Theorem caucvgprprlemval 7951

Description: Lemma for caucvgprpr 7975. 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 7587	. . . . 5 |- <N C_ ( N. X. N. )
2	1	brel 4784	. . . 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 7950	. . . 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 4096	. . . . 5 |- ( a = A -> ( a <N b <-> A <N b ) )
10		fveq2 5648	. . . . . . 7 |- ( a = A -> ( F ` a ) = ( F ` A ) )
11		opeq1 3867	. . . . . . . . . . . . 13 |- ( a = A -> <. a , 1o >. = <. A , 1o >. )
12	11	eceq1d 6781	. . . . . . . . . . . 12 |- ( a = A -> [ <. a , 1o >. ] ~Q = [ <. A , 1o >. ] ~Q )
13	12	fveq2d 5652	. . . . . . . . . . 11 |- ( a = A -> ( *Q ` [ <. a , 1o >. ] ~Q ) = ( *Q ` [ <. A , 1o >. ] ~Q ) )
14	13	breq2d 4105	. . . . . . . . . 10 |- ( a = A -> ( l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) <-> l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) ) )
15	14	abbidv 2350	. . . . . . . . 9 |- ( a = A -> { l | l <Q ( *Q ` [ <. a , 1o >. ] ~Q ) } = { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } )
16	13	breq1d 4103	. . . . . . . . . 10 |- ( a = A -> ( ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u <-> ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u ) )
17	16	abbidv 2350	. . . . . . . . 9 |- ( a = A -> { u | ( *Q ` [ <. a , 1o >. ] ~Q ) <Q u } = { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } )
18	15, 17	opeq12d 3875	. . . . . . . 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 6044	. . . . . . 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 4106	. . . . . 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 6046	. . . . . . 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 4105	. . . . . 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 473	. . . . 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 4097	. . . . 5 |- ( b = B -> ( A <N b <-> A <N B ) )
26		fveq2 5648	. . . . . . . 8 |- ( b = B -> ( F ` b ) = ( F ` B ) )
27	26	oveq1d 6043	. . . . . . 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 4105	. . . . . 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 4103	. . . . . 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 473	. . . . 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 2924	. . 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 4096	. . . . . . 7 |- ( l = p -> ( l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) <-> p <Q ( *Q ` [ <. A , 1o >. ] ~Q ) ) )
35	34	cbvabv 2357	. . . . . 6 |- { l | l <Q ( *Q ` [ <. A , 1o >. ] ~Q ) } = { p | p <Q ( *Q ` [ <. A , 1o >. ] ~Q ) }
36		breq2 4097	. . . . . . 7 |- ( u = q -> ( ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u <-> ( *Q ` [ <. A , 1o >. ] ~Q ) <Q q ) )
37	36	cbvabv 2357	. . . . . 6 |- { u | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q u } = { q | ( *Q ` [ <. A , 1o >. ] ~Q ) <Q q }
38	35, 37	opeq12i 3872	. . . . 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 6039	. . . 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 4101	. . 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 6039	. . . 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 4101	. . 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 460	. 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 } >. ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   {cab 2217   A.wral 2511   <.cop 3676   class class class wbr 4093   -->wf 5329   ` cfv 5333  (class class class)co 6028   1oc1o 6618   [cec 6743   N.cnpi 7535   <N clti 7538   ~Q ceq 7542   *Qcrq 7547   <Q cltq 7548   P.cnp 7554   +P. cpp 7556   <P cltp 7558

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fv 5341  df-ov 6031  df-ec 6747  df-lti 7570

This theorem is referenced by:  caucvgprprlemnkltj  7952  caucvgprprlemnjltk  7954  caucvgprprlemnbj  7956