Theorem caucvgsrlemfv 7592

Description: Lemma for caucvgsr 7603. Coercing sequence value from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)

Hypotheses

Ref	Expression
caucvgsr.f	|- ( ph -> F : N. --> R. )
caucvgsr.cau	|- ( ph -> A. n e. N. A. k e. N. ( n <N k -> ( ( F ` n ) <R ( ( F ` k ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) /\ ( F ` k ) <R ( ( F ` n ) +R [ <. ( <. { l | l <Q ( *Q ` [ <. n , 1o >. ] ~Q ) } , { u | ( *Q ` [ <. n , 1o >. ] ~Q ) <Q u } >. +P. 1P ) , 1P >. ] ~R ) ) ) )
caucvgsrlemgt1.gt1	|- ( ph -> A. m e. N. 1R <R ( F ` m ) )
caucvgsrlemf.xfr	|- G = ( x e. N. |-> ( iota_ y e. P. ( F ` x ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) )

Assertion

Ref	Expression
caucvgsrlemfv	|- ( ( ph /\ A e. N. ) -> [ <. ( ( G ` A ) +P. 1P ) , 1P >. ] ~R = ( F ` A ) )

Distinct variable groups:   A, m   x, A, y   m, F   x, F, y   ph, x

Allowed substitution hints:   ph( y, u, k, m, n, l)   A( u, k, n, l)   F( u, k, n, l)   G( x, y, u, k, m, n, l)

Proof of Theorem caucvgsrlemfv

Step	Hyp	Ref	Expression
1		caucvgsrlemf.xfr	. . . . . . 7 |- G = ( x e. N. |-> ( iota_ y e. P. ( F ` x ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) )
2	1	a1i 9	. . . . . 6 |- ( ( ph /\ A e. N. ) -> G = ( x e. N. |-> ( iota_ y e. P. ( F ` x ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) ) )
3		fveq2 5414	. . . . . . . . 9 |- ( x = A -> ( F ` x ) = ( F ` A ) )
4	3	eqeq1d 2146	. . . . . . . 8 |- ( x = A -> ( ( F ` x ) = [ <. ( y +P. 1P ) , 1P >. ] ~R <-> ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) )
5	4	riotabidv 5725	. . . . . . 7 |- ( x = A -> ( iota_ y e. P. ( F ` x ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) = ( iota_ y e. P. ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) )
6	5	adantl 275	. . . . . 6 |- ( ( ( ph /\ A e. N. ) /\ x = A ) -> ( iota_ y e. P. ( F ` x ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) = ( iota_ y e. P. ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) )
7		simpr 109	. . . . . 6 |- ( ( ph /\ A e. N. ) -> A e. N. )
8		caucvgsr.f	. . . . . . 7 |- ( ph -> F : N. --> R. )
9		caucvgsrlemgt1.gt1	. . . . . . 7 |- ( ph -> A. m e. N. 1R <R ( F ` m ) )
10	8, 9	caucvgsrlemcl 7590	. . . . . 6 |- ( ( ph /\ A e. N. ) -> ( iota_ y e. P. ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) e. P. )
11	2, 6, 7, 10	fvmptd 5495	. . . . 5 |- ( ( ph /\ A e. N. ) -> ( G ` A ) = ( iota_ y e. P. ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) )
12	11	oveq1d 5782	. . . 4 |- ( ( ph /\ A e. N. ) -> ( ( G ` A ) +P. 1P ) = ( ( iota_ y e. P. ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) +P. 1P ) )
13	12	opeq1d 3706	. . 3 |- ( ( ph /\ A e. N. ) -> <. ( ( G ` A ) +P. 1P ) , 1P >. = <. ( ( iota_ y e. P. ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) +P. 1P ) , 1P >. )
14	13	eceq1d 6458	. 2 |- ( ( ph /\ A e. N. ) -> [ <. ( ( G ` A ) +P. 1P ) , 1P >. ] ~R = [ <. ( ( iota_ y e. P. ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) +P. 1P ) , 1P >. ] ~R )
15		eqcom 2139	. . . . . . 7 |- ( ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R <-> [ <. ( y +P. 1P ) , 1P >. ] ~R = ( F ` A ) )
16	15	a1i 9	. . . . . 6 |- ( y e. P. -> ( ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R <-> [ <. ( y +P. 1P ) , 1P >. ] ~R = ( F ` A ) ) )
17	16	riotabiia 5740	. . . . 5 |- ( iota_ y e. P. ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) = ( iota_ y e. P. [ <. ( y +P. 1P ) , 1P >. ] ~R = ( F ` A ) )
18	17	oveq1i 5777	. . . 4 |- ( ( iota_ y e. P. ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) +P. 1P ) = ( ( iota_ y e. P. [ <. ( y +P. 1P ) , 1P >. ] ~R = ( F ` A ) ) +P. 1P )
19	18	opeq1i 3703	. . 3 |- <. ( ( iota_ y e. P. ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) +P. 1P ) , 1P >. = <. ( ( iota_ y e. P. [ <. ( y +P. 1P ) , 1P >. ] ~R = ( F ` A ) ) +P. 1P ) , 1P >.
20		eceq1 6457	. . 3 |- ( <. ( ( iota_ y e. P. ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) +P. 1P ) , 1P >. = <. ( ( iota_ y e. P. [ <. ( y +P. 1P ) , 1P >. ] ~R = ( F ` A ) ) +P. 1P ) , 1P >. -> [ <. ( ( iota_ y e. P. ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) +P. 1P ) , 1P >. ] ~R = [ <. ( ( iota_ y e. P. [ <. ( y +P. 1P ) , 1P >. ] ~R = ( F ` A ) ) +P. 1P ) , 1P >. ] ~R )
21	19, 20	mp1i 10	. 2 |- ( ( ph /\ A e. N. ) -> [ <. ( ( iota_ y e. P. ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) +P. 1P ) , 1P >. ] ~R = [ <. ( ( iota_ y e. P. [ <. ( y +P. 1P ) , 1P >. ] ~R = ( F ` A ) ) +P. 1P ) , 1P >. ] ~R )
22	8	ffvelrnda 5548	. . 3 |- ( ( ph /\ A e. N. ) -> ( F ` A ) e. R. )
23		0lt1sr 7566	. . . 4 |- 0R <R 1R
24		fveq2 5414	. . . . . . 7 |- ( m = A -> ( F ` m ) = ( F ` A ) )
25	24	breq2d 3936	. . . . . 6 |- ( m = A -> ( 1R <R ( F ` m ) <-> 1R <R ( F ` A ) ) )
26	25	rspcv 2780	. . . . 5 |- ( A e. N. -> ( A. m e. N. 1R <R ( F ` m ) -> 1R <R ( F ` A ) ) )
27	9, 26	mpan9 279	. . . 4 |- ( ( ph /\ A e. N. ) -> 1R <R ( F ` A ) )
28		ltsosr 7565	. . . . 5 |- <R Or R.
29		ltrelsr 7539	. . . . 5 |- <R C_ ( R. X. R. )
30	28, 29	sotri 4929	. . . 4 |- ( ( 0R <R 1R /\ 1R <R ( F ` A ) ) -> 0R <R ( F ` A ) )
31	23, 27, 30	sylancr 410	. . 3 |- ( ( ph /\ A e. N. ) -> 0R <R ( F ` A ) )
32		prsrriota 7589	. . 3 |- ( ( ( F ` A ) e. R. /\ 0R <R ( F ` A ) ) -> [ <. ( ( iota_ y e. P. [ <. ( y +P. 1P ) , 1P >. ] ~R = ( F ` A ) ) +P. 1P ) , 1P >. ] ~R = ( F ` A ) )
33	22, 31, 32	syl2anc 408	. 2 |- ( ( ph /\ A e. N. ) -> [ <. ( ( iota_ y e. P. [ <. ( y +P. 1P ) , 1P >. ] ~R = ( F ` A ) ) +P. 1P ) , 1P >. ] ~R = ( F ` A ) )
34	14, 21, 33	3eqtrd 2174	1 |- ( ( ph /\ A e. N. ) -> [ <. ( ( G ` A ) +P. 1P ) , 1P >. ] ~R = ( F ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   {cab 2123   A.wral 2414   <.cop 3525   class class class wbr 3924   |-> cmpt 3984   -->wf 5114   ` cfv 5118   iota_crio 5722  (class class class)co 5767   1oc1o 6299   [cec 6420   N.cnpi 7073   <N clti 7076   ~Q ceq 7080   *Qcrq 7085   <Q cltq 7086   P.cnp 7092   1Pc1p 7093   +P. cpp 7094   ~R cer 7097   R.cnr 7098   0Rc0r 7099   1Rc1r 7100   +R cplr 7102   <R cltr 7104

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-i1p 7268  df-iplp 7269  df-iltp 7271  df-enr 7527  df-nr 7528  df-ltr 7531  df-0r 7532  df-1r 7533

This theorem is referenced by:  caucvgsrlemcau  7594  caucvgsrlembound  7595  caucvgsrlemgt1  7596