Theorem caucvgprprlemval 8055

Description: Lemma for caucvgprpr 8079. Cauchy condition expressed in terms of classes. (Contributed by Jim Kingdon, 3-Mar-2021.)

Hypotheses

Ref	Expression
caucvgprpr.f	|- ( ph -> F : N. --> P. )
caucvgprpr.cau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )

Assertion

Ref	Expression
caucvgprprlemval	|- ( ( ph /\ A <N B ) -> ( ( F ` A ) <P ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q q } >. ) /\ ( F ` B ) <P ( ( F ` A ) +P. <. { p | p <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q q } >. ) ) )

Distinct variable groups:   A, l   u, A   A, p, l   A, q, u   k, F, n   k, l, n   u, k, n

Allowed substitution hints:   ph( u, k, n, q, p, l)   A( k, n)   B( u, k, n, q, p, l)   F( u, q, p, l)

Proof of Theorem caucvgprprlemval

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltrelpi 7691	. . . . 5 |- <N C_ ( N. X. N. )
2	1	brel 4827	. . . 4 |- ( A <N B -> ( A e. N. /\ B e. N. ) )
3	2	adantl 277	. . 3 |- ( ( ph /\ A <N B ) -> ( A e. N. /\ B e. N. ) )
4		caucvgprpr.f	. . . . 5 |- ( ph -> F : N. --> P. )
5		caucvgprpr.cau	. . . . 5 |- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <P ( ( F ` k ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` k ) <P ( ( F ` n ) +P. <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. ) ) ) )
6	4, 5	caucvgprprlemcbv 8054	. . . 4 |- ( ph -> A. a e. N. A. b e. N. ( a <N b -> ( ( F ` a ) <P ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` b ) <P ( ( F ` a ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) ) ) )
7	6	adantr 276	. . 3 |- ( ( ph /\ A <N B ) -> A. a e. N. A. b e. N. ( a <N b -> ( ( F ` a ) <P ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` b ) <P ( ( F ` a ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) ) ) )
8		simpr 110	. . 3 |- ( ( ph /\ A <N B ) -> A <N B )
9		breq1 4133	. . . . 5 |- ( a = A -> ( a <N b <-> A <N b ) )
10		fveq2 5695	. . . . . . 7 |- ( a = A -> ( F ` a ) = ( F ` A ) )
11		opeq1 3904	. . . . . . . . . . . . 13 |- ( a = A -> <. a , 1o >. = <. A , 1o >. )
12	11	eceq1d 6843	. . . . . . . . . . . 12 |- ( a = A -> [ <. a , 1o >. ] ~Q = [ <. A , 1o >. ] ~Q )
13	12	fveq2d 5699	. . . . . . . . . . 11 |- ( a = A -> ( *Q ` [ <. a , 1o >. ] ~Q ) = ( *Q ` [ <. A , 1o >. ] ~Q ) )
14	13	breq2d 4142	. . . . . . . . . 10 |- ( a = A -> ( l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) <-> l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) ) )
15	14	abbidv 2358	. . . . . . . . 9 |- ( a = A -> { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } = { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } )
16	13	breq1d 4140	. . . . . . . . . 10 |- ( a = A -> ( ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u <-> ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u ) )
17	16	abbidv 2358	. . . . . . . . 9 |- ( a = A -> { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } = { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } )
18	15, 17	opeq12d 3912	. . . . . . . 8 |- ( a = A -> <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. = <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. )
19	18	oveq2d 6101	. . . . . . 7 |- ( a = A -> ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) = ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) )
20	10, 19	breq12d 4143	. . . . . 6 |- ( a = A -> ( ( F ` a ) <P ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) <-> ( F ` A ) <P ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) ) )
21	10, 18	oveq12d 6103	. . . . . . 7 |- ( a = A -> ( ( F ` a ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) = ( ( F ` A ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) )
22	21	breq2d 4142	. . . . . 6 |- ( a = A -> ( ( F ` b ) <P ( ( F ` a ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) <-> ( F ` b ) <P ( ( F ` A ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) ) )
23	20, 22	anbi12d 477	. . . . 5 |- ( a = A -> ( ( ( F ` a ) <P ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` b ) <P ( ( F ` a ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) ) <-> ( ( F ` A ) <P ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` b ) <P ( ( F ` A ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) ) ) )
24	9, 23	imbi12d 234	. . . 4 |- ( a = A -> ( ( a <N b -> ( ( F ` a ) <P ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` b ) <P ( ( F ` a ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) ) ) <-> ( A <N b -> ( ( F ` A ) <P ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` b ) <P ( ( F ` A ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) ) ) ) )
25		breq2 4134	. . . . 5 |- ( b = B -> ( A <N b <-> A <N B ) )
26		fveq2 5695	. . . . . . . 8 |- ( b = B -> ( F ` b ) = ( F ` B ) )
27	26	oveq1d 6100	. . . . . . 7 |- ( b = B -> ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) = ( ( F ` B ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) )
28	27	breq2d 4142	. . . . . 6 |- ( b = B -> ( ( F ` A ) <P ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) <-> ( F ` A ) <P ( ( F ` B ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) ) )
29	26	breq1d 4140	. . . . . 6 |- ( b = B -> ( ( F ` b ) <P ( ( F ` A ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) <-> ( F ` B ) <P ( ( F ` A ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) ) )
30	28, 29	anbi12d 477	. . . . 5 |- ( b = B -> ( ( ( F ` A ) <P ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` b ) <P ( ( F ` A ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) ) <-> ( ( F ` A ) <P ( ( F ` B ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` B ) <P ( ( F ` A ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) ) ) )
31	25, 30	imbi12d 234	. . . 4 |- ( b = B -> ( ( A <N b -> ( ( F ` A ) <P ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` b ) <P ( ( F ` A ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) ) ) <-> ( A <N B -> ( ( F ` A ) <P ( ( F ` B ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` B ) <P ( ( F ` A ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) ) ) ) )
32	24, 31	rspc2v 2943	. . 3 |- ( ( A e. N. /\ B e. N. ) -> ( A. a e. N. A. b e. N. ( a <N b -> ( ( F ` a ) <P ( ( F ` b ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` b ) <P ( ( F ` a ) +P. <. { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } >. ) ) ) -> ( A <N B -> ( ( F ` A ) <P ( ( F ` B ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` B ) <P ( ( F ` A ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) ) ) ) )
33	3, 7, 8, 32	syl3c 63	. 2 |- ( ( ph /\ A <N B ) -> ( ( F ` A ) <P ( ( F ` B ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` B ) <P ( ( F ` A ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) ) )
34		breq1 4133	. . . . . . 7 |- ( l = p -> ( l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. A , 1o >. ] ~Q ) ) )
35	34	cbvabv 2365	. . . . . 6 |- { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. A , 1o >. ] ~Q ) }
36		breq2 4134	. . . . . . 7 |- ( u = q -> ( ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u <-> ( *Q ` [ <. A , 1o >. ] ~Q ) <Q q ) )
37	36	cbvabv 2365	. . . . . 6 |- { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } = { q | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q q }
38	35, 37	opeq12i 3909	. . . . 5 |- <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. = <. { p | p <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q q } >.
39	38	oveq2i 6096	. . . 4 |- ( ( F ` B ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) = ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q q } >. )
40	39	breq2i 4138	. . 3 |- ( ( F ` A ) <P ( ( F ` B ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) <-> ( F ` A ) <P ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q q } >. ) )
41	38	oveq2i 6096	. . . 4 |- ( ( F ` A ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) = ( ( F ` A ) +P. <. { p | p <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q q } >. )
42	41	breq2i 4138	. . 3 |- ( ( F ` B ) <P ( ( F ` A ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) <-> ( F ` B ) <P ( ( F ` A ) +P. <. { p | p <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q q } >. ) )
43	40, 42	anbi12i 464	. 2 |- ( ( ( F ` A ) <P ( ( F ` B ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) /\ ( F ` B ) <P ( ( F ` A ) +P. <. { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } >. ) ) <-> ( ( F ` A ) <P ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q q } >. ) /\ ( F ` B ) <P ( ( F ` A ) +P. <. { p | p <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q q } >. ) ) )
44	33, 43	sylib 122	1 |- ( ( ph /\ A <N B ) -> ( ( F ` A ) <P ( ( F ` B ) +P. <. { p | p <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q q } >. ) /\ ( F ` B ) <P ( ( F ` A ) +P. <. { p | p <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } , { q | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q q } >. ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {cab 2224   A.wral 2528   <.cop 3712   class class class wbr 4130   -->wf 5373   ` cfv 5377  (class class class)co 6085   1oc1o 6680   [cec 6805   N.cnpi 7639   <N clti 7642   ~Q ceq 7646   *Qcrq 7651   <Q cltq 7652   P.cnp 7658   +P. cpp 7660   <P cltp 7662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-cnv 4782  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fv 5385  df-ov 6088  df-ec 6809  df-lti 7674

This theorem is used by:  caucvgprprlemnkltj  8056  caucvgprprlemnjltk  8058  caucvgprprlemnbj  8060