Theorem caucvgsrlemfv 7599

Description: Lemma for caucvgsr 7610. 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 5421	. . . . . . . . 9 |- ( x = A -> ( F ` x ) = ( F ` A ) )
4	3	eqeq1d 2148	. . . . . . . 8 |- ( x = A -> ( ( F ` x ) = [ <. ( y +P. 1P ) , 1P >. ] ~R <-> ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) )
5	4	riotabidv 5732	. . . . . . 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 7597	. . . . . 6 |- ( ( ph /\ A e. N. ) -> ( iota_ y e. P. ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) e. P. )
11	2, 6, 7, 10	fvmptd 5502	. . . . 5 |- ( ( ph /\ A e. N. ) -> ( G ` A ) = ( iota_ y e. P. ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) )
12	11	oveq1d 5789	. . . 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 3711	. . 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 6465	. 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 2141	. . . . . . 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 5747	. . . . 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 5784	. . . 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 3708	. . 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 6464	. . 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 5555	. . 3 |- ( ( ph /\ A e. N. ) -> ( F ` A ) e. R. )
23		0lt1sr 7573	. . . 4 |- 0R <R 1R
24		fveq2 5421	. . . . . . 7 |- ( m = A -> ( F ` m ) = ( F ` A ) )
25	24	breq2d 3941	. . . . . 6 |- ( m = A -> ( 1R <R ( F ` m ) <-> 1R <R ( F ` A ) ) )
26	25	rspcv 2785	. . . . 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 7572	. . . . 5 |- <R Or R.
29		ltrelsr 7546	. . . . 5 |- <R C_ ( R. X. R. )
30	28, 29	sotri 4934	. . . 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 7596	. . 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 2176	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 2125   A.wral 2416   <.cop 3530   class class class wbr 3929   |-> cmpt 3989   -->wf 5119   ` cfv 5123   iota_crio 5729  (class class class)co 5774   1oc1o 6306   [cec 6427   N.cnpi 7080   <N clti 7083   ~Q ceq 7087   *Qcrq 7092   <Q cltq 7093   P.cnp 7099   1Pc1p 7100   +P. cpp 7101   ~R cer 7104   R.cnr 7105   0Rc0r 7106   1Rc1r 7107   +R cplr 7109   <R cltr 7111

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 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502

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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7112  df-pli 7113  df-mi 7114  df-lti 7115  df-plpq 7152  df-mpq 7153  df-enq 7155  df-nqqs 7156  df-plqqs 7157  df-mqqs 7158  df-1nqqs 7159  df-rq 7160  df-ltnqqs 7161  df-enq0 7232  df-nq0 7233  df-0nq0 7234  df-plq0 7235  df-mq0 7236  df-inp 7274  df-i1p 7275  df-iplp 7276  df-iltp 7278  df-enr 7534  df-nr 7535  df-ltr 7538  df-0r 7539  df-1r 7540

This theorem is referenced by:  caucvgsrlemcau  7601  caucvgsrlembound  7602  caucvgsrlemgt1  7603