Theorem caucvgsrlemfv 8010

Description: Lemma for caucvgsr 8021. 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 5639	. . . . . . . . 9 |- ( x = A -> ( F ` x ) = ( F ` A ) )
4	3	eqeq1d 2240	. . . . . . . 8 |- ( x = A -> ( ( F ` x ) = [ <. ( y +P. 1P ) , 1P >. ] ~R <-> ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) )
5	4	riotabidv 5972	. . . . . . 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 277	. . . . . 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 110	. . . . . 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 8008	. . . . . 6 |- ( ( ph /\ A e. N. ) -> ( iota_ y e. P. ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) e. P. )
11	2, 6, 7, 10	fvmptd 5727	. . . . 5 |- ( ( ph /\ A e. N. ) -> ( G ` A ) = ( iota_ y e. P. ( F ` A ) = [ <. ( y +P. 1P ) , 1P >. ] ~R ) )
12	11	oveq1d 6032	. . . 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 3868	. . 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 6737	. 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 2233	. . . . . . 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 5989	. . . . 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 6027	. . . 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 3865	. . 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 6736	. . 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	ffvelcdmda 5782	. . 3 |- ( ( ph /\ A e. N. ) -> ( F ` A ) e. R. )
23		0lt1sr 7984	. . . 4 |- 0R <R 1R
24		fveq2 5639	. . . . . . 7 |- ( m = A -> ( F ` m ) = ( F ` A ) )
25	24	breq2d 4100	. . . . . 6 |- ( m = A -> ( 1R <R ( F ` m ) <-> 1R <R ( F ` A ) ) )
26	25	rspcv 2906	. . . . 5 |- ( A e. N. -> ( A. m e. N. 1R <R ( F ` m ) -> 1R <R ( F ` A ) ) )
27	9, 26	mpan9 281	. . . 4 |- ( ( ph /\ A e. N. ) -> 1R <R ( F ` A ) )
28		ltsosr 7983	. . . . 5 |- <R Or R.
29		ltrelsr 7957	. . . . 5 |- <R C_ ( R. X. R. )
30	28, 29	sotri 5132	. . . 4 |- ( ( 0R <R 1R /\ 1R <R ( F ` A ) ) -> 0R <R ( F ` A ) )
31	23, 27, 30	sylancr 414	. . 3 |- ( ( ph /\ A e. N. ) -> 0R <R ( F ` A ) )
32		prsrriota 8007	. . 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 411	. 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 2268	1 |- ( ( ph /\ A e. N. ) -> [ <. ( ( G ` A ) +P. 1P ) , 1P >. ] ~R = ( F ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1397   e. wcel 2202   {cab 2217   A.wral 2510   <.cop 3672   class class class wbr 4088   |-> cmpt 4150   -->wf 5322   ` cfv 5326   iota_crio 5969  (class class class)co 6017   1oc1o 6574   [cec 6699   N.cnpi 7491   <N clti 7494   ~Q ceq 7498   *Qcrq 7503   <Q cltq 7504   P.cnp 7510   1Pc1p 7511   +P. cpp 7512   ~R cer 7515   R.cnr 7516   0Rc0r 7517   1Rc1r 7518   +R cplr 7520   <R cltr 7522

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-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-eprel 4386  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-1o 6581  df-2o 6582  df-oadd 6585  df-omul 6586  df-er 6701  df-ec 6703  df-qs 6707  df-ni 7523  df-pli 7524  df-mi 7525  df-lti 7526  df-plpq 7563  df-mpq 7564  df-enq 7566  df-nqqs 7567  df-plqqs 7568  df-mqqs 7569  df-1nqqs 7570  df-rq 7571  df-ltnqqs 7572  df-enq0 7643  df-nq0 7644  df-0nq0 7645  df-plq0 7646  df-mq0 7647  df-inp 7685  df-i1p 7686  df-iplp 7687  df-iltp 7689  df-enr 7945  df-nr 7946  df-ltr 7949  df-0r 7950  df-1r 7951

This theorem is referenced by:  caucvgsrlemcau  8012  caucvgsrlembound  8013  caucvgsrlemgt1  8014