Theorem fourierdlem14 38095

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 38083	. . . . . . . . . . 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 215	. . . . . . . . 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 466	. . . . . . . 8 |- ( ph -> V e. ( RR ^m ( 0 ... M ) ) )
8		elmapi 7511	. . . . . . . 8 |- ( V e. ( RR ^m ( 0 ... M ) ) -> V : ( 0 ... M ) --> RR )
9	7, 8	syl 17	. . . . . . 7 |- ( ph -> V : ( 0 ... M ) --> RR )
10	9	fnvinran 37398	. . . . . 6 |- ( ( ph /\ i e. ( 0 ... M ) ) -> ( V ` i ) e. RR )
11		fourierdlem14.x	. . . . . . 7 |- ( ph -> X e. RR )
12	11	adantr 472	. . . . . 6 |- ( ( ph /\ i e. ( 0 ... M ) ) -> X e. RR )
13	10, 12	resubcld 10068	. . . . 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 6061	. . . 4 |- ( ph -> Q : ( 0 ... M ) --> RR )
16		reex 9648	. . . . . 6 |- RR e. _V
17	16	a1i 11	. . . . 5 |- ( ph -> RR e. _V )
18		ovex 6336	. . . . . 6 |- ( 0 ... M ) e. _V
19	18	a1i 11	. . . . 5 |- ( ph -> ( 0 ... M ) e. _V )
20	17, 19	elmapd 7504	. . . 4 |- ( ph -> ( Q e. ( RR ^m ( 0 ... M ) ) <-> Q : ( 0 ... M ) --> RR ) )
21	15, 20	mpbird 240	. . 3 |- ( ph -> Q e. ( RR ^m ( 0 ... M ) ) )
22	14	a1i 11	. . . . . 6 |- ( ph -> Q = ( i e. ( 0 ... M ) |-> ( ( V ` i ) - X ) ) )
23		fveq2 5879	. . . . . . . 8 |- ( i = 0 -> ( V ` i ) = ( V ` 0 ) )
24	23	oveq1d 6323	. . . . . . 7 |- ( i = 0 -> ( ( V ` i ) - X ) = ( ( V ` 0 ) - X ) )
25	24	adantl 473	. . . . . 6 |- ( ( ph /\ i = 0 ) -> ( ( V ` i ) - X ) = ( ( V ` 0 ) - X ) )
26		0zd 10973	. . . . . . . . 9 |- ( ph -> 0 e. ZZ )
27	2	nnzd 11062	. . . . . . . . 9 |- ( ph -> M e. ZZ )
28	26, 27, 26	3jca 1210	. . . . . . . 8 |- ( ph -> ( 0 e. ZZ /\ M e. ZZ /\ 0 e. ZZ ) )
29		0le0 10721	. . . . . . . . 9 |- 0 <_ 0
30	29	a1i 11	. . . . . . . 8 |- ( ph -> 0 <_ 0 )
31		0red 9662	. . . . . . . . 9 |- ( ph -> 0 e. RR )
32	2	nnred 10646	. . . . . . . . 9 |- ( ph -> M e. RR )
33	2	nngt0d 10675	. . . . . . . . 9 |- ( ph -> 0 < M )
34	31, 32, 33	ltled 9800	. . . . . . . 8 |- ( ph -> 0 <_ M )
35	28, 30, 34	jca32 544	. . . . . . 7 |- ( ph -> ( ( 0 e. ZZ /\ M e. ZZ /\ 0 e. ZZ ) /\ ( 0 <_ 0 /\ 0 <_ M ) ) )
36		elfz2 11817	. . . . . . 7 |- ( 0 e. ( 0 ... M ) <-> ( ( 0 e. ZZ /\ M e. ZZ /\ 0 e. ZZ ) /\ ( 0 <_ 0 /\ 0 <_ M ) ) )
37	35, 36	sylibr 217	. . . . . 6 |- ( ph -> 0 e. ( 0 ... M ) )
38	9, 37	ffvelrnd 6038	. . . . . . 7 |- ( ph -> ( V ` 0 ) e. RR )
39	38, 11	resubcld 10068	. . . . . 6 |- ( ph -> ( ( V ` 0 ) - X ) e. RR )
40	22, 25, 37, 39	fvmptd 5969	. . . . 5 |- ( ph -> ( Q ` 0 ) = ( ( V ` 0 ) - X ) )
41	6	simprd 470	. . . . . . . 8 |- ( ph -> ( ( ( V ` 0 ) = ( A + X ) /\ ( V ` M ) = ( B + X ) ) /\ A. i e. ( 0..^ M ) ( V ` i ) < ( V ` ( i + 1 ) ) ) )
42	41	simpld 466	. . . . . . 7 |- ( ph -> ( ( V ` 0 ) = ( A + X ) /\ ( V ` M ) = ( B + X ) ) )
43	42	simpld 466	. . . . . 6 |- ( ph -> ( V ` 0 ) = ( A + X ) )
44	43	oveq1d 6323	. . . . 5 |- ( ph -> ( ( V ` 0 ) - X ) = ( ( A + X ) - X ) )
45		fourierdlem14.1	. . . . . . 7 |- ( ph -> A e. RR )
46	45	recnd 9687	. . . . . 6 |- ( ph -> A e. CC )
47	11	recnd 9687	. . . . . 6 |- ( ph -> X e. CC )
48	46, 47	pncand 10006	. . . . 5 |- ( ph -> ( ( A + X ) - X ) = A )
49	40, 44, 48	3eqtrd 2509	. . . 4 |- ( ph -> ( Q ` 0 ) = A )
50		fveq2 5879	. . . . . . . 8 |- ( i = M -> ( V ` i ) = ( V ` M ) )
51	50	oveq1d 6323	. . . . . . 7 |- ( i = M -> ( ( V ` i ) - X ) = ( ( V ` M ) - X ) )
52	51	adantl 473	. . . . . 6 |- ( ( ph /\ i = M ) -> ( ( V ` i ) - X ) = ( ( V ` M ) - X ) )
53	26, 27, 27	3jca 1210	. . . . . . . 8 |- ( ph -> ( 0 e. ZZ /\ M e. ZZ /\ M e. ZZ ) )
54	32	leidd 10201	. . . . . . . 8 |- ( ph -> M <_ M )
55	53, 34, 54	jca32 544	. . . . . . 7 |- ( ph -> ( ( 0 e. ZZ /\ M e. ZZ /\ M e. ZZ ) /\ ( 0 <_ M /\ M <_ M ) ) )
56		elfz2 11817	. . . . . . 7 |- ( M e. ( 0 ... M ) <-> ( ( 0 e. ZZ /\ M e. ZZ /\ M e. ZZ ) /\ ( 0 <_ M /\ M <_ M ) ) )
57	55, 56	sylibr 217	. . . . . 6 |- ( ph -> M e. ( 0 ... M ) )
58	9, 57	ffvelrnd 6038	. . . . . . 7 |- ( ph -> ( V ` M ) e. RR )
59	58, 11	resubcld 10068	. . . . . 6 |- ( ph -> ( ( V ` M ) - X ) e. RR )
60	22, 52, 57, 59	fvmptd 5969	. . . . 5 |- ( ph -> ( Q ` M ) = ( ( V ` M ) - X ) )
61	42	simprd 470	. . . . . 6 |- ( ph -> ( V ` M ) = ( B + X ) )
62	61	oveq1d 6323	. . . . 5 |- ( ph -> ( ( V ` M ) - X ) = ( ( B + X ) - X ) )
63		fourierdlem14.2	. . . . . . 7 |- ( ph -> B e. RR )
64	63	recnd 9687	. . . . . 6 |- ( ph -> B e. CC )
65	64, 47	pncand 10006	. . . . 5 |- ( ph -> ( ( B + X ) - X ) = B )
66	60, 62, 65	3eqtrd 2509	. . . 4 |- ( ph -> ( Q ` M ) = B )
67	49, 66	jca 541	. . 3 |- ( ph -> ( ( Q ` 0 ) = A /\ ( Q ` M ) = B ) )
68		elfzofz 11962	. . . . . . 7 |- ( i e. ( 0..^ M ) -> i e. ( 0 ... M ) )
69	68, 10	sylan2 482	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( V ` i ) e. RR )
70	9	adantr 472	. . . . . . 7 |- ( ( ph /\ i e. ( 0..^ M ) ) -> V : ( 0 ... M ) --> RR )
71		fzofzp1 12037	. . . . . . . 8 |- ( i e. ( 0..^ M ) -> ( i + 1 ) e. ( 0 ... M ) )
72	71	adantl 473	. . . . . . 7 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( i + 1 ) e. ( 0 ... M ) )
73	70, 72	ffvelrnd 6038	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( V ` ( i + 1 ) ) e. RR )
74	11	adantr 472	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> X e. RR )
75	41	simprd 470	. . . . . . 7 |- ( ph -> A. i e. ( 0..^ M ) ( V ` i ) < ( V ` ( i + 1 ) ) )
76	75	r19.21bi 2776	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( V ` i ) < ( V ` ( i + 1 ) ) )
77	69, 73, 74, 76	ltsub1dd 10246	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( V ` i ) - X ) < ( ( V ` ( i + 1 ) ) - X ) )
78	68	adantl 473	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> i e. ( 0 ... M ) )
79	68, 13	sylan2 482	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( V ` i ) - X ) e. RR )
80	14	fvmpt2 5972	. . . . . 6 |- ( ( i e. ( 0 ... M ) /\ ( ( V ` i ) - X ) e. RR ) -> ( Q ` i ) = ( ( V ` i ) - X ) )
81	78, 79, 80	syl2anc 673	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( Q ` i ) = ( ( V ` i ) - X ) )
82		fveq2 5879	. . . . . . . . . 10 |- ( i = j -> ( V ` i ) = ( V ` j ) )
83	82	oveq1d 6323	. . . . . . . . 9 |- ( i = j -> ( ( V ` i ) - X ) = ( ( V ` j ) - X ) )
84	83	cbvmptv 4488	. . . . . . . 8 |- ( i e. ( 0 ... M ) |-> ( ( V ` i ) - X ) ) = ( j e. ( 0 ... M ) |-> ( ( V ` j ) - X ) )
85	14, 84	eqtri 2493	. . . . . . 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 5879	. . . . . . . 8 |- ( j = ( i + 1 ) -> ( V ` j ) = ( V ` ( i + 1 ) ) )
88	87	oveq1d 6323	. . . . . . 7 |- ( j = ( i + 1 ) -> ( ( V ` j ) - X ) = ( ( V ` ( i + 1 ) ) - X ) )
89	88	adantl 473	. . . . . 6 |- ( ( ( ph /\ i e. ( 0..^ M ) ) /\ j = ( i + 1 ) ) -> ( ( V ` j ) - X ) = ( ( V ` ( i + 1 ) ) - X ) )
90	73, 74	resubcld 10068	. . . . . 6 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( ( V ` ( i + 1 ) ) - X ) e. RR )
91	86, 89, 72, 90	fvmptd 5969	. . . . 5 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( Q ` ( i + 1 ) ) = ( ( V ` ( i + 1 ) ) - X ) )
92	77, 81, 91	3brtr4d 4426	. . . 4 |- ( ( ph /\ i e. ( 0..^ M ) ) -> ( Q ` i ) < ( Q ` ( i + 1 ) ) )
93	92	ralrimiva 2809	. . 3 |- ( ph -> A. i e. ( 0..^ M ) ( Q ` i ) < ( Q ` ( i + 1 ) ) )
94	21, 67, 93	jca32 544	. 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 38083	. . 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 240	1 |- ( ph -> Q e. ( O ` M ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   A.wral 2756   {crab 2760   _Vcvv 3031   class class class wbr 4395   |-> cmpt 4454   -->wf 5585   ` cfv 5589  (class class class)co 6308   ^m cmap 7490   RRcr 9556   0cc0 9557   1c1 9558   + caddc 9560   < clt 9693   <_ cle 9694   - cmin 9880   NNcn 10631   ZZcz 10961   ...cfz 11810  ..^cfzo 11942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943

This theorem is referenced by:  fourierdlem74  38156  fourierdlem75  38157  fourierdlem84  38166  fourierdlem85  38167  fourierdlem88  38170  fourierdlem103  38185  fourierdlem104  38186