Theorem fourierdlem13 40337

Description: Value of V in terms of value of Q. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fourierdlem13.a	|- ( ph -> A e. RR )
fourierdlem13.b	|- ( ph -> B e. RR )
fourierdlem13.x	|- ( ph -> X e. RR )
fourierdlem13.p	|- P = ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = ( A + X ) /\ ( p ` m ) = ( B + X ) ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } )
fourierdlem13.m	|- ( ph -> M e. NN )
fourierdlem13.v	|- ( ph -> V e. ( P ` M ) )
fourierdlem13.i	|- ( ph -> I e. ( 0 ... M ) )
fourierdlem13.q	|- Q = ( i e. ( 0 ... M ) |-> ( ( V ` i ) - X ) )

Assertion

Ref	Expression
fourierdlem13	|- ( ph -> ( ( Q ` I ) = ( ( V ` I ) - X ) /\ ( V ` I ) = ( X + ( Q ` I ) ) ) )

Distinct variable groups:   A, m, p   B, m, p   i, I   i, M, m, p   i, V, p   i, X, m, p   ph, i

Allowed substitution hints:   ph( m, p)   A( i)   B( i)   P( i, m, p)   Q( i, m, p)   I( m, p)   V( m)

Proof of Theorem fourierdlem13

Step	Hyp	Ref	Expression
1		fourierdlem13.q	. . . 4 |- Q = ( i e. ( 0 ... M ) |-> ( ( V ` i ) - X ) )
2	1	a1i 11	. . 3 |- ( ph -> Q = ( i e. ( 0 ... M ) |-> ( ( V ` i ) - X ) ) )
3		simpr 477	. . . . 5 |- ( ( ph /\ i = I ) -> i = I )
4	3	fveq2d 6195	. . . 4 |- ( ( ph /\ i = I ) -> ( V ` i ) = ( V ` I ) )
5	4	oveq1d 6665	. . 3 |- ( ( ph /\ i = I ) -> ( ( V ` i ) - X ) = ( ( V ` I ) - X ) )
6		fourierdlem13.i	. . 3 |- ( ph -> I e. ( 0 ... M ) )
7		fourierdlem13.v	. . . . . . . 8 |- ( ph -> V e. ( P ` M ) )
8		fourierdlem13.m	. . . . . . . . 9 |- ( ph -> M e. NN )
9		fourierdlem13.p	. . . . . . . . . 10 |- P = ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = ( A + X ) /\ ( p ` m ) = ( B + X ) ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } )
10	9	fourierdlem2 40326	. . . . . . . . 9 |- ( M e. NN -> ( V e. ( P ` M ) <-> ( V e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( V ` 0 ) = ( A + X ) /\ ( V ` M ) = ( B + X ) ) /\ A. i e. ( 0..^ M ) ( V ` i ) < ( V ` ( i + 1 ) ) ) ) ) )
11	8, 10	syl 17	. . . . . . . 8 |- ( ph -> ( V e. ( P ` M ) <-> ( V e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( V ` 0 ) = ( A + X ) /\ ( V ` M ) = ( B + X ) ) /\ A. i e. ( 0..^ M ) ( V ` i ) < ( V ` ( i + 1 ) ) ) ) ) )
12	7, 11	mpbid 222	. . . . . . 7 |- ( ph -> ( V e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( V ` 0 ) = ( A + X ) /\ ( V ` M ) = ( B + X ) ) /\ A. i e. ( 0..^ M ) ( V ` i ) < ( V ` ( i + 1 ) ) ) ) )
13	12	simpld 475	. . . . . 6 |- ( ph -> V e. ( RR ^m ( 0 ... M ) ) )
14		elmapi 7879	. . . . . 6 |- ( V e. ( RR ^m ( 0 ... M ) ) -> V : ( 0 ... M ) --> RR )
15	13, 14	syl 17	. . . . 5 |- ( ph -> V : ( 0 ... M ) --> RR )
16	15, 6	ffvelrnd 6360	. . . 4 |- ( ph -> ( V ` I ) e. RR )
17		fourierdlem13.x	. . . 4 |- ( ph -> X e. RR )
18	16, 17	resubcld 10458	. . 3 |- ( ph -> ( ( V ` I ) - X ) e. RR )
19	2, 5, 6, 18	fvmptd 6288	. 2 |- ( ph -> ( Q ` I ) = ( ( V ` I ) - X ) )
20	19	oveq2d 6666	. . 3 |- ( ph -> ( X + ( Q ` I ) ) = ( X + ( ( V ` I ) - X ) ) )
21	17	recnd 10068	. . . 4 |- ( ph -> X e. CC )
22	16	recnd 10068	. . . 4 |- ( ph -> ( V ` I ) e. CC )
23	21, 22	pncan3d 10395	. . 3 |- ( ph -> ( X + ( ( V ` I ) - X ) ) = ( V ` I ) )
24	20, 23	eqtr2d 2657	. 2 |- ( ph -> ( V ` I ) = ( X + ( Q ` I ) ) )
25	19, 24	jca 554	1 |- ( ph -> ( ( Q ` I ) = ( ( V ` I ) - X ) /\ ( V ` I ) = ( X + ( Q ` I ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   - cmin 10266   NNcn 11020   ...cfz 12326  ..^cfzo 12465

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-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  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

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-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-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-1st 7168  df-2nd 7169  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-ltxr 10079  df-sub 10268  df-neg 10269

This theorem is referenced by:  fourierdlem72  40395  fourierdlem103  40426  fourierdlem104  40427