Theorem caucvgprprlemell 7868

Description: Lemma for caucvgprpr 7895. 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 6007	. . . . . . . 8 |- ( l = X -> ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) = ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) )
2	1	breq2d 4094	. . . . . . 7 |- ( l = X -> ( p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <-> p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) ) )
3	2	abbidv 2347	. . . . . 6 |- ( l = X -> { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } = { p | p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } )
4	1	breq1d 4092	. . . . . . 7 |- ( l = X -> ( ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q <-> ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q ) )
5	4	abbidv 2347	. . . . . 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 3864	. . . . 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 4092	. . . 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 2531	. . 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 5629	. . . 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 7546	. . . . . 6 |- Q. e. _V
12	11	rabex 4227	. . . . 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 4227	. . . . 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 6290	. . . 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 2250	. . 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 2962	. 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 3856	. . . . . . . . . . . 12 |- ( r = a -> <. r , 1o >. = <. a , 1o >. )
18	17	eceq1d 6714	. . . . . . . . . . 11 |- ( r = a -> [ <. r , 1o >. ] ~Q = [ <. a , 1o >. ] ~Q )
19	18	fveq2d 5630	. . . . . . . . . 10 |- ( r = a -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. a , 1o >. ] ~Q ) )
20	19	oveq2d 6016	. . . . . . . . 9 |- ( r = a -> ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) = ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
21	20	breq2d 4094	. . . . . . . 8 |- ( r = a -> ( p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <-> p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) )
22	21	abbidv 2347	. . . . . . 7 |- ( r = a -> { p | p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } = { p | p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } )
23	20	breq1d 4092	. . . . . . . 8 |- ( r = a -> ( ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q <-> ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q ) )
24	23	abbidv 2347	. . . . . . 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 3864	. . . . . 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 5626	. . . . . 6 |- ( r = a -> ( F ` r ) = ( F ` a ) )
27	25, 26	breq12d 4095	. . . . 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 2766	. . . 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 3856	. . . . . . . . . . . 12 |- ( a = b -> <. a , 1o >. = <. b , 1o >. )
30	29	eceq1d 6714	. . . . . . . . . . 11 |- ( a = b -> [ <. a , 1o >. ] ~Q = [ <. b , 1o >. ] ~Q )
31	30	fveq2d 5630	. . . . . . . . . 10 |- ( a = b -> ( *Q ` [ <. a , 1o >. ] ~Q ) = ( *Q ` [ <. b , 1o >. ] ~Q ) )
32	31	oveq2d 6016	. . . . . . . . 9 |- ( a = b -> ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
33	32	breq2d 4094	. . . . . . . 8 |- ( a = b -> ( p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <-> p <Q ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) ) )
34	33	abbidv 2347	. . . . . . 7 |- ( a = b -> { p | p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } = { p | p <Q ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } )
35	32	breq1d 4092	. . . . . . . 8 |- ( a = b -> ( ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q <-> ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q ) )
36	35	abbidv 2347	. . . . . . 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 3864	. . . . . 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 5626	. . . . . 6 |- ( a = b -> ( F ` a ) = ( F ` b ) )
39	37, 38	breq12d 4095	. . . . 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 2766	. . . 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 457	. 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 ) ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   {cab 2215   E.wrex 2509   {crab 2512   <.cop 3669   class class class wbr 4082   ` cfv 5317  (class class class)co 6000   1stc1st 6282   1oc1o 6553   [cec 6676   N.cnpi 7455   ~Q ceq 7462   Q.cnq 7463   +Q cplq 7465   *Qcrq 7467   <Q cltq 7468   +P. cpp 7476   <P cltp 7478

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-1st 6284  df-ec 6680  df-qs 6684  df-ni 7487  df-nqqs 7531

This theorem is referenced by:  caucvgprprlemopl  7880  caucvgprprlemlol  7881  caucvgprprlemdisj  7885  caucvgprprlemloc  7886