Theorem cauappcvgprlemladdfl 7656

Description: Lemma for cauappcvgprlemladd 7659. 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 7654	. . . . . 6 |- ( ph -> L e. P. )
6		cauappcvgprlemladd.s	. . . . . . 7 |- ( ph -> S e. Q. )
7		nqprlu 7548	. . . . . . 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 7469	. . . . . . 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 7376	. . . . . . 7 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
11	9, 10	genpelvl 7513	. . . . . 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 411	. . . . 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 296	. . . 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 5884	. . . . . . . . . . . . . . . 16 |- ( l = s -> ( l +Q q ) = ( s +Q q ) )
15	14	breq1d 4015	. . . . . . . . . . . . . . 15 |- ( l = s -> ( ( l +Q q ) <Q ( F ` q ) <-> ( s +Q q ) <Q ( F ` q ) ) )
16	15	rexbidv 2478	. . . . . . . . . . . . . 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 5520	. . . . . . . . . . . . . . 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 7364	. . . . . . . . . . . . . . . . 17 |- Q. e. _V
19	18	rabex 4149	. . . . . . . . . . . . . . . 16 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
20	18	rabex 4149	. . . . . . . . . . . . . . . 16 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
21	19, 20	op1st 6149	. . . . . . . . . . . . . . 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 2198	. . . . . . . . . . . . . 14 |- ( 1st ` L ) = { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) }
23	16, 22	elrab2 2898	. . . . . . . . . . . . 13 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. q e. Q. ( s +Q q ) <Q ( F ` q ) ) )
24	23	biimpi 120	. . . . . . . . . . . 12 |- ( s e. ( 1st ` L ) -> ( s e. Q. /\ E. q e. Q. ( s +Q q ) <Q ( F ` q ) ) )
25	24	ad2antrl 490	. . . . . . . . . . 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 276	. . . . . . . . . 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 112	. . . . . . . . 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 2742	. . . . . . . . . . . . . . 15 |- t e. _V
29		breq1 4008	. . . . . . . . . . . . . . 15 |- ( l = t -> ( l <Q S <-> t <Q S ) )
30		ltnqex 7550	. . . . . . . . . . . . . . . 16 |- { l | l <Q S } e. _V
31		gtnqex 7551	. . . . . . . . . . . . . . . 16 |- { u | S <Q u } e. _V
32	30, 31	op1st 6149	. . . . . . . . . . . . . . 15 |- ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) = { l | l <Q S }
33	28, 29, 32	elab2 2887	. . . . . . . . . . . . . 14 |- ( t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) <-> t <Q S )
34	33	biimpi 120	. . . . . . . . . . . . 13 |- ( t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) -> t <Q S )
35	34	ad2antll 491	. . . . . . . . . . . 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 276	. . . . . . . . . . 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 7366	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
38	37	brel 4680	. . . . . . . . . . 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 112	. . . . . . . . 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 7376	. . . . . . . . 9 |- ( ( s e. Q. /\ t e. Q. ) -> ( s +Q t ) e. Q. )
42	27, 40, 41	syl2anc 411	. . . . . . . 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 2240	. . . . . . . . 9 |- ( r = ( s +Q t ) -> ( r e. Q. <-> ( s +Q t ) e. Q. ) )
44	43	adantl 277	. . . . . . . 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 167	. . . . . . 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 114	. . . . . . . 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 488	. . . . . . . . . . . . 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 528	. . . . . . . . . . . . 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 488	. . . . . . . . . . . . 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 7382	. . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
51	50	adantl 277	. . . . . . . . . . . . 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 7383	. . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( ( f +Q g ) +Q h ) = ( f +Q ( g +Q h ) ) )
53	52	adantl 277	. . . . . . . . . . . . 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 6057	. . . . . . . . . . . 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 110	. . . . . . . . . . . . . 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 488	. . . . . . . . . . . . . 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 4680	. . . . . . . . . . . . . . 15 |- ( ( s +Q q ) <Q ( F ` q ) -> ( ( s +Q q ) e. Q. /\ ( F ` q ) e. Q. ) )
58		lt2addnq 7405	. . . . . . . . . . . . . . 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 290	. . . . . . . . . . . . . 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 433	. . . . . . . . . . . . 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 477	. . . . . . . . . . . 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 4029	. . . . . . . . . . 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 5884	. . . . . . . . . . . . 13 |- ( r = ( s +Q t ) -> ( r +Q q ) = ( ( s +Q t ) +Q q ) )
64	63	breq1d 4015	. . . . . . . . . . . 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 493	. . . . . . . . . . 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 167	. . . . . . . . . 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 115	. . . . . . . . 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 2579	. . . . . . . 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 5884	. . . . . . . . . 10 |- ( l = r -> ( l +Q q ) = ( r +Q q ) )
71	70	breq1d 4015	. . . . . . . . 9 |- ( l = r -> ( ( l +Q q ) <Q ( ( F ` q ) +Q S ) <-> ( r +Q q ) <Q ( ( F ` q ) +Q S ) ) )
72	71	rexbidv 2478	. . . . . . . 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 4149	. . . . . . . . 9 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } e. _V
74	18	rabex 4149	. . . . . . . . 9 |- { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } e. _V
75	73, 74	op1st 6149	. . . . . . . 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 2898	. . . . . . 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 417	. . . . . 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 115	. . . . 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 2602	. . . 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 115	. 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 3163	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 104   <-> wb 105   /\ w3a 978   = wceq 1353   e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   {crab 2459   C_ wss 3131   <.cop 3597   class class class wbr 4005   -->wf 5214   ` cfv 5218  (class class class)co 5877   1stc1st 6141   Q.cnq 7281   +Q cplq 7283   <Q cltq 7286   P.cnp 7292   +P. cpp 7294

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-eprel 4291  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-1o 6419  df-oadd 6423  df-omul 6424  df-er 6537  df-ec 6539  df-qs 6543  df-ni 7305  df-pli 7306  df-mi 7307  df-lti 7308  df-plpq 7345  df-mpq 7346  df-enq 7348  df-nqqs 7349  df-plqqs 7350  df-mqqs 7351  df-1nqqs 7352  df-rq 7353  df-ltnqqs 7354  df-inp 7467  df-iplp 7469

This theorem is referenced by:  cauappcvgprlemladdru  7657  cauappcvgprlemladd  7659