Theorem caucvgprprlemelu 7648

Description: Lemma for caucvgprpr 7674. 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 3993	. . . . . . 7 |- ( u = X -> ( p <Q u <-> p <Q X ) )
2	1	abbidv 2288	. . . . . 6 |- ( u = X -> { p | p <Q u } = { p | p <Q X } )
3		breq1 3992	. . . . . . 7 |- ( u = X -> ( u <Q q <-> X <Q q ) )
4	3	abbidv 2288	. . . . . 6 |- ( u = X -> { q | u <Q q } = { q | X <Q q } )
5	2, 4	opeq12d 3773	. . . . 5 |- ( u = X -> <. { p | p <Q u } , { q | u <Q q } >. = <. { p | p <Q X } , { q | X <Q q } >. )
6	5	breq2d 4001	. . . 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 2471	. . 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 5499	. . . 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 7325	. . . . . 6 |- Q. e. _V
11	10	rabex 4133	. . . . 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 4133	. . . . 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 6126	. . . 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 2191	. . 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 2889	. 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 5496	. . . . . . 7 |- ( r = a -> ( F ` r ) = ( F ` a ) )
17		opeq1 3765	. . . . . . . . . . . 12 |- ( r = a -> <. r , 1o >. = <. a , 1o >. )
18	17	eceq1d 6549	. . . . . . . . . . 11 |- ( r = a -> [ <. r , 1o >. ] ~Q = [ <. a , 1o >. ] ~Q )
19	18	fveq2d 5500	. . . . . . . . . 10 |- ( r = a -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. a , 1o >. ] ~Q ) )
20	19	breq2d 4001	. . . . . . . . 9 |- ( r = a -> ( p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
21	20	abbidv 2288	. . . . . . . 8 |- ( r = a -> { p | p <Q ( *Q ` [ <. r , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } )
22	19	breq1d 3999	. . . . . . . . 9 |- ( r = a -> ( ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. a , 1o >. ] ~Q ) <Q q ) )
23	22	abbidv 2288	. . . . . . . 8 |- ( r = a -> { q | ( *Q ` [ <. r , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q q } )
24	21, 23	opeq12d 3773	. . . . . . 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 5871	. . . . . 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 3999	. . . . 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 2697	. . . 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 5496	. . . . . . 7 |- ( a = b -> ( F ` a ) = ( F ` b ) )
29		opeq1 3765	. . . . . . . . . . . 12 |- ( a = b -> <. a , 1o >. = <. b , 1o >. )
30	29	eceq1d 6549	. . . . . . . . . . 11 |- ( a = b -> [ <. a , 1o >. ] ~Q = [ <. b , 1o >. ] ~Q )
31	30	fveq2d 5500	. . . . . . . . . 10 |- ( a = b -> ( *Q ` [ <. a , 1o >. ] ~Q ) = ( *Q ` [ <. b , 1o >. ] ~Q ) )
32	31	breq2d 4001	. . . . . . . . 9 |- ( a = b -> ( p <Q ( *Q ` [ <. a , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
33	32	abbidv 2288	. . . . . . . 8 |- ( a = b -> { p | p <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. b , 1o >. ] ~Q ) } )
34	31	breq1d 3999	. . . . . . . . 9 |- ( a = b -> ( ( *Q ` [ <. a , 1o >. ] ~Q ) <Q q <-> ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q ) )
35	34	abbidv 2288	. . . . . . . 8 |- ( a = b -> { q | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q q } = { q | ( *Q ` [ <. b , 1o >. ] ~Q ) <Q q } )
36	33, 35	opeq12d 3773	. . . . . . 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 5871	. . . . . 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 3999	. . . . 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 2697	. . . 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 454	. 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 1348   e. wcel 2141   {cab 2156   E.wrex 2449   {crab 2452   <.cop 3586   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   2ndc2nd 6118   1oc1o 6388   [cec 6511   N.cnpi 7234   ~Q ceq 7241   Q.cnq 7242   +Q cplq 7244   *Qcrq 7246   <Q cltq 7247   +P. cpp 7255   <P cltp 7257

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-2nd 6120  df-ec 6515  df-qs 6519  df-ni 7266  df-nqqs 7310

This theorem is referenced by:  caucvgprprlemopu  7661  caucvgprprlemupu  7662  caucvgprprlemdisj  7664  caucvgprprlemloc  7665