Theorem cauappcvgprlemladdfl 6753

Description: Lemma for cauappcvgprlemladd 6756. 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 6751	. . . . . 6 |- ( ph -> L e. P. )
6		cauappcvgprlemladd.s	. . . . . . 7 |- ( ph -> S e. Q. )
7		nqprlu 6645	. . . . . . 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 6566	. . . . . . 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 6473	. . . . . . 7 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
11	9, 10	genpelvl 6610	. . . . . 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 391	. . . . 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 280	. . . 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 5519	. . . . . . . . . . . . . . . 16 |- ( l = s -> ( l +Q q ) = ( s +Q q ) )
15	14	breq1d 3774	. . . . . . . . . . . . . . 15 |- ( l = s -> ( ( l +Q q ) <Q ( F ` q ) <-> ( s +Q q ) <Q ( F ` q ) ) )
16	15	rexbidv 2327	. . . . . . . . . . . . . 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 5181	. . . . . . . . . . . . . . 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 6461	. . . . . . . . . . . . . . . . 17 |- Q. e. _V
19	18	rabex 3901	. . . . . . . . . . . . . . . 16 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
20	18	rabex 3901	. . . . . . . . . . . . . . . 16 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
21	19, 20	op1st 5773	. . . . . . . . . . . . . . 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 2060	. . . . . . . . . . . . . 14 |- ( 1st ` L ) = { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) }
23	16, 22	elrab2 2700	. . . . . . . . . . . . 13 |- ( s e. ( 1st ` L ) <-> ( s e. Q. /\ E. q e. Q. ( s +Q q ) <Q ( F ` q ) ) )
24	23	biimpi 113	. . . . . . . . . . . 12 |- ( s e. ( 1st ` L ) -> ( s e. Q. /\ E. q e. Q. ( s +Q q ) <Q ( F ` q ) ) )
25	24	ad2antrl 459	. . . . . . . . . . 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 261	. . . . . . . . . 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 105	. . . . . . . . 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 2560	. . . . . . . . . . . . . . 15 |- t e. _V
29		breq1 3767	. . . . . . . . . . . . . . 15 |- ( l = t -> ( l <Q S <-> t <Q S ) )
30		ltnqex 6647	. . . . . . . . . . . . . . . 16 |- { l | l <Q S } e. _V
31		gtnqex 6648	. . . . . . . . . . . . . . . 16 |- { u | S <Q u } e. _V
32	30, 31	op1st 5773	. . . . . . . . . . . . . . 15 |- ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) = { l | l <Q S }
33	28, 29, 32	elab2 2690	. . . . . . . . . . . . . 14 |- ( t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) <-> t <Q S )
34	33	biimpi 113	. . . . . . . . . . . . 13 |- ( t e. ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) -> t <Q S )
35	34	ad2antll 460	. . . . . . . . . . . 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 261	. . . . . . . . . . 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 6463	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
38	37	brel 4392	. . . . . . . . . . 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 105	. . . . . . . . 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 6473	. . . . . . . . 9 |- ( ( s e. Q. /\ t e. Q. ) -> ( s +Q t ) e. Q. )
42	27, 40, 41	syl2anc 391	. . . . . . . 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 2100	. . . . . . . . 9 |- ( r = ( s +Q t ) -> ( r e. Q. <-> ( s +Q t ) e. Q. ) )
44	43	adantl 262	. . . . . . . 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 156	. . . . . . 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 107	. . . . . . . 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 457	. . . . . . . . . . . . 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 482	. . . . . . . . . . . . 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 457	. . . . . . . . . . . . 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 6479	. . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
51	50	adantl 262	. . . . . . . . . . . . 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 6480	. . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( ( f +Q g ) +Q h ) = ( f +Q ( g +Q h ) ) )
53	52	adantl 262	. . . . . . . . . . . . 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 5681	. . . . . . . . . . . 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 103	. . . . . . . . . . . . . 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 457	. . . . . . . . . . . . . 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 4392	. . . . . . . . . . . . . . 15 |- ( ( s +Q q ) <Q ( F ` q ) -> ( ( s +Q q ) e. Q. /\ ( F ` q ) e. Q. ) )
58		lt2addnq 6502	. . . . . . . . . . . . . . 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 274	. . . . . . . . . . . . . 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 409	. . . . . . . . . . . . 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 446	. . . . . . . . . . . 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 3786	. . . . . . . . . . 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 5519	. . . . . . . . . . . . 13 |- ( r = ( s +Q t ) -> ( r +Q q ) = ( ( s +Q t ) +Q q ) )
64	63	breq1d 3774	. . . . . . . . . . . 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 462	. . . . . . . . . . 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 156	. . . . . . . . . 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 108	. . . . . . . . 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 2421	. . . . . . . 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 5519	. . . . . . . . . 10 |- ( l = r -> ( l +Q q ) = ( r +Q q ) )
71	70	breq1d 3774	. . . . . . . . 9 |- ( l = r -> ( ( l +Q q ) <Q ( ( F ` q ) +Q S ) <-> ( r +Q q ) <Q ( ( F ` q ) +Q S ) ) )
72	71	rexbidv 2327	. . . . . . . 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 3901	. . . . . . . . 9 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } e. _V
74	18	rabex 3901	. . . . . . . . 9 |- { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } e. _V
75	73, 74	op1st 5773	. . . . . . . 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 2700	. . . . . . 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 394	. . . . . 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 108	. . . . 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 2440	. . . 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 108	. 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 2951	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 97   <-> wb 98   /\ w3a 885   = wceq 1243   e. wcel 1393   {cab 2026   A.wral 2306   E.wrex 2307   {crab 2310   C_ wss 2917   <.cop 3378   class class class wbr 3764   -->wf 4898   ` cfv 4902  (class class class)co 5512   1stc1st 5765   Q.cnq 6378   +Q cplq 6380   <Q cltq 6383   P.cnp 6389   +P. cpp 6391

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-eprel 4026  df-id 4030  df-po 4033  df-iso 4034  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-1o 6001  df-oadd 6005  df-omul 6006  df-er 6106  df-ec 6108  df-qs 6112  df-ni 6402  df-pli 6403  df-mi 6404  df-lti 6405  df-plpq 6442  df-mpq 6443  df-enq 6445  df-nqqs 6446  df-plqqs 6447  df-mqqs 6448  df-1nqqs 6449  df-rq 6450  df-ltnqqs 6451  df-inp 6564  df-iplp 6566

This theorem is referenced by:  cauappcvgprlemladdru  6754  cauappcvgprlemladd  6756