Theorem cauappcvgprlemladdfl 7364

Description: Lemma for cauappcvgprlemladd 7367. 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 7362	. . . . . 6 |- ( ph -> L e. P. )
6		cauappcvgprlemladd.s	. . . . . . 7 |- ( ph -> S e. Q. )
7		nqprlu 7256	. . . . . . 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 7177	. . . . . . 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 7084	. . . . . . 7 |- ( ( g e. Q. /\ h e. Q. ) -> ( g +Q h ) e. Q. )
11	9, 10	genpelvl 7221	. . . . . 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 406	. . . . 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 292	. . . 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 5713	. . . . . . . . . . . . . . . 16 |- ( l = s -> ( l +Q q ) = ( s +Q q ) )
15	14	breq1d 3885	. . . . . . . . . . . . . . 15 |- ( l = s -> ( ( l +Q q ) <Q ( F ` q ) <-> ( s +Q q ) <Q ( F ` q ) ) )
16	15	rexbidv 2397	. . . . . . . . . . . . . 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 5356	. . . . . . . . . . . . . . 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 7072	. . . . . . . . . . . . . . . . 17 |- Q. e. _V
19	18	rabex 4012	. . . . . . . . . . . . . . . 16 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) } e. _V
20	18	rabex 4012	. . . . . . . . . . . . . . . 16 |- { u e. Q. | E. q e. Q. ( ( F ` q ) +Q q ) <Q u } e. _V
21	19, 20	op1st 5975	. . . . . . . . . . . . . . 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 2120	. . . . . . . . . . . . . 14 |- ( 1st ` L ) = { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( F ` q ) }
23	16, 22	elrab2 2796	. . . . . . . . . . . . 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 477	. . . . . . . . . . 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 272	. . . . . . . . . 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 2644	. . . . . . . . . . . . . . 15 |- t e. _V
29		breq1 3878	. . . . . . . . . . . . . . 15 |- ( l = t -> ( l <Q S <-> t <Q S ) )
30		ltnqex 7258	. . . . . . . . . . . . . . . 16 |- { l | l <Q S } e. _V
31		gtnqex 7259	. . . . . . . . . . . . . . . 16 |- { u | S <Q u } e. _V
32	30, 31	op1st 5975	. . . . . . . . . . . . . . 15 |- ( 1st ` <. { l | l <Q S } , { u | S <Q u } >. ) = { l | l <Q S }
33	28, 29, 32	elab2 2785	. . . . . . . . . . . . . 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 478	. . . . . . . . . . . 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 272	. . . . . . . . . . 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 7074	. . . . . . . . . . . 12 |- <Q C_ ( Q. X. Q. )
38	37	brel 4529	. . . . . . . . . . 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 7084	. . . . . . . . 9 |- ( ( s e. Q. /\ t e. Q. ) -> ( s +Q t ) e. Q. )
42	27, 40, 41	syl2anc 406	. . . . . . . 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 2162	. . . . . . . . 9 |- ( r = ( s +Q t ) -> ( r e. Q. <-> ( s +Q t ) e. Q. ) )
44	43	adantl 273	. . . . . . . 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 475	. . . . . . . . . . . . 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 500	. . . . . . . . . . . . 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 475	. . . . . . . . . . . . 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 7090	. . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ g e. Q. ) -> ( f +Q g ) = ( g +Q f ) )
51	50	adantl 273	. . . . . . . . . . . . 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 7091	. . . . . . . . . . . . . 14 |- ( ( f e. Q. /\ g e. Q. /\ h e. Q. ) -> ( ( f +Q g ) +Q h ) = ( f +Q ( g +Q h ) ) )
53	52	adantl 273	. . . . . . . . . . . . 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 5883	. . . . . . . . . . . 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 475	. . . . . . . . . . . . . 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 4529	. . . . . . . . . . . . . . 15 |- ( ( s +Q q ) <Q ( F ` q ) -> ( ( s +Q q ) e. Q. /\ ( F ` q ) e. Q. ) )
58		lt2addnq 7113	. . . . . . . . . . . . . . 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 286	. . . . . . . . . . . . . 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 427	. . . . . . . . . . . . 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 464	. . . . . . . . . . . 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 3897	. . . . . . . . . . 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 5713	. . . . . . . . . . . . 13 |- ( r = ( s +Q t ) -> ( r +Q q ) = ( ( s +Q t ) +Q q ) )
64	63	breq1d 3885	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 2493	. . . . . . . 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 5713	. . . . . . . . . 10 |- ( l = r -> ( l +Q q ) = ( r +Q q ) )
71	70	breq1d 3885	. . . . . . . . 9 |- ( l = r -> ( ( l +Q q ) <Q ( ( F ` q ) +Q S ) <-> ( r +Q q ) <Q ( ( F ` q ) +Q S ) ) )
72	71	rexbidv 2397	. . . . . . . 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 4012	. . . . . . . . 9 |- { l e. Q. | E. q e. Q. ( l +Q q ) <Q ( ( F ` q ) +Q S ) } e. _V
74	18	rabex 4012	. . . . . . . . 9 |- { u e. Q. | E. q e. Q. ( ( ( F ` q ) +Q q ) +Q S ) <Q u } e. _V
75	73, 74	op1st 5975	. . . . . . . 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 2796	. . . . . . 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 411	. . . . . 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 2516	. . . 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 3053	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 930   = wceq 1299   e. wcel 1448   {cab 2086   A.wral 2375   E.wrex 2376   {crab 2379   C_ wss 3021   <.cop 3477   class class class wbr 3875   -->wf 5055   ` cfv 5059  (class class class)co 5706   1stc1st 5967   Q.cnq 6989   +Q cplq 6991   <Q cltq 6994   P.cnp 7000   +P. cpp 7002

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440

This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-eprel 4149  df-id 4153  df-po 4156  df-iso 4157  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-recs 6132  df-irdg 6197  df-1o 6243  df-oadd 6247  df-omul 6248  df-er 6359  df-ec 6361  df-qs 6365  df-ni 7013  df-pli 7014  df-mi 7015  df-lti 7016  df-plpq 7053  df-mpq 7054  df-enq 7056  df-nqqs 7057  df-plqqs 7058  df-mqqs 7059  df-1nqqs 7060  df-rq 7061  df-ltnqqs 7062  df-inp 7175  df-iplp 7177

This theorem is referenced by:  cauappcvgprlemladdru  7365  cauappcvgprlemladd  7367