Theorem cauappcvgprlemladdfl 7596

Description: Lemma for cauappcvgprlemladd 7599. The forward subset relationship for the lower 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
cauappcvgprlemladdfl	|- ( ph -> ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) C_ ( 1st ` <. { 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 cauappcvgprlemladdfl

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 7594	. . . . . 6 |- ( ph -> L e. P. )
6		cauappcvgprlemladd.s	. . . . . . 7 |- ( ph -> S e. Q. )
7		nqprlu 7488	. . . . . . 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 7409	. . . . . . 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 7316	. . . . . . 7 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
11	9, 10	genpelvl 7453	. . . . . 6 |- ( ( L e. P. /\ <. { l | l <Q S } , { u | S <Q u } >. e. P. ) -> ( r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) <-> E. s e. ( 1st ` L ) E. t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) r = ( s +Q t ) ) )
12	5, 8, 11	syl2anc 409	. . . . 5 |- ( ph -> ( r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) <-> E. s e. ( 1st ` L ) E. t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) r = ( s +Q t ) ) )
13	12	biimpa 294	. . . 4 |- ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> E. s e. ( 1st ` L ) E. t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) r = ( s +Q t ) )
14		oveq1 5849	. . . . . . . . . . . . . . . 16 |- ( l = s -> ( l +Q q ) = ( s +Q q ) )
15	14	breq1d 3992	. . . . . . . . . . . . . . 15 |- ( l = s -> ( ( l +Q q ) <Q ( F ` q ) <-> ( s +Q q ) <Q ( F ` q ) ) )
16	15	rexbidv 2467	. . . . . . . . . . . . . 14 |- ( l = s -> ( E. q e. Q. ( l +Q q ) <Q ( F ` q ) <-> E. q e. Q. ( s +Q q ) <Q ( F ` q ) ) )
17	4	fveq2i 5489	. . . . . . . . . . . . . . 15 |- ( 1st ` L ) = ( 1st ` <. { 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 } >. )
18		nqex 7304	. . . . . . . . . . . . . . . . 17 |- Q. e. _V
19	18	rabex 4126	. . . . . . . . . . . . . . . 16 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
20	18	rabex 4126	. . . . . . . . . . . . . . . 16 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
21	19, 20	op1st 6114	. . . . . . . . . . . . . . 15 |- ( 1st ` <. { 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 } >. ) = { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) }
22	17, 21	eqtri 2186	. . . . . . . . . . . . . 14 |- ( 1st ` L ) = { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) }
23	16, 22	elrab2 2885	. . . . . . . . . . . . 13 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. q e. Q. ( s +Q q ) <Q ( F ` q ) ) )
24	23	biimpi 119	. . . . . . . . . . . 12 |- ( s e. ( 1st ` L ) -> ( s e. Q. /\ E. q e. Q. ( s +Q q ) <Q ( F ` q ) ) )
25	24	ad2antrl 482	. . . . . . . . . . 11 |- ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> ( s e. Q. /\ E. q e. Q. ( s +Q q ) <Q ( F ` q ) ) )
26	25	adantr 274	. . . . . . . . . 10 |- ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( s e. Q. /\ E. q e. Q. ( s +Q q ) <Q ( F ` q ) ) )
27	26	simpld 111	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> s e. Q. )
28		vex 2729	. . . . . . . . . . . . . . 15 |- t e. _V
29		breq1 3985	. . . . . . . . . . . . . . 15 |- ( l = t -> ( l <Q S <-> t <Q S ) )
30		ltnqex 7490	. . . . . . . . . . . . . . . 16 |- { l | l <Q S } e. _V
31		gtnqex 7491	. . . . . . . . . . . . . . . 16 |- { u | S <Q u } e. _V
32	30, 31	op1st 6114	. . . . . . . . . . . . . . 15 |- ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) = { l | l <Q S }
33	28, 29, 32	elab2 2874	. . . . . . . . . . . . . 14 |- ( t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) <-> t <Q S )
34	33	biimpi 119	. . . . . . . . . . . . 13 |- ( t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) -> t <Q S )
35	34	ad2antll 483	. . . . . . . . . . . 12 |- ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> t <Q S )
36	35	adantr 274	. . . . . . . . . . 11 |- ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> t <Q S )
37		ltrelnq 7306	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
38	37	brel 4656	. . . . . . . . . . 11 |- ( t <Q S -> ( t e. Q. /\ S e. Q. ) )
39	36, 38	syl 14	. . . . . . . . . 10 |- ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( t e. Q. /\ S e. Q. ) )
40	39	simpld 111	. . . . . . . . 9 |- ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> t e. Q. )
41		addclnq 7316	. . . . . . . . 9 |- ( ( s e. Q. /\ t e. Q. ) -> ( s +Q t ) e. Q. )
42	27, 40, 41	syl2anc 409	. . . . . . . 8 |- ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( s +Q t ) e. Q. )
43		eleq1 2229	. . . . . . . . 9 |- ( r = ( s +Q t ) -> ( r e. Q. <-> ( s +Q t ) e. Q. ) )
44	43	adantl 275	. . . . . . . 8 |- ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( r e. Q. <-> ( s +Q t ) e. Q. ) )
45	42, 44	mpbird 166	. . . . . . 7 |- ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> r e. Q. )
46	26	simprd 113	. . . . . . . 8 |- ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> E. q e. Q. ( s +Q q ) <Q ( F ` q ) )
47	27	ad2antrr 480	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ q e. Q. ) /\ ( s +Q q ) <Q ( F ` q ) ) -> s e. Q. )
48		simplr 520	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ q e. Q. ) /\ ( s +Q q ) <Q ( F ` q ) ) -> q e. Q. )
49	40	ad2antrr 480	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ q e. Q. ) /\ ( s +Q q ) <Q ( F ` q ) ) -> t e. Q. )
50		addcomnqg 7322	. . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
51	50	adantl 275	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ q e. Q. ) /\ ( s +Q q ) <Q ( F ` q ) ) /\ ( f e. Q. /\ g e. Q. ) ) -> ( f +Q g ) = ( g +Q f ) )
52		addassnqg 7323	. . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( ( f +Q g ) +Q h ) = ( f +Q ( g +Q h ) ) )
53	52	adantl 275	. . . . . . . . . . . . 13 |- ( ( ( ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ q e. Q. ) /\ ( s +Q q ) <Q ( F ` q ) ) /\ ( f e. Q. /\ g e. Q. /\ h e. Q. ) ) -> ( ( f +Q g ) +Q h ) = ( f +Q ( g +Q h ) ) )
54	47, 48, 49, 51, 53	caov32d 6022	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ q e. Q. ) /\ ( s +Q q ) <Q ( F ` q ) ) -> ( ( s +Q q ) +Q t ) = ( ( s +Q t ) +Q q ) )
55		simpr 109	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ ( s +Q q ) <Q ( F ` q ) ) -> ( s +Q q ) <Q ( F ` q ) )
56	35	ad2antrr 480	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ ( s +Q q ) <Q ( F ` q ) ) -> t <Q S )
57	37	brel 4656	. . . . . . . . . . . . . . 15 |- ( ( s +Q q ) <Q ( F ` q ) -> ( ( s +Q q ) e. Q. /\ ( F ` q ) e. Q. ) )
58		lt2addnq 7345	. . . . . . . . . . . . . . 15 |- ( ( ( ( s +Q q ) e. Q. /\ ( F ` q ) e. Q. ) /\ ( t e. Q. /\ S e. Q. ) ) -> ( ( ( s +Q q ) <Q ( F ` q ) /\ t <Q S ) -> ( ( s +Q q ) +Q t ) <Q ( ( F ` q ) +Q S ) ) )
59	57, 39, 58	syl2anr 288	. . . . . . . . . . . . . 14 |- ( ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ ( s +Q q ) <Q ( F ` q ) ) -> ( ( ( s +Q q ) <Q ( F ` q ) /\ t <Q S ) -> ( ( s +Q q ) +Q t ) <Q ( ( F ` q ) +Q S ) ) )
60	55, 56, 59	mp2and 430	. . . . . . . . . . . . 13 |- ( ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ ( s +Q q ) <Q ( F ` q ) ) -> ( ( s +Q q ) +Q t ) <Q ( ( F ` q ) +Q S ) )
61	60	adantlr 469	. . . . . . . . . . . 12 |- ( ( ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ q e. Q. ) /\ ( s +Q q ) <Q ( F ` q ) ) -> ( ( s +Q q ) +Q t ) <Q ( ( F ` q ) +Q S ) )
62	54, 61	eqbrtrrd 4006	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ q e. Q. ) /\ ( s +Q q ) <Q ( F ` q ) ) -> ( ( s +Q t ) +Q q ) <Q ( ( F ` q ) +Q S ) )
63		oveq1 5849	. . . . . . . . . . . . 13 |- ( r = ( s +Q t ) -> ( r +Q q ) = ( ( s +Q t ) +Q q ) )
64	63	breq1d 3992	. . . . . . . . . . . 12 |- ( r = ( s +Q t ) -> ( ( r +Q q ) <Q ( ( F ` q ) +Q S ) <-> ( ( s +Q t ) +Q q ) <Q ( ( F ` q ) +Q S ) ) )
65	64	ad3antlr 485	. . . . . . . . . . 11 |- ( ( ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ q e. Q. ) /\ ( s +Q q ) <Q ( F ` q ) ) -> ( ( r +Q q ) <Q ( ( F ` q ) +Q S ) <-> ( ( s +Q t ) +Q q ) <Q ( ( F ` q ) +Q S ) ) )
66	62, 65	mpbird 166	. . . . . . . . . 10 |- ( ( ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ q e. Q. ) /\ ( s +Q q ) <Q ( F ` q ) ) -> ( r +Q q ) <Q ( ( F ` q ) +Q S ) )
67	66	ex 114	. . . . . . . . 9 |- ( ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) /\ q e. Q. ) -> ( ( s +Q q ) <Q ( F ` q ) -> ( r +Q q ) <Q ( ( F ` q ) +Q S ) ) )
68	67	reximdva 2568	. . . . . . . 8 |- ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> ( E. q e. Q. ( s +Q q ) <Q ( F ` q ) -> E. q e. Q. ( r +Q q ) <Q ( ( F ` q ) +Q S ) ) )
69	46, 68	mpd 13	. . . . . . 7 |- ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> E. q e. Q. ( r +Q q ) <Q ( ( F ` q ) +Q S ) )
70		oveq1 5849	. . . . . . . . . 10 |- ( l = r -> ( l +Q q ) = ( r +Q q ) )
71	70	breq1d 3992	. . . . . . . . 9 |- ( l = r -> ( ( l +Q q ) <Q ( ( F ` q ) +Q S ) <-> ( r +Q q ) <Q ( ( F ` q ) +Q S ) ) )
72	71	rexbidv 2467	. . . . . . . 8 |- ( l = r -> ( E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) <-> E. q e. Q. ( r +Q q ) <Q ( ( F ` q ) +Q S ) ) )
73	18	rabex 4126	. . . . . . . . 9 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } e. _V
74	18	rabex 4126	. . . . . . . . 9 |- { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } e. _V
75	73, 74	op1st 6114	. . . . . . . 8 |- ( 1st ` <. { 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 } >. ) = { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) }
76	72, 75	elrab2 2885	. . . . . . 7 |- ( r e. ( 1st ` <. { 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. ( r +Q q ) <Q ( ( F ` q ) +Q S ) ) )
77	45, 69, 76	sylanbrc 414	. . . . . 6 |- ( ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ r = ( s +Q t ) ) -> r e. ( 1st ` <. { 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 } >. ) )
78	77	ex 114	. . . . 5 |- ( ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) /\ ( s e. ( 1st ` L ) /\ t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> ( r = ( s +Q t ) -> r e. ( 1st ` <. { 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 } >. ) ) )
79	78	rexlimdvva 2591	. . . 4 |- ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> ( E. s e. ( 1st ` L ) E. t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) r = ( s +Q t ) -> r e. ( 1st ` <. { 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 } >. ) ) )
80	13, 79	mpd 13	. . 3 |- ( ( ph /\ r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) ) -> r e. ( 1st ` <. { 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 } >. ) )
81	80	ex 114	. 2 |- ( ph -> ( r e. ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) -> r e. ( 1st ` <. { 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 } >. ) ) )
82	81	ssrdv 3148	1 |- ( ph -> ( 1st ` ( L +P. <. { l | l <Q S } , { u | S <Q u } >. ) ) C_ ( 1st ` <. { 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 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   {cab 2151   A.wral 2444   E.wrex 2445   {crab 2448   C_ wss 3116   <.cop 3579   class class class wbr 3982   -->wf 5184   ` cfv 5188  (class class class)co 5842   1stc1st 6106   Q.cnq 7221   +Q cplq 7223   <Q cltq 7226   P.cnp 7232   +P. cpp 7234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-inp 7407  df-iplp 7409

This theorem is referenced by:  cauappcvgprlemladdru  7597  cauappcvgprlemladd  7599