Theorem caucvgprprlemelu 7905

Description: Lemma for caucvgprpr 7931. 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 4092	. . . . . . 7 |- ( u = X -> ( p <Q u <-> p <Q X ) )
2	1	abbidv 2349	. . . . . 6 |- ( u = X -> { p | p <Q u } = { p | p <Q X } )
3		breq1 4091	. . . . . . 7 |- ( u = X -> ( u <Q q <-> X <Q q ) )
4	3	abbidv 2349	. . . . . 6 |- ( u = X -> { q | u <Q q } = { q | X <Q q } )
5	2, 4	opeq12d 3870	. . . . 5 |- ( u = X -> <. { p | p <Q u } , { q | u <Q q } >. = <. { p | p <Q X } , { q | X <Q q } >. )
6	5	breq2d 4100	. . . 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 2533	. . 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 5642	. . . 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 7582	. . . . . 6 |- Q. e. _V
11	10	rabex 4234	. . . . 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 4234	. . . . 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 6309	. . . 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 2252	. . 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 2965	. 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 5639	. . . . . . 7 |- ( r = a -> ( F ` r ) = ( F ` a ) )
17		opeq1 3862	. . . . . . . . . . . 12 |- ( r = a -> <. r , 1o >. = <. a , 1o >. )
18	17	eceq1d 6737	. . . . . . . . . . 11 |- ( r = a -> [ <. r , 1o >. ] ~Q = [ <. a , 1o >. ] ~Q )
19	18	fveq2d 5643	. . . . . . . . . 10 |- ( r = a -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. a , 1o >. ] ~Q ) )
20	19	breq2d 4100	. . . . . . . . 9 |- ( r = a -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
21	20	abbidv 2349	. . . . . . . 8 |- ( r = a -> { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } )
22	19	breq1d 4098	. . . . . . . . 9 |- ( r = a -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. a , 1o >. ] ~Q ) <Q q ) )
23	22	abbidv 2349	. . . . . . . 8 |- ( r = a -> { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q q } )
24	21, 23	opeq12d 3870	. . . . . . 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 6035	. . . . . 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 4098	. . . . 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 2768	. . . 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 5639	. . . . . . 7 |- ( a = b -> ( F ` a ) = ( F ` b ) )
29		opeq1 3862	. . . . . . . . . . . 12 |- ( a = b -> <. a , 1o >. = <. b , 1o >. )
30	29	eceq1d 6737	. . . . . . . . . . 11 |- ( a = b -> [ <. a , 1o >. ] ~Q = [ <. b , 1o >. ] ~Q )
31	30	fveq2d 5643	. . . . . . . . . 10 |- ( a = b -> ( *Q ` [ <. a , 1o >. ] ~Q ) = ( *Q ` [ <. b , 1o >. ] ~Q ) )
32	31	breq2d 4100	. . . . . . . . 9 |- ( a = b -> ( p <Q ( *Q ` [ <. a , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
33	32	abbidv 2349	. . . . . . . 8 |- ( a = b -> { p | p <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } )
34	31	breq1d 4098	. . . . . . . . 9 |- ( a = b -> ( ( *Q ` [ <. a , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q ) )
35	34	abbidv 2349	. . . . . . . 8 |- ( a = b -> { q | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } )
36	33, 35	opeq12d 3870	. . . . . . 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 6035	. . . . . 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 4098	. . . . 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 2768	. . . 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 184	. . 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 457	. 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 184	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 104   <-> wb 105   = wceq 1397   e. wcel 2202   {cab 2217   E.wrex 2511   {crab 2514   <.cop 3672   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   2ndc2nd 6301   1oc1o 6574   [cec 6699   N.cnpi 7491   ~Q ceq 7498   Q.cnq 7499   +Q cplq 7501   *Qcrq 7503   <Q cltq 7504   +P. cpp 7512   <P cltp 7514

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-2nd 6303  df-ec 6703  df-qs 6707  df-ni 7523  df-nqqs 7567

This theorem is referenced by:  caucvgprprlemopu  7918  caucvgprprlemupu  7919  caucvgprprlemdisj  7921  caucvgprprlemloc  7922