Theorem knoppcnlem7 32489

Description: Lemma for knoppcn 32494. (Contributed by Asger C. Ipsen, 4-Apr-2021.) (Revised by Asger C. Ipsen, 5-Jul-2021.)

Hypotheses

Ref	Expression
knoppcnlem7.t	|- T = ( x e. RR |-> ( abs ` ( ( |_ ` ( x + ( 1 / 2 ) ) ) - x ) ) )
knoppcnlem7.f	|- F = ( y e. RR |-> ( n e. NN0 |-> ( ( C ^ n ) x. ( T ` ( ( ( 2 x. N ) ^ n ) x. y ) ) ) ) )
knoppcnlem7.n	|- ( ph -> N e. NN )
knoppcnlem7.1	|- ( ph -> C e. RR )
knoppcnlem7.2	|- ( ph -> M e. NN0 )

Assertion

Ref	Expression
knoppcnlem7	|- ( ph -> ( seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ` M ) = ( w e. RR |-> ( seq 0 ( + , ( F ` w ) ) ` M ) ) )

Distinct variable groups:   m, F, w, z   m, M, w   ph, m, w

Allowed substitution hints:   ph( x, y, z, n)   C( x, y, z, w, m, n)   T( x, y, z, w, m, n)   F( x, y, n)   M( x, y, z, n)   N( x, y, z, w, m, n)

Proof of Theorem knoppcnlem7

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reex 10027	. . 3 |- RR e. _V
2	1	a1i 11	. 2 |- ( ph -> RR e. _V )
3		knoppcnlem7.2	. . 3 |- ( ph -> M e. NN0 )
4		elnn0uz 11725	. . 3 |- ( M e. NN0 <-> M e. ( ZZ>= ` 0 ) )
5	3, 4	sylib 208	. 2 |- ( ph -> M e. ( ZZ>= ` 0 ) )
6		eqid 2622	. . . 4 |- ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) = ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) )
7	6	a1i 11	. . 3 |- ( ( ph /\ k e. ( 0 ... M ) ) -> ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) = ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) )
8		fveq2 6191	. . . . . . 7 |- ( z = w -> ( F ` z ) = ( F ` w ) )
9	8	fveq1d 6193	. . . . . 6 |- ( z = w -> ( ( F ` z ) ` m ) = ( ( F ` w ) ` m ) )
10	9	cbvmptv 4750	. . . . 5 |- ( z e. RR |-> ( ( F ` z ) ` m ) ) = ( w e. RR |-> ( ( F ` w ) ` m ) )
11	10	a1i 11	. . . 4 |- ( ( ( ph /\ k e. ( 0 ... M ) ) /\ m = k ) -> ( z e. RR |-> ( ( F ` z ) ` m ) ) = ( w e. RR |-> ( ( F ` w ) ` m ) ) )
12		fveq2 6191	. . . . . 6 |- ( m = k -> ( ( F ` w ) ` m ) = ( ( F ` w ) ` k ) )
13	12	mpteq2dv 4745	. . . . 5 |- ( m = k -> ( w e. RR |-> ( ( F ` w ) ` m ) ) = ( w e. RR |-> ( ( F ` w ) ` k ) ) )
14	13	adantl 482	. . . 4 |- ( ( ( ph /\ k e. ( 0 ... M ) ) /\ m = k ) -> ( w e. RR |-> ( ( F ` w ) ` m ) ) = ( w e. RR |-> ( ( F ` w ) ` k ) ) )
15	11, 14	eqtrd 2656	. . 3 |- ( ( ( ph /\ k e. ( 0 ... M ) ) /\ m = k ) -> ( z e. RR |-> ( ( F ` z ) ` m ) ) = ( w e. RR |-> ( ( F ` w ) ` k ) ) )
16		elfznn0 12433	. . . 4 |- ( k e. ( 0 ... M ) -> k e. NN0 )
17	16	adantl 482	. . 3 |- ( ( ph /\ k e. ( 0 ... M ) ) -> k e. NN0 )
18	1	mptex 6486	. . . 4 |- ( w e. RR |-> ( ( F ` w ) ` k ) ) e. _V
19	18	a1i 11	. . 3 |- ( ( ph /\ k e. ( 0 ... M ) ) -> ( w e. RR |-> ( ( F ` w ) ` k ) ) e. _V )
20	7, 15, 17, 19	fvmptd 6288	. 2 |- ( ( ph /\ k e. ( 0 ... M ) ) -> ( ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ` k ) = ( w e. RR |-> ( ( F ` w ) ` k ) ) )
21	2, 5, 20	seqof 12858	1 |- ( ph -> ( seq 0 ( oF + , ( m e. NN0 |-> ( z e. RR |-> ( ( F ` z ) ` m ) ) ) ) ` M ) = ( w e. RR |-> ( seq 0 ( + , ( F ` w ) ) ` M ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   ` cfv 5888  (class class class)co 6650   oFcof 6895   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   / cdiv 10684   NNcn 11020   2c2 11070   NN0cn0 11292   ZZ>=cuz 11687   ...cfz 12326   |_cfl 12591   seqcseq 12801   ^cexp 12860   abscabs 13974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802

This theorem is referenced by:  knoppcnlem8  32490  knoppcnlem9  32491  knoppcnlem11  32493  knoppndvlem4  32506