Theorem caucvgprprlemell 6875

Description: Lemma for caucvgprpr 6902. 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 5539	. . . . . . . 8 |- ( l = X -> ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) = ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) )
2	1	breq2d 3797	. . . . . . 7 |- ( l = X -> ( p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <-> p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) ) )
3	2	abbidv 2196	. . . . . 6 |- ( l = X -> { p | p <Q ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } = { p | p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } )
4	1	breq1d 3795	. . . . . . 7 |- ( l = X -> ( ( l +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q <-> ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q ) )
5	4	abbidv 2196	. . . . . 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 3578	. . . . 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 3795	. . . 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 2369	. . 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 5201	. . . 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 6553	. . . . . 6 |- Q. e. _V
12	11	rabex 3922	. . . . 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 3922	. . . . 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 5793	. . . 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 2101	. . 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 2751	. 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 3570	. . . . . . . . . . . 12 |- ( r = a -> <. r , 1o >. = <. a , 1o >. )
18	17	eceq1d 6165	. . . . . . . . . . 11 |- ( r = a -> [ <. r , 1o >. ] ~Q = [ <. a , 1o >. ] ~Q )
19	18	fveq2d 5202	. . . . . . . . . 10 |- ( r = a -> ( *Q ` [ <. r , 1o >. ] ~Q ) = ( *Q ` [ <. a , 1o >. ] ~Q ) )
20	19	oveq2d 5548	. . . . . . . . 9 |- ( r = a -> ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) = ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) )
21	20	breq2d 3797	. . . . . . . 8 |- ( r = a -> ( p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <-> p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) ) )
22	21	abbidv 2196	. . . . . . 7 |- ( r = a -> { p | p <Q ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) } = { p | p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } )
23	20	breq1d 3795	. . . . . . . 8 |- ( r = a -> ( ( X +Q ( *Q ` [ <. r , 1o >. ] ~Q ) ) <Q q <-> ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q ) )
24	23	abbidv 2196	. . . . . . 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 3578	. . . . . 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 5198	. . . . . 6 |- ( r = a -> ( F ` r ) = ( F ` a ) )
27	25, 26	breq12d 3798	. . . . 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 2578	. . . 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 3570	. . . . . . . . . . . 12 |- ( a = b -> <. a , 1o >. = <. b , 1o >. )
30	29	eceq1d 6165	. . . . . . . . . . 11 |- ( a = b -> [ <. a , 1o >. ] ~Q = [ <. b , 1o >. ] ~Q )
31	30	fveq2d 5202	. . . . . . . . . 10 |- ( a = b -> ( *Q ` [ <. a , 1o >. ] ~Q ) = ( *Q ` [ <. b , 1o >. ] ~Q ) )
32	31	oveq2d 5548	. . . . . . . . 9 |- ( a = b -> ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) = ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) )
33	32	breq2d 3797	. . . . . . . 8 |- ( a = b -> ( p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <-> p <Q ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) ) )
34	33	abbidv 2196	. . . . . . 7 |- ( a = b -> { p | p <Q ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) } = { p | p <Q ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) } )
35	32	breq1d 3795	. . . . . . . 8 |- ( a = b -> ( ( X +Q ( *Q ` [ <. a , 1o >. ] ~Q ) ) <Q q <-> ( X +Q ( *Q ` [ <. b , 1o >. ] ~Q ) ) <Q q ) )
36	35	abbidv 2196	. . . . . . 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 3578	. . . . . 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 5198	. . . . . 6 |- ( a = b -> ( F ` a ) = ( F ` b ) )
39	37, 38	breq12d 3798	. . . . 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 2578	. . . 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 182	. . 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 444	. 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 182	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 102   <-> wb 103   = wceq 1284   e. wcel 1433   {cab 2067   E.wrex 2349   {crab 2352   <.cop 3401   class class class wbr 3785   ` cfv 4922  (class class class)co 5532   1stc1st 5785   1oc1o 6017   [cec 6127   N.cnpi 6462   ~Q ceq 6469   Q.cnq 6470   +Q cplq 6472   *Qcrq 6474   <Q cltq 6475   +P. cpp 6483   <P cltp 6485

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-iinf 4329

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-1st 5787  df-ec 6131  df-qs 6135  df-ni 6494  df-nqqs 6538

This theorem is referenced by:  caucvgprprlemopl  6887  caucvgprprlemlol  6888  caucvgprprlemdisj  6892  caucvgprprlemloc  6893