Theorem caucvgprprlemell 8052

Description: Lemma for caucvgprpr 8079. Membership in the lower cut of the putative limit. (Contributed by Jim Kingdon, 21-Jan-2021.)

Hypothesis

Ref	Expression
caucvgprprlemell.lim	|- L = <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.

Assertion

Ref	Expression
caucvgprprlemell	|- ( X e. ( 1st ` L ) <-> ( X e. Q. /\ E. b e. N. <. { p | p <Q ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) )

Distinct variable groups:   F, b   F, l, r   u, F, r   X, b, p   X, l, r, p   u, X, p   X, q, b   q, l, r   u, q

Allowed substitution hints:   F( q, p)   L( u, r, q, p, b, l)

Proof of Theorem caucvgprprlemell

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 6092	. . . . . . . 8 |- ( l = X -> ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) = ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) )
2	1	breq2d 4142	. . . . . . 7 |- ( l = X -> ( p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <-> p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) ) )
3	2	abbidv 2358	. . . . . 6 |- ( l = X -> { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } = { p | p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } )
4	1	breq1d 4140	. . . . . . 7 |- ( l = X -> ( ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q <-> ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q ) )
5	4	abbidv 2358	. . . . . 6 |- ( l = X -> { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } = { q | ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } )
6	3, 5	opeq12d 3912	. . . . 5 |- ( l = X -> <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. = <. { p | p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. )
7	6	breq1d 4140	. . . 4 |- ( l = X -> ( <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) <-> <. { p | p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) ) )
8	7	rexbidv 2551	. . 3 |- ( l = X -> ( E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) <-> E. r e. N. <. { p | p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) ) )
9		caucvgprprlemell.lim	. . . . 5 |- L = <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >.
10	9	fveq2i 5698	. . . 4 |- ( 1st ` L ) = ( 1st ` <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >. )
11		nqex 7730	. . . . . 6 |- Q. e. _V
12	11	rabex 4280	. . . . 5 |- { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } e. _V
13	11	rabex 4280	. . . . 5 |- { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } e. _V
14	12, 13	op1st 6380	. . . 4 |- ( 1st ` <. { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) } , { u e. Q. | E. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q u } , { q | u <Q q } >. } >. ) = { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) }
15	10, 14	eqtri 2259	. . 3 |- ( 1st ` L ) = { l e. Q. | E. r e. N. <. { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) }
16	8, 15	elrab2 2985	. 2 |- ( X e. ( 1st ` L ) <-> ( X e. Q. /\ E. r e. N. <. { p | p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) ) )
17		opeq1 3904	. . . . . . . . . . . 12 |- ( r = a -> <. r , 1o >. = <. a , 1o >. )
18	17	eceq1d 6843	. . . . . . . . . . 11 |- ( r = a -> [ <. r , 1o >. ] ~Q = [ <. a , 1o >. ] ~Q )
19	18	fveq2d 5699	. . . . . . . . . 10 |- ( r = a -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. a , 1o >. ] ~Q ) )
20	19	oveq2d 6101	. . . . . . . . 9 |- ( r = a -> ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) = ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
21	20	breq2d 4142	. . . . . . . 8 |- ( r = a -> ( p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <-> p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) )
22	21	abbidv 2358	. . . . . . 7 |- ( r = a -> { p | p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } = { p | p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } )
23	20	breq1d 4140	. . . . . . . 8 |- ( r = a -> ( ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q <-> ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q ) )
24	23	abbidv 2358	. . . . . . 7 |- ( r = a -> { q | ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } = { q | ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q } )
25	22, 24	opeq12d 3912	. . . . . 6 |- ( r = a -> <. { p | p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. = <. { p | p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q } >. )
26		fveq2 5695	. . . . . 6 |- ( r = a -> ( F ` r ) = ( F ` a ) )
27	25, 26	breq12d 4143	. . . . 5 |- ( r = a -> ( <. { p | p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) <-> <. { p | p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` a ) ) )
28	27	cbvrexv 2787	. . . 4 |- ( E. r e. N. <. { p | p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) <-> E. a e. N. <. { p | p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` a ) )
29		opeq1 3904	. . . . . . . . . . . 12 |- ( a = b -> <. a , 1o >. = <. b , 1o >. )
30	29	eceq1d 6843	. . . . . . . . . . 11 |- ( a = b -> [ <. a , 1o >. ] ~Q = [ <. b , 1o >. ] ~Q )
31	30	fveq2d 5699	. . . . . . . . . 10 |- ( a = b -> ( *Q ` [ <. a , 1o >. ] ~Q ) = ( *Q ` [ <. b , 1o >. ] ~Q ) )
32	31	oveq2d 6101	. . . . . . . . 9 |- ( a = b -> ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
33	32	breq2d 4142	. . . . . . . 8 |- ( a = b -> ( p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <-> p <Q ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) ) )
34	33	abbidv 2358	. . . . . . 7 |- ( a = b -> { p | p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } = { p | p <Q ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } )
35	32	breq1d 4140	. . . . . . . 8 |- ( a = b -> ( ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q <-> ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q ) )
36	35	abbidv 2358	. . . . . . 7 |- ( a = b -> { q | ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q } = { q | ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } )
37	34, 36	opeq12d 3912	. . . . . 6 |- ( a = b -> <. { p | p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q } >. = <. { p | p <Q ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. )
38		fveq2 5695	. . . . . 6 |- ( a = b -> ( F ` a ) = ( F ` b ) )
39	37, 38	breq12d 4143	. . . . 5 |- ( a = b -> ( <. { p | p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` a ) <-> <. { p | p <Q ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) )
40	39	cbvrexv 2787	. . . 4 |- ( E. a e. N. <. { p | p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` a ) <-> E. b e. N. <. { p | p <Q ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) )
41	28, 40	bitri 184	. . 3 |- ( E. r e. N. <. { p | p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) <-> E. b e. N. <. { p | p <Q ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) )
42	41	anbi2i 461	. 2 |- ( ( X e. Q. /\ E. r e. N. <. { p | p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` r ) ) <-> ( X e. Q. /\ E. b e. N. <. { p | p <Q ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) )
43	16, 42	bitri 184	1 |- ( X e. ( 1st ` L ) <-> ( X e. Q. /\ E. b e. N. <. { p | p <Q ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } , { q | ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q } >. <P ( F ` b ) ) )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {cab 2224   E.wrex 2529   {crab 2532   <.cop 3712   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   1stc1st 6372   1oc1o 6680   [cec 6805   N.cnpi 7639   ~Q ceq 7646   Q.cnq 7647   +Q cplq 7649   *Qcrq 7651   <Q cltq 7652   +P. cpp 7660   <P cltp 7662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-1st 6374  df-ec 6809  df-qs 6813  df-ni 7671  df-nqqs 7715

This theorem is used by:  caucvgprprlemopl  8064  caucvgprprlemlol  8065  caucvgprprlemdisj  8069  caucvgprprlemloc  8070