Theorem fourierdlem14 40338

Description: Given the partition V, Q is the partition shifted to the left by X. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fourierdlem14.1	|- ( ph -> A e. RR )
fourierdlem14.2	|- ( ph -> B e. RR )
fourierdlem14.x	|- ( ph -> X e. RR )
fourierdlem14.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 ) ) ) } )
fourierdlem14.o	|- O = ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = A /\ ( p ` m ) = B ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } )
fourierdlem14.m	|- ( ph -> M e. NN )
fourierdlem14.v	|- ( ph -> V e. ( P ` M ) )
fourierdlem14.q	|- Q = ( i e. ( 0 ... M ) |-> ( ( V ` i ) - X ) )

Assertion

Ref	Expression
fourierdlem14	|- ( ph -> Q e. ( O ` M ) )

Distinct variable groups:   A, m, p   B, m, p   i, M, m, p   Q, i, 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( m)   O( i, m, p)   V( m)

Proof of Theorem fourierdlem14

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem14.v	. . . . . . . . . 10 |- ( ph -> V e. ( P ` M ) )
2		fourierdlem14.m	. . . . . . . . . . 11 |- ( ph -> M e. NN )
3		fourierdlem14.p	. . . . . . . . . . . 12 |- 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 ) ) ) } )
4	3	fourierdlem2 40326	. . . . . . . . . . 11 |- ( 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 ) ) ) ) ) )
5	2, 4	syl 17	. . . . . . . . . 10 |- ( 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 ) ) ) ) ) )
6	1, 5	mpbid 222	. . . . . . . . 9 |- ( 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 ) ) ) ) )
7	6	simpld 475	. . . . . . . 8 |- ( ph -> V e. ( RR ^m ( 0 ... M ) ) )
8		elmapi 7879	. . . . . . . 8 |- ( V e. ( RR ^m ( 0 ... M ) ) -> V : ( 0 ... M ) --> RR )
9	7, 8	syl 17	. . . . . . 7 |- ( ph -> V : ( 0 ... M ) --> RR )
10	9	ffvelrnda 6359	. . . . . 6 |- ( ( ph /\ i e. ( 0 ... M ) ) -> ( V ` i ) e. RR )
11		fourierdlem14.x	. . . . . . 7 |- ( ph -> X e. RR )
12	11	adantr 481	. . . . . 6 |- ( ( ph /\ i e. ( 0 ... M ) ) -> X e. RR )
13	10, 12	resubcld 10458	. . . . 5 |- ( ( ph /\ i e. ( 0 ... M ) ) -> ( ( V ` i ) - X ) e. RR )
14		fourierdlem14.q	. . . . 5 |- Q = ( i e. ( 0 ... M ) |-> ( ( V ` i ) - X ) )
15	13, 14	fmptd 6385	. . . 4 |- ( ph -> Q : ( 0 ... M ) --> RR )
16		reex 10027	. . . . . 6 |- RR e. _V
17	16	a1i 11	. . . . 5 |- ( ph -> RR e. _V )
18		ovex 6678	. . . . . 6 |- ( 0 ... M ) e. _V
19	18	a1i 11	. . . . 5 |- ( ph -> ( 0 ... M ) e. _V )
20	17, 19	elmapd 7871	. . . 4 |- ( ph -> ( Q e. ( RR ^m ( 0 ... M ) ) <-> Q : ( 0 ... M ) --> RR ) )
21	15, 20	mpbird 247	. . 3 |- ( ph -> Q e. ( RR ^m ( 0 ... M ) ) )
22	14	a1i 11	. . . . . 6 |- ( ph -> Q = ( i e. ( 0 ... M ) |-> ( ( V ` i ) - X ) ) )
23		fveq2 6191	. . . . . . . 8 |- ( i = 0 -> ( V ` i ) = ( V ` 0 ) )
24	23	oveq1d 6665	. . . . . . 7 |- ( i = 0 -> ( ( V ` i ) - X ) = ( ( V ` 0 ) - X ) )
25	24	adantl 482	. . . . . 6 |- ( ( ph /\ i = 0 ) -> ( ( V ` i ) - X ) = ( ( V ` 0 ) - X ) )
26		0zd 11389	. . . . . . . . 9 |- ( ph -> 0 e. ZZ )
27	2	nnzd 11481	. . . . . . . . 9 |- ( ph -> M e. ZZ )
28	26, 27, 26	3jca 1242	. . . . . . . 8 |- ( ph -> ( 0 e. ZZ /\ M e. ZZ /\ 0 e. ZZ ) )
29		0le0 11110	. . . . . . . . 9 |- 0 <_ 0
30	29	a1i 11	. . . . . . . 8 |- ( ph -> 0 <_ 0 )
31		0red 10041	. . . . . . . . 9 |- ( ph -> 0 e. RR )
32	2	nnred 11035	. . . . . . . . 9 |- ( ph -> M e. RR )
33	2	nngt0d 11064	. . . . . . . . 9 |- ( ph -> 0 < M )
34	31, 32, 33	ltled 10185	. . . . . . . 8 |- ( ph -> 0 <_ M )
35	28, 30, 34	jca32 558	. . . . . . 7 |- ( ph -> ( ( 0 e. ZZ /\ M e. ZZ /\ 0 e. ZZ ) /\ ( 0 <_ 0 /\ 0 <_ M ) ) )
36		elfz2 12333	. . . . . . 7 |- ( 0 e. ( 0 ... M ) <-> ( ( 0 e. ZZ /\ M e. ZZ /\ 0 e. ZZ ) /\ ( 0 <_ 0 /\ 0 <_ M ) ) )
37	35, 36	sylibr 224	. . . . . 6 |- ( ph -> 0 e. ( 0 ... M ) )
38	9, 37	ffvelrnd 6360	. . . . . . 7 |- ( ph -> ( V ` 0 ) e. RR )
39	38, 11	resubcld 10458	. . . . . 6 |- ( ph -> ( ( V ` 0 ) - X ) e. RR )
40	22, 25, 37, 39	fvmptd 6288	. . . . 5 |- ( ph -> ( Q ` 0 ) = ( ( V ` 0 ) - X ) )
41	6	simprd 479	. . . . . . . 8 |- ( ph -> ( ( ( V ` 0 ) = ( A + X ) /\ ( V ` M ) = ( B + X ) ) /\ A. i e. ( 0..^ M ) ( V ` i ) < ( V ` ( i + 1 ) ) ) )
42	41	simpld 475	. . . . . . 7 |- ( ph -> ( ( V ` 0 ) = ( A + X ) /\ ( V ` M ) = ( B + X ) ) )
43	42	simpld 475	. . . . . 6 |- ( ph -> ( V ` 0 ) = ( A + X ) )
44	43	oveq1d 6665	. . . . 5 |- ( ph -> ( ( V ` 0 ) - X ) = ( ( A + X ) - X ) )
45		fourierdlem14.1	. . . . . . 7 |- ( ph -> A e. RR )
46	45	recnd 10068	. . . . . 6 |- ( ph -> A e. CC )
47	11	recnd 10068	. . . . . 6 |- ( ph -> X e. CC )
48	46, 47	pncand 10393	. . . . 5 |- ( ph -> ( ( A + X ) - X ) = A )
49	40, 44, 48	3eqtrd 2660	. . . 4 |- ( ph -> ( Q ` 0 ) = A )
50		fveq2 6191	. . . . . . . 8 |- ( i = M -> ( V ` i ) = ( V ` M ) )
51	50	oveq1d 6665	. . . . . . 7 |- ( i = M -> ( ( V ` i ) - X ) = ( ( V ` M ) - X ) )
52	51	adantl 482	. . . . . 6 |- ( ( ph /\ i = M ) -> ( ( V ` i ) - X ) = ( ( V ` M ) - X ) )
53	26, 27, 27	3jca 1242	. . . . . . . 8 |- ( ph -> ( 0 e. ZZ /\ M e. ZZ /\ M e. ZZ ) )
54	32	leidd 10594	. . . . . . . 8 |- ( ph -> M <_ M )
55	53, 34, 54	jca32 558	. . . . . . 7 |- ( ph -> ( ( 0 e. ZZ /\ M e. ZZ /\ M e. ZZ ) /\ ( 0 <_ M /\ M <_ M ) ) )
56		elfz2 12333	. . . . . . 7 |- ( M e. ( 0 ... M ) <-> ( ( 0 e. ZZ /\ M e. ZZ /\ M e. ZZ ) /\ ( 0 <_ M /\ M <_ M ) ) )
57	55, 56	sylibr 224	. . . . . 6 |- ( ph -> M e. ( 0 ... M ) )
58	9, 57	ffvelrnd 6360	. . . . . . 7 |- ( ph -> ( V ` M ) e. RR )
59	58, 11	resubcld 10458	. . . . . 6 |- ( ph -> ( ( V ` M ) - X ) e. RR )
60	22, 52, 57, 59	fvmptd 6288	. . . . 5 |- ( ph -> ( Q ` M ) = ( ( V ` M ) - X ) )
61	42	simprd 479	. . . . . 6 |- ( ph -> ( V ` M ) = ( B + X ) )
62	61	oveq1d 6665	. . . . 5 |- ( ph -> ( ( V ` M ) - X ) = ( ( B + X ) - X ) )
63		fourierdlem14.2	. . . . . . 7 |- ( ph -> B e. RR )
64	63	recnd 10068	. . . . . 6 |- ( ph -> B e. CC )
65	64, 47	pncand 10393	. . . . 5 |- ( ph -> ( ( B + X ) - X ) = B )
66	60, 62, 65	3eqtrd 2660	. . . 4 |- ( ph -> ( Q ` M ) = B )
67	49, 66	jca 554	. . 3 |- ( ph -> ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) )
68		elfzofz 12485	. . . . . . 7 |- ( i e. ( 0..^ M ) -> i e. ( 0 ... M ) )
69	68, 10	sylan2 491	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( V ` i ) e. RR )
70	9	adantr 481	. . . . . . 7 |- ( ( ph /\ i e. ( 0..^ M ) ) -> V : ( 0 ... M ) --> RR )
71		fzofzp1 12565	. . . . . . . 8 |- ( i e. ( 0..^ M ) -> ( i + 1 ) e. ( 0 ... M ) )
72	71	adantl 482	. . . . . . 7 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( i + 1 ) e. ( 0 ... M ) )
73	70, 72	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( V ` ( i + 1 ) ) e. RR )
74	11	adantr 481	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> X e. RR )
75	41	simprd 479	. . . . . . 7 |- ( ph -> A. i e. ( 0..^ M ) ( V ` i ) < ( V ` ( i + 1 ) ) )
76	75	r19.21bi 2932	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( V ` i ) < ( V ` ( i + 1 ) ) )
77	69, 73, 74, 76	ltsub1dd 10639	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( V ` i ) - X ) < ( ( V ` ( i + 1 ) ) - X ) )
78	68	adantl 482	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> i e. ( 0 ... M ) )
79	68, 13	sylan2 491	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( V ` i ) - X ) e. RR )
80	14	fvmpt2 6291	. . . . . 6 |- ( ( i e. ( 0 ... M ) /\ ( ( V ` i ) - X ) e. RR ) -> ( Q ` i ) = ( ( V ` i ) - X ) )
81	78, 79, 80	syl2anc 693	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( Q ` i ) = ( ( V ` i ) - X ) )
82		fveq2 6191	. . . . . . . . . 10 |- ( i = j -> ( V ` i ) = ( V ` j ) )
83	82	oveq1d 6665	. . . . . . . . 9 |- ( i = j -> ( ( V ` i ) - X ) = ( ( V ` j ) - X ) )
84	83	cbvmptv 4750	. . . . . . . 8 |- ( i e. ( 0 ... M ) |-> ( ( V ` i ) - X ) ) = ( j e. ( 0 ... M ) |-> ( ( V ` j ) - X ) )
85	14, 84	eqtri 2644	. . . . . . 7 |- Q = ( j e. ( 0 ... M ) |-> ( ( V ` j ) - X ) )
86	85	a1i 11	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> Q = ( j e. ( 0 ... M ) |-> ( ( V ` j ) - X ) ) )
87		fveq2 6191	. . . . . . . 8 |- ( j = ( i + 1 ) -> ( V ` j ) = ( V ` ( i + 1 ) ) )
88	87	oveq1d 6665	. . . . . . 7 |- ( j = ( i + 1 ) -> ( ( V ` j ) - X ) = ( ( V ` ( i + 1 ) ) - X ) )
89	88	adantl 482	. . . . . 6 |- ( ( ( ph /\ i e. ( 0..^ M ) ) /\ j = ( i + 1 ) ) -> ( ( V ` j ) - X ) = ( ( V ` ( i + 1 ) ) - X ) )
90	73, 74	resubcld 10458	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( V ` ( i + 1 ) ) - X ) e. RR )
91	86, 89, 72, 90	fvmptd 6288	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( Q ` ( i + 1 ) ) = ( ( V ` ( i + 1 ) ) - X ) )
92	77, 81, 91	3brtr4d 4685	. . . 4 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( Q ` i ) < ( Q ` ( i + 1 ) ) )
93	92	ralrimiva 2966	. . 3 |- ( ph -> A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) )
94	21, 67, 93	jca32 558	. 2 |- ( ph -> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) )
95		fourierdlem14.o	. . . 4 |- O = ( m e. NN |-> { p e. ( RR ^m ( 0 ... m ) ) | ( ( ( p ` 0 ) = A /\ ( p ` m ) = B ) /\ A. i e. ( 0..^ m ) ( p ` i ) < ( p ` ( i + 1 ) ) ) } )
96	95	fourierdlem2 40326	. . 3 |- ( M e. NN -> ( Q e. ( O ` M ) <-> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) ) )
97	2, 96	syl 17	. 2 |- ( ph -> ( Q e. ( O ` M ) <-> ( Q e. ( RR ^m ( 0 ... M ) ) /\ ( ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) /\ A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) ) ) ) )
98	94, 97	mpbird 247	1 |- ( ph -> Q e. ( O ` M ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   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   <_ cle 10075   - cmin 10266   NNcn 11020   ZZcz 11377   ...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-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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  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-fzo 12466

This theorem is referenced by:  fourierdlem74  40397  fourierdlem75  40398  fourierdlem84  40407  fourierdlem85  40408  fourierdlem88  40411  fourierdlem103  40426  fourierdlem104  40427