Theorem axcaucvglemval 8160

Description: Lemma for axcaucvg 8163. Value of sequence when mapping to N. and R.. (Contributed by Jim Kingdon, 10-Jul-2021.)

Hypotheses

Ref	Expression
axcaucvg.n	|- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }
axcaucvg.f	|- ( ph -> F : N --> RR )
axcaucvg.cau	|- ( ph -> A. n e. N A. k e. N ( n <RR k -> ( ( F ` n ) <RR ( ( F ` k ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) /\ ( F ` k ) <RR ( ( F ` n ) + ( iota_ r e. RR ( n x. r ) = 1 ) ) ) ) )
axcaucvg.g	|- G = ( j e. N. |-> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) )

Assertion

Ref	Expression
axcaucvglemval	|- ( ( ph /\ J e. N. ) -> ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. ( G ` J ) , 0R >. )

Distinct variable groups:   j, F, z   z, G   j, J, l, u, z   ph, j   y, l, u   x, y

Allowed substitution hints:   ph( x, y, z, u, k, n, r, l)   F( x, y, u, k, n, r, l)   G( x, y, u, j, k, n, r, l)   J( x, y, k, n, r)   N( x, y, z, u, j, k, n, r, l)

Proof of Theorem axcaucvglemval

Step	Hyp	Ref	Expression
1		axcaucvg.g	. . . . 5 |- G = ( j e. N. |-> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) )
2	1	a1i 9	. . . 4 |- ( ( ph /\ J e. N. ) -> G = ( j e. N. |-> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) ) )
3		opeq1 3867	. . . . . . . . . . . . . . . 16 |- ( j = J -> <. j , 1o >. = <. J , 1o >. )
4	3	eceq1d 6781	. . . . . . . . . . . . . . 15 |- ( j = J -> [ <. j , 1o >. ] ~Q = [ <. J , 1o >. ] ~Q )
5	4	breq2d 4105	. . . . . . . . . . . . . 14 |- ( j = J -> ( l <Q [ <. j , 1o >. ] ~Q <-> l <Q [ <. J , 1o >. ] ~Q ) )
6	5	abbidv 2350	. . . . . . . . . . . . 13 |- ( j = J -> { l | l <Q [ <. j , 1o >. ] ~Q } = { l | l <Q [ <. J , 1o >. ] ~Q } )
7	4	breq1d 4103	. . . . . . . . . . . . . 14 |- ( j = J -> ( [ <. j , 1o >. ] ~Q <Q u <-> [ <. J , 1o >. ] ~Q <Q u ) )
8	7	abbidv 2350	. . . . . . . . . . . . 13 |- ( j = J -> { u | [ <. j , 1o >. ] ~Q <Q u } = { u | [ <. J , 1o >. ] ~Q <Q u } )
9	6, 8	opeq12d 3875	. . . . . . . . . . . 12 |- ( j = J -> <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. = <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. )
10	9	oveq1d 6043	. . . . . . . . . . 11 |- ( j = J -> ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) = ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) )
11	10	opeq1d 3873	. . . . . . . . . 10 |- ( j = J -> <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. = <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. )
12	11	eceq1d 6781	. . . . . . . . 9 |- ( j = J -> [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R = [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R )
13	12	opeq1d 3873	. . . . . . . 8 |- ( j = J -> <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. = <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. )
14	13	fveq2d 5652	. . . . . . 7 |- ( j = J -> ( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) )
15	14	eqeq1d 2240	. . . . . 6 |- ( j = J -> ( ( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. <-> ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) )
16	15	riotabidv 5983	. . . . 5 |- ( j = J -> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) = ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) )
17	16	adantl 277	. . . 4 |- ( ( ( ph /\ J e. N. ) /\ j = J ) -> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. j , 1o >. ] ~Q } , { u | [ <. j , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) = ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) )
18		simpr 110	. . . 4 |- ( ( ph /\ J e. N. ) -> J e. N. )
19		axcaucvg.n	. . . . 5 |- N = |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) }
20		axcaucvg.f	. . . . 5 |- ( ph -> F : N --> RR )
21	19, 20	axcaucvglemcl 8158	. . . 4 |- ( ( ph /\ J e. N. ) -> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) e. R. )
22	2, 17, 18, 21	fvmptd 5736	. . 3 |- ( ( ph /\ J e. N. ) -> ( G ` J ) = ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) )
23	22	eqcomd 2237	. 2 |- ( ( ph /\ J e. N. ) -> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) = ( G ` J ) )
24	22, 21	eqeltrd 2308	. . 3 |- ( ( ph /\ J e. N. ) -> ( G ` J ) e. R. )
25	20	adantr 276	. . . . . 6 |- ( ( ph /\ J e. N. ) -> F : N --> RR )
26		pitonn 8111	. . . . . . . 8 |- ( J e. N. -> <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. |^| { x | ( 1 e. x /\ A. y e. x ( y + 1 ) e. x ) } )
27	26, 19	eleqtrrdi 2325	. . . . . . 7 |- ( J e. N. -> <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. N )
28	27	adantl 277	. . . . . 6 |- ( ( ph /\ J e. N. ) -> <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. e. N )
29	25, 28	ffvelcdmd 5791	. . . . 5 |- ( ( ph /\ J e. N. ) -> ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) e. RR )
30		elrealeu 8092	. . . . 5 |- ( ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) e. RR <-> E! z e. R. <. z , 0R >. = ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) )
31	29, 30	sylib 122	. . . 4 |- ( ( ph /\ J e. N. ) -> E! z e. R. <. z , 0R >. = ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) )
32		eqcom 2233	. . . . 5 |- ( <. z , 0R >. = ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <-> ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. )
33	32	reubii 2721	. . . 4 |- ( E! z e. R. <. z , 0R >. = ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) <-> E! z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. )
34	31, 33	sylib 122	. . 3 |- ( ( ph /\ J e. N. ) -> E! z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. )
35		opeq1 3867	. . . . 5 |- ( z = ( G ` J ) -> <. z , 0R >. = <. ( G ` J ) , 0R >. )
36	35	eqeq2d 2243	. . . 4 |- ( z = ( G ` J ) -> ( ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. <-> ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. ( G ` J ) , 0R >. ) )
37	36	riota2 6005	. . 3 |- ( ( ( G ` J ) e. R. /\ E! z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) -> ( ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. ( G ` J ) , 0R >. <-> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) = ( G ` J ) ) )
38	24, 34, 37	syl2anc 411	. 2 |- ( ( ph /\ J e. N. ) -> ( ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. ( G ` J ) , 0R >. <-> ( iota_ z e. R. ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. z , 0R >. ) = ( G ` J ) ) )
39	23, 38	mpbird 167	1 |- ( ( ph /\ J e. N. ) -> ( F ` <. [ <. ( <. { l | l <Q [ <. J , 1o >. ] ~Q } , { u | [ <. J , 1o >. ] ~Q <Q u } >. +P. 1P ) , 1P >. ] ~R , 0R >. ) = <. ( G ` J ) , 0R >. )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2202   {cab 2217   A.wral 2511   E!wreu 2513   <.cop 3676   |^|cint 3933   class class class wbr 4093   |-> cmpt 4155   -->wf 5329   ` cfv 5333   iota_crio 5980  (class class class)co 6028   1oc1o 6618   [cec 6743   N.cnpi 7535   ~Q ceq 7542   <Q cltq 7548   1Pc1p 7555   +P. cpp 7556   ~R cer 7559   R.cnr 7560   0Rc0r 7561   RRcr 8074   1c1 8076   + caddc 8078   <RR cltrr 8079   x. cmul 8080

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 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-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-eprel 4392  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-1o 6625  df-2o 6626  df-oadd 6629  df-omul 6630  df-er 6745  df-ec 6747  df-qs 6751  df-ni 7567  df-pli 7568  df-mi 7569  df-lti 7570  df-plpq 7607  df-mpq 7608  df-enq 7610  df-nqqs 7611  df-plqqs 7612  df-mqqs 7613  df-1nqqs 7614  df-rq 7615  df-ltnqqs 7616  df-enq0 7687  df-nq0 7688  df-0nq0 7689  df-plq0 7690  df-mq0 7691  df-inp 7729  df-i1p 7730  df-iplp 7731  df-enr 7989  df-nr 7990  df-plr 7991  df-0r 7994  df-1r 7995  df-c 8081  df-1 8083  df-r 8085  df-add 8086

This theorem is referenced by:  axcaucvglemcau  8161  axcaucvglemres  8162