Theorem cauappcvgprlemladdfl 7750

Description: Lemma for cauappcvgprlemladd 7753. 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 7748	. . . . . 6 |- ( ph -> L e. P. )
6		cauappcvgprlemladd.s	. . . . . . 7 |- ( ph -> S e. Q. )
7		nqprlu 7642	. . . . . . 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 7563	. . . . . . 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 7470	. . . . . . 7 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
11	9, 10	genpelvl 7607	. . . . . 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 5941	. . . . . . . . . . . . . . . 16 |- ( l = s -> ( l +Q q ) = ( s +Q q ) )
15	14	breq1d 4053	. . . . . . . . . . . . . . 15 |- ( l = s -> ( ( l +Q q ) <Q ( F ` q ) <-> ( s +Q q ) <Q ( F ` q ) ) )
16	15	rexbidv 2506	. . . . . . . . . . . . . 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 5573	. . . . . . . . . . . . . . 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 7458	. . . . . . . . . . . . . . . . 17 |- Q. e. _V
19	18	rabex 4187	. . . . . . . . . . . . . . . 16 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
20	18	rabex 4187	. . . . . . . . . . . . . . . 16 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
21	19, 20	op1st 6222	. . . . . . . . . . . . . . 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 2225	. . . . . . . . . . . . . 14 |- ( 1st ` L ) = { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) }
23	16, 22	elrab2 2931	. . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . 15 |- t e. _V
29		breq1 4046	. . . . . . . . . . . . . . 15 |- ( l = t -> ( l <Q S <-> t <Q S ) )
30		ltnqex 7644	. . . . . . . . . . . . . . . 16 |- { l | l <Q S } e. _V
31		gtnqex 7645	. . . . . . . . . . . . . . . 16 |- { u | S <Q u } e. _V
32	30, 31	op1st 6222	. . . . . . . . . . . . . . 15 |- ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) = { l | l <Q S }
33	28, 29, 32	elab2 2920	. . . . . . . . . . . . . 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 7460	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
38	37	brel 4725	. . . . . . . . . . 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 7470	. . . . . . . . 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 2267	. . . . . . . . 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 7476	. . . . . . . . . . . . . 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 7477	. . . . . . . . . . . . . 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 6117	. . . . . . . . . . . 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 4725	. . . . . . . . . . . . . . 15 |- ( ( s +Q q ) <Q ( F ` q ) -> ( ( s +Q q ) e. Q. /\ ( F ` q ) e. Q. ) )
58		lt2addnq 7499	. . . . . . . . . . . . . . 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 4067	. . . . . . . . . . 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 5941	. . . . . . . . . . . . 13 |- ( r = ( s +Q t ) -> ( r +Q q ) = ( ( s +Q t ) +Q q ) )
64	63	breq1d 4053	. . . . . . . . . . . 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 2607	. . . . . . . 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 5941	. . . . . . . . . 10 |- ( l = r -> ( l +Q q ) = ( r +Q q ) )
71	70	breq1d 4053	. . . . . . . . 9 |- ( l = r -> ( ( l +Q q ) <Q ( ( F ` q ) +Q S ) <-> ( r +Q q ) <Q ( ( F ` q ) +Q S ) ) )
72	71	rexbidv 2506	. . . . . . . 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 4187	. . . . . . . . 9 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } e. _V
74	18	rabex 4187	. . . . . . . . 9 |- { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } e. _V
75	73, 74	op1st 6222	. . . . . . . 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 2931	. . . . . . 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 2630	. . . 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 3198	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 980   = wceq 1372   e. wcel 2175   {cab 2190   A.wral 2483   E.wrex 2484   {crab 2487   C_ wss 3165   <.cop 3635   class class class wbr 4043   -->wf 5264   ` cfv 5268  (class class class)co 5934   1stc1st 6214   Q.cnq 7375   +Q cplq 7377   <Q cltq 7380   P.cnp 7386   +P. cpp 7388

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4478  ax-setind 4583  ax-iinf 4634

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-eprel 4334  df-id 4338  df-po 4341  df-iso 4342  df-iord 4411  df-on 4413  df-suc 4416  df-iom 4637  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-ov 5937  df-oprab 5938  df-mpo 5939  df-1st 6216  df-2nd 6217  df-recs 6381  df-irdg 6446  df-1o 6492  df-oadd 6496  df-omul 6497  df-er 6610  df-ec 6612  df-qs 6616  df-ni 7399  df-pli 7400  df-mi 7401  df-lti 7402  df-plpq 7439  df-mpq 7440  df-enq 7442  df-nqqs 7443  df-plqqs 7444  df-mqqs 7445  df-1nqqs 7446  df-rq 7447  df-ltnqqs 7448  df-inp 7561  df-iplp 7563

This theorem is referenced by:  cauappcvgprlemladdru  7751  cauappcvgprlemladd  7753