Theorem caucvgprprlemelu 7494

Description: Lemma for caucvgprpr 7520. Membership in the upper cut of the putative limit. (Contributed by Jim Kingdon, 28-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
caucvgprprlemelu	|- ( X e. ( 2nd ` L ) <-> ( X e. Q. /\ E. b e. N. ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q X } , { q | X <Q q } >. ) )

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 caucvgprprlemelu

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 3933	. . . . . . 7 |- ( u = X -> ( p <Q u <-> p <Q X ) )
2	1	abbidv 2257	. . . . . 6 |- ( u = X -> { p | p <Q u } = { p | p <Q X } )
3		breq1 3932	. . . . . . 7 |- ( u = X -> ( u <Q q <-> X <Q q ) )
4	3	abbidv 2257	. . . . . 6 |- ( u = X -> { q | u <Q q } = { q | X <Q q } )
5	2, 4	opeq12d 3713	. . . . 5 |- ( u = X -> <. { p | p <Q u } , { q | u <Q q } >. = <. { p | p <Q X } , { q | X <Q q } >. )
6	5	breq2d 3941	. . . 4 |- ( u = X -> ( ( ( 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 } >. <-> ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q X } , { q | X <Q q } >. ) )
7	6	rexbidv 2438	. . 3 |- ( u = X -> ( 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. r e. N. ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q X } , { q | X <Q q } >. ) )
8		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 } >. } >.
9	8	fveq2i 5424	. . . 4 |- ( 2nd ` L ) = ( 2nd ` <. { 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		nqex 7171	. . . . . 6 |- Q. e. _V
11	10	rabex 4072	. . . . 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
12	10	rabex 4072	. . . . 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
13	11, 12	op2nd 6045	. . . 4 |- ( 2nd ` <. { 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 } >. } >. ) = { 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 } >. }
14	9, 13	eqtri 2160	. . 3 |- ( 2nd ` L ) = { 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 } >. }
15	7, 14	elrab2 2843	. 2 |- ( X e. ( 2nd ` L ) <-> ( X 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 X } , { q | X <Q q } >. ) )
16		fveq2 5421	. . . . . . 7 |- ( r = a -> ( F ` r ) = ( F ` a ) )
17		opeq1 3705	. . . . . . . . . . . 12 |- ( r = a -> <. r , 1o >. = <. a , 1o >. )
18	17	eceq1d 6465	. . . . . . . . . . 11 |- ( r = a -> [ <. r , 1o >. ] ~Q = [ <. a , 1o >. ] ~Q )
19	18	fveq2d 5425	. . . . . . . . . 10 |- ( r = a -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. a , 1o >. ] ~Q ) )
20	19	breq2d 3941	. . . . . . . . 9 |- ( r = a -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
21	20	abbidv 2257	. . . . . . . 8 |- ( r = a -> { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } )
22	19	breq1d 3939	. . . . . . . . 9 |- ( r = a -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. a , 1o >. ] ~Q ) <Q q ) )
23	22	abbidv 2257	. . . . . . . 8 |- ( r = a -> { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q q } )
24	21, 23	opeq12d 3713	. . . . . . 7 |- ( r = a -> <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. = <. { p | p <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q q } >. )
25	16, 24	oveq12d 5792	. . . . . 6 |- ( r = a -> ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) = ( ( F ` a ) +P. <. { p | p <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q q } >. ) )
26	25	breq1d 3939	. . . . 5 |- ( r = a -> ( ( ( F ` r ) +P. <. { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q X } , { q | X <Q q } >. <-> ( ( F ` a ) +P. <. { p | p <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q X } , { q | X <Q q } >. ) )
27	26	cbvrexv 2655	. . . 4 |- ( 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 X } , { q | X <Q q } >. <-> E. a e. N. ( ( F ` a ) +P. <. { p | p <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q X } , { q | X <Q q } >. )
28		fveq2 5421	. . . . . . 7 |- ( a = b -> ( F ` a ) = ( F ` b ) )
29		opeq1 3705	. . . . . . . . . . . 12 |- ( a = b -> <. a , 1o >. = <. b , 1o >. )
30	29	eceq1d 6465	. . . . . . . . . . 11 |- ( a = b -> [ <. a , 1o >. ] ~Q = [ <. b , 1o >. ] ~Q )
31	30	fveq2d 5425	. . . . . . . . . 10 |- ( a = b -> ( *Q ` [ <. a , 1o >. ] ~Q ) = ( *Q ` [ <. b , 1o >. ] ~Q ) )
32	31	breq2d 3941	. . . . . . . . 9 |- ( a = b -> ( p <Q ( *Q ` [ <. a , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
33	32	abbidv 2257	. . . . . . . 8 |- ( a = b -> { p | p <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } )
34	31	breq1d 3939	. . . . . . . . 9 |- ( a = b -> ( ( *Q ` [ <. a , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q ) )
35	34	abbidv 2257	. . . . . . . 8 |- ( a = b -> { q | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } )
36	33, 35	opeq12d 3713	. . . . . . 7 |- ( a = b -> <. { p | p <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q q } >. = <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. )
37	28, 36	oveq12d 5792	. . . . . 6 |- ( a = b -> ( ( F ` a ) +P. <. { p | p <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q q } >. ) = ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) )
38	37	breq1d 3939	. . . . 5 |- ( a = b -> ( ( ( F ` a ) +P. <. { p | p <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q X } , { q | X <Q q } >. <-> ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q X } , { q | X <Q q } >. ) )
39	38	cbvrexv 2655	. . . 4 |- ( E. a e. N. ( ( F ` a ) +P. <. { p | p <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q X } , { q | X <Q q } >. <-> E. b e. N. ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q X } , { q | X <Q q } >. )
40	27, 39	bitri 183	. . 3 |- ( 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 X } , { q | X <Q q } >. <-> E. b e. N. ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q X } , { q | X <Q q } >. )
41	40	anbi2i 452	. 2 |- ( ( X 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 X } , { q | X <Q q } >. ) <-> ( X e. Q. /\ E. b e. N. ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q X } , { q | X <Q q } >. ) )
42	15, 41	bitri 183	1 |- ( X e. ( 2nd ` L ) <-> ( X e. Q. /\ E. b e. N. ( ( F ` b ) +P. <. { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } >. ) <P <. { p | p <Q X } , { q | X <Q q } >. ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   {cab 2125   E.wrex 2417   {crab 2420   <.cop 3530   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   2ndc2nd 6037   1oc1o 6306   [cec 6427   N.cnpi 7080   ~Q ceq 7087   Q.cnq 7088   +Q cplq 7090   *Qcrq 7092   <Q cltq 7093   +P. cpp 7101   <P cltp 7103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-2nd 6039  df-ec 6431  df-qs 6435  df-ni 7112  df-nqqs 7156

This theorem is referenced by:  caucvgprprlemopu  7507  caucvgprprlemupu  7508  caucvgprprlemdisj  7510  caucvgprprlemloc  7511