ndc2nd 6302   Q.cnq 7500   +Q cplq 7502   <Q cltq 7505   P.cnp 7511   +P. cpp 7513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-irdg 6536  df-1o 6582  df-oadd 6586  df-omul 6587  df-er 6702  df-ec 6704  df-qs 6708  df-ni 7524  df-pli 7525  df-mi 7526  df-lti 7527  df-plpq 7564  df-mpq 7565  df-enq 7567  df-nqqs 7568  df-plqqs 7569  df-mqqs 7570  df-1nqqs 7571  df-rq 7572  df-ltnqqs 7573  df-inp 7686  df-iplp 7688

This theorem is referenced by:  cauappcvgprlemladdrl  7877  cauappcvgprlemladd  7878