Theorem caucvgsrlemfv 7732

Description: Lemma for caucvgsr 7743. 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 5486	. . . . . . . . 9 |- ( x = A -> ( F ` x ) = ( F ` A ) )
4	3	eqeq1d 2174	. . . . . . . 8 |- ( x = A -> ( ( F ` x ) = [ <. ( y +P. 1P ) , 1P >. ] ~R <-> ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) )
5	4	riotabidv 5800	. . . . . . 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 7730	. . . . . 6 |- ( ( ph /\ A e. N. ) -> ( iota_ y e. P. ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) e. P. )
11	2, 6, 7, 10	fvmptd 5567	. . . . 5 |- ( ( ph /\ A e. N. ) -> ( G ` A ) = ( iota_ y e. P. ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) )
12	11	oveq1d 5857	. . . 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 3764	. . 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 6537	. 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 2167	. . . . . . 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 5815	. . . . 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 5852	. . . 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 3761	. . 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 6536	. . 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 5620	. . 3 |- ( ( ph /\ A e. N. ) -> ( F ` A ) e. R. )
23		0lt1sr 7706	. . . 4 |- 0R <R 1R
24		fveq2 5486	. . . . . . 7 |- ( m = A -> ( F ` m ) = ( F ` A ) )
25	24	breq2d 3994	. . . . . 6 |- ( m = A -> ( 1R <R ( F ` m ) <-> 1R <R ( F ` A ) ) )
26	25	rspcv 2826	. . . . 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 7705	. . . . 5 |- <R Or R.
29		ltrelsr 7679	. . . . 5 |- <R C_ ( R. X. R. )
30	28, 29	sotri 4999	. . . 4 |- ( ( 0R <R 1R /\ 1R <R ( F ` A ) ) -> 0R <R ( F ` A ) )
31	23, 27, 30	sylancr 411	. . 3 |- ( ( ph /\ A e. N. ) -> 0R <R ( F ` A ) )
32		prsrriota 7729	. . 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 409	. 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 2202	1 |- ( ( ph /\ A e. N. ) -> [ <. ( ( G ` A ) +P. 1P ) , 1P >. ] ~R = ( F ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   {cab 2151   A.wral 2444   <.cop 3579   class class class wbr 3982   |-> cmpt 4043   -->wf 5184   ` cfv 5188   iota_crio 5797  (class class class)co 5842   1oc1o 6377   [cec 6499   N.cnpi 7213   <N clti 7216   ~Q ceq 7220   *Qcrq 7225   <Q cltq 7226   P.cnp 7232   1Pc1p 7233   +P. cpp 7234   ~R cer 7237   R.cnr 7238   0Rc0r 7239   1Rc1r 7240   +R cplr 7242   <R cltr 7244

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-eprel 4267  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-1o 6384  df-2o 6385  df-oadd 6388  df-omul 6389  df-er 6501  df-ec 6503  df-qs 6507  df-ni 7245  df-pli 7246  df-mi 7247  df-lti 7248  df-plpq 7285  df-mpq 7286  df-enq 7288  df-nqqs 7289  df-plqqs 7290  df-mqqs 7291  df-1nqqs 7292  df-rq 7293  df-ltnqqs 7294  df-enq0 7365  df-nq0 7366  df-0nq0 7367  df-plq0 7368  df-mq0 7369  df-inp 7407  df-i1p 7408  df-iplp 7409  df-iltp 7411  df-enr 7667  df-nr 7668  df-ltr 7671  df-0r 7672  df-1r 7673

This theorem is referenced by:  caucvgsrlemcau  7734  caucvgsrlembound  7735  caucvgsrlemgt1  7736