Theorem caucvgprprlemell 7606

Description: Lemma for caucvgprpr 7633. 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 5832	. . . . . . . 8 |- ( l = X -> ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) = ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) )
2	1	breq2d 3978	. . . . . . 7 |- ( l = X -> ( p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <-> p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) ) )
3	2	abbidv 2275	. . . . . 6 |- ( l = X -> { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } = { p | p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } )
4	1	breq1d 3976	. . . . . . 7 |- ( l = X -> ( ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q <-> ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q ) )
5	4	abbidv 2275	. . . . . 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 3750	. . . . 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 3976	. . . 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 2458	. . 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 5472	. . . 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 7284	. . . . . 6 |- Q. e. _V
12	11	rabex 4109	. . . . 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 4109	. . . . 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 6095	. . . 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 2178	. . 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 2871	. 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 3742	. . . . . . . . . . . 12 |- ( r = a -> <. r , 1o >. = <. a , 1o >. )
18	17	eceq1d 6517	. . . . . . . . . . 11 |- ( r = a -> [ <. r , 1o >. ] ~Q = [ <. a , 1o >. ] ~Q )
19	18	fveq2d 5473	. . . . . . . . . 10 |- ( r = a -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. a , 1o >. ] ~Q ) )
20	19	oveq2d 5841	. . . . . . . . 9 |- ( r = a -> ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) = ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
21	20	breq2d 3978	. . . . . . . 8 |- ( r = a -> ( p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <-> p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) )
22	21	abbidv 2275	. . . . . . 7 |- ( r = a -> { p | p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } = { p | p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } )
23	20	breq1d 3976	. . . . . . . 8 |- ( r = a -> ( ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q <-> ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q ) )
24	23	abbidv 2275	. . . . . . 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 3750	. . . . . 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 5469	. . . . . 6 |- ( r = a -> ( F ` r ) = ( F ` a ) )
27	25, 26	breq12d 3979	. . . . 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 2681	. . . 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 3742	. . . . . . . . . . . 12 |- ( a = b -> <. a , 1o >. = <. b , 1o >. )
30	29	eceq1d 6517	. . . . . . . . . . 11 |- ( a = b -> [ <. a , 1o >. ] ~Q = [ <. b , 1o >. ] ~Q )
31	30	fveq2d 5473	. . . . . . . . . 10 |- ( a = b -> ( *Q ` [ <. a , 1o >. ] ~Q ) = ( *Q ` [ <. b , 1o >. ] ~Q ) )
32	31	oveq2d 5841	. . . . . . . . 9 |- ( a = b -> ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
33	32	breq2d 3978	. . . . . . . 8 |- ( a = b -> ( p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <-> p <Q ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) ) )
34	33	abbidv 2275	. . . . . . 7 |- ( a = b -> { p | p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } = { p | p <Q ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } )
35	32	breq1d 3976	. . . . . . . 8 |- ( a = b -> ( ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q <-> ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q ) )
36	35	abbidv 2275	. . . . . . 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 3750	. . . . . 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 5469	. . . . . 6 |- ( a = b -> ( F ` a ) = ( F ` b ) )
39	37, 38	breq12d 3979	. . . . 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 2681	. . . 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 183	. . 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 453	. 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 183	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 103   <-> wb 104   = wceq 1335   e. wcel 2128   {cab 2143   E.wrex 2436   {crab 2439   <.cop 3563   class class class wbr 3966   ` cfv 5171  (class class class)co 5825   1stc1st 6087   1oc1o 6357   [cec 6479   N.cnpi 7193   ~Q ceq 7200   Q.cnq 7201   +Q cplq 7203   *Qcrq 7205   <Q cltq 7206   +P. cpp 7214   <P cltp 7216

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4080  ax-sep 4083  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-iinf 4548

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-int 3809  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-id 4254  df-iom 4551  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-ov 5828  df-1st 6089  df-ec 6483  df-qs 6487  df-ni 7225  df-nqqs 7269

This theorem is referenced by:  caucvgprprlemopl  7618  caucvgprprlemlol  7619  caucvgprprlemdisj  7623  caucvgprprlemloc  7624