Theorem fourierdlem35 40359

Description: There is a single point in ( A (,] B ) that's distant from X a multiple integer of T. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fourierdlem35.a	|- ( ph -> A e. RR )
fourierdlem35.b	|- ( ph -> B e. RR )
fourierdlem35.altb	|- ( ph -> A < B )
fourierdlem35.t	|- T = ( B - A )
fourierdlem35.5	|- ( ph -> X e. RR )
fourierdlem35.i	|- ( ph -> I e. ZZ )
fourierdlem35.j	|- ( ph -> J e. ZZ )
fourierdlem35.iel	|- ( ph -> ( X + ( I x. T ) ) e. ( A (,] B ) )
fourierdlem35.jel	|- ( ph -> ( X + ( J x. T ) ) e. ( A (,] B ) )

Assertion

Ref	Expression
fourierdlem35	|- ( ph -> I = J )

Proof of Theorem fourierdlem35

Step	Hyp	Ref	Expression
1		neqne 2802	. . 3 |- ( -. I = J -> I =/= J )
2		fourierdlem35.a	. . . . . . . 8 |- ( ph -> A e. RR )
3	2	adantr 481	. . . . . . 7 |- ( ( ph /\ I < J ) -> A e. RR )
4		fourierdlem35.b	. . . . . . . 8 |- ( ph -> B e. RR )
5	4	adantr 481	. . . . . . 7 |- ( ( ph /\ I < J ) -> B e. RR )
6		fourierdlem35.altb	. . . . . . . 8 |- ( ph -> A < B )
7	6	adantr 481	. . . . . . 7 |- ( ( ph /\ I < J ) -> A < B )
8		fourierdlem35.t	. . . . . . 7 |- T = ( B - A )
9		fourierdlem35.5	. . . . . . . 8 |- ( ph -> X e. RR )
10	9	adantr 481	. . . . . . 7 |- ( ( ph /\ I < J ) -> X e. RR )
11		fourierdlem35.i	. . . . . . . 8 |- ( ph -> I e. ZZ )
12	11	adantr 481	. . . . . . 7 |- ( ( ph /\ I < J ) -> I e. ZZ )
13		fourierdlem35.j	. . . . . . . 8 |- ( ph -> J e. ZZ )
14	13	adantr 481	. . . . . . 7 |- ( ( ph /\ I < J ) -> J e. ZZ )
15		simpr 477	. . . . . . 7 |- ( ( ph /\ I < J ) -> I < J )
16		iocssicc 12261	. . . . . . . . 9 |- ( A (,] B ) C_ ( A [,] B )
17		fourierdlem35.iel	. . . . . . . . 9 |- ( ph -> ( X + ( I x. T ) ) e. ( A (,] B ) )
18	16, 17	sseldi 3601	. . . . . . . 8 |- ( ph -> ( X + ( I x. T ) ) e. ( A [,] B ) )
19	18	adantr 481	. . . . . . 7 |- ( ( ph /\ I < J ) -> ( X + ( I x. T ) ) e. ( A [,] B ) )
20		fourierdlem35.jel	. . . . . . . . 9 |- ( ph -> ( X + ( J x. T ) ) e. ( A (,] B ) )
21	16, 20	sseldi 3601	. . . . . . . 8 |- ( ph -> ( X + ( J x. T ) ) e. ( A [,] B ) )
22	21	adantr 481	. . . . . . 7 |- ( ( ph /\ I < J ) -> ( X + ( J x. T ) ) e. ( A [,] B ) )
23	3, 5, 7, 8, 10, 12, 14, 15, 19, 22	fourierdlem6 40330	. . . . . 6 |- ( ( ph /\ I < J ) -> J = ( I + 1 ) )
24	23	orcd 407	. . . . 5 |- ( ( ph /\ I < J ) -> ( J = ( I + 1 ) \/ I = ( J + 1 ) ) )
25	24	adantlr 751	. . . 4 |- ( ( ( ph /\ I =/= J ) /\ I < J ) -> ( J = ( I + 1 ) \/ I = ( J + 1 ) ) )
26		simpll 790	. . . . 5 |- ( ( ( ph /\ I =/= J ) /\ -. I < J ) -> ph )
27	13	zred 11482	. . . . . . 7 |- ( ph -> J e. RR )
28	26, 27	syl 17	. . . . . 6 |- ( ( ( ph /\ I =/= J ) /\ -. I < J ) -> J e. RR )
29	11	zred 11482	. . . . . . 7 |- ( ph -> I e. RR )
30	26, 29	syl 17	. . . . . 6 |- ( ( ( ph /\ I =/= J ) /\ -. I < J ) -> I e. RR )
31		id 22	. . . . . . . 8 |- ( I =/= J -> I =/= J )
32	31	necomd 2849	. . . . . . 7 |- ( I =/= J -> J =/= I )
33	32	ad2antlr 763	. . . . . 6 |- ( ( ( ph /\ I =/= J ) /\ -. I < J ) -> J =/= I )
34		simpr 477	. . . . . 6 |- ( ( ( ph /\ I =/= J ) /\ -. I < J ) -> -. I < J )
35	28, 30, 33, 34	lttri5d 39513	. . . . 5 |- ( ( ( ph /\ I =/= J ) /\ -. I < J ) -> J < I )
36	2	adantr 481	. . . . . . 7 |- ( ( ph /\ J < I ) -> A e. RR )
37	4	adantr 481	. . . . . . 7 |- ( ( ph /\ J < I ) -> B e. RR )
38	6	adantr 481	. . . . . . 7 |- ( ( ph /\ J < I ) -> A < B )
39	9	adantr 481	. . . . . . 7 |- ( ( ph /\ J < I ) -> X e. RR )
40	13	adantr 481	. . . . . . 7 |- ( ( ph /\ J < I ) -> J e. ZZ )
41	11	adantr 481	. . . . . . 7 |- ( ( ph /\ J < I ) -> I e. ZZ )
42		simpr 477	. . . . . . 7 |- ( ( ph /\ J < I ) -> J < I )
43	21	adantr 481	. . . . . . 7 |- ( ( ph /\ J < I ) -> ( X + ( J x. T ) ) e. ( A [,] B ) )
44	18	adantr 481	. . . . . . 7 |- ( ( ph /\ J < I ) -> ( X + ( I x. T ) ) e. ( A [,] B ) )
45	36, 37, 38, 8, 39, 40, 41, 42, 43, 44	fourierdlem6 40330	. . . . . 6 |- ( ( ph /\ J < I ) -> I = ( J + 1 ) )
46	45	olcd 408	. . . . 5 |- ( ( ph /\ J < I ) -> ( J = ( I + 1 ) \/ I = ( J + 1 ) ) )
47	26, 35, 46	syl2anc 693	. . . 4 |- ( ( ( ph /\ I =/= J ) /\ -. I < J ) -> ( J = ( I + 1 ) \/ I = ( J + 1 ) ) )
48	25, 47	pm2.61dan 832	. . 3 |- ( ( ph /\ I =/= J ) -> ( J = ( I + 1 ) \/ I = ( J + 1 ) ) )
49	1, 48	sylan2 491	. 2 |- ( ( ph /\ -. I = J ) -> ( J = ( I + 1 ) \/ I = ( J + 1 ) ) )
50	2	rexrd 10089	. . . . . . . 8 |- ( ph -> A e. RR* )
51	4	rexrd 10089	. . . . . . . 8 |- ( ph -> B e. RR* )
52		iocleub 39725	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ ( X + ( J x. T ) ) e. ( A (,] B ) ) -> ( X + ( J x. T ) ) <_ B )
53	50, 51, 20, 52	syl3anc 1326	. . . . . . 7 |- ( ph -> ( X + ( J x. T ) ) <_ B )
54	53	adantr 481	. . . . . 6 |- ( ( ph /\ J = ( I + 1 ) ) -> ( X + ( J x. T ) ) <_ B )
55	2	adantr 481	. . . . . . . . 9 |- ( ( ph /\ J = ( I + 1 ) ) -> A e. RR )
56	4, 2	resubcld 10458	. . . . . . . . . . . . 13 |- ( ph -> ( B - A ) e. RR )
57	8, 56	syl5eqel 2705	. . . . . . . . . . . 12 |- ( ph -> T e. RR )
58	29, 57	remulcld 10070	. . . . . . . . . . 11 |- ( ph -> ( I x. T ) e. RR )
59	9, 58	readdcld 10069	. . . . . . . . . 10 |- ( ph -> ( X + ( I x. T ) ) e. RR )
60	59	adantr 481	. . . . . . . . 9 |- ( ( ph /\ J = ( I + 1 ) ) -> ( X + ( I x. T ) ) e. RR )
61	57	adantr 481	. . . . . . . . 9 |- ( ( ph /\ J = ( I + 1 ) ) -> T e. RR )
62		iocgtlb 39724	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* /\ ( X + ( I x. T ) ) e. ( A (,] B ) ) -> A < ( X + ( I x. T ) ) )
63	50, 51, 17, 62	syl3anc 1326	. . . . . . . . . 10 |- ( ph -> A < ( X + ( I x. T ) ) )
64	63	adantr 481	. . . . . . . . 9 |- ( ( ph /\ J = ( I + 1 ) ) -> A < ( X + ( I x. T ) ) )
65	55, 60, 61, 64	ltadd1dd 10638	. . . . . . . 8 |- ( ( ph /\ J = ( I + 1 ) ) -> ( A + T ) < ( ( X + ( I x. T ) ) + T ) )
66	8	eqcomi 2631	. . . . . . . . . . 11 |- ( B - A ) = T
67	4	recnd 10068	. . . . . . . . . . . 12 |- ( ph -> B e. CC )
68	2	recnd 10068	. . . . . . . . . . . 12 |- ( ph -> A e. CC )
69	57	recnd 10068	. . . . . . . . . . . 12 |- ( ph -> T e. CC )
70	67, 68, 69	subaddd 10410	. . . . . . . . . . 11 |- ( ph -> ( ( B - A ) = T <-> ( A + T ) = B ) )
71	66, 70	mpbii 223	. . . . . . . . . 10 |- ( ph -> ( A + T ) = B )
72	71	eqcomd 2628	. . . . . . . . 9 |- ( ph -> B = ( A + T ) )
73	72	adantr 481	. . . . . . . 8 |- ( ( ph /\ J = ( I + 1 ) ) -> B = ( A + T ) )
74	9	recnd 10068	. . . . . . . . . . 11 |- ( ph -> X e. CC )
75	58	recnd 10068	. . . . . . . . . . 11 |- ( ph -> ( I x. T ) e. CC )
76	74, 75, 69	addassd 10062	. . . . . . . . . 10 |- ( ph -> ( ( X + ( I x. T ) ) + T ) = ( X + ( ( I x. T ) + T ) ) )
77	76	adantr 481	. . . . . . . . 9 |- ( ( ph /\ J = ( I + 1 ) ) -> ( ( X + ( I x. T ) ) + T ) = ( X + ( ( I x. T ) + T ) ) )
78	29	recnd 10068	. . . . . . . . . . . . 13 |- ( ph -> I e. CC )
79	78, 69	adddirp1d 10066	. . . . . . . . . . . 12 |- ( ph -> ( ( I + 1 ) x. T ) = ( ( I x. T ) + T ) )
80	79	eqcomd 2628	. . . . . . . . . . 11 |- ( ph -> ( ( I x. T ) + T ) = ( ( I + 1 ) x. T ) )
81	80	oveq2d 6666	. . . . . . . . . 10 |- ( ph -> ( X + ( ( I x. T ) + T ) ) = ( X + ( ( I + 1 ) x. T ) ) )
82	81	adantr 481	. . . . . . . . 9 |- ( ( ph /\ J = ( I + 1 ) ) -> ( X + ( ( I x. T ) + T ) ) = ( X + ( ( I + 1 ) x. T ) ) )
83		oveq1 6657	. . . . . . . . . . . 12 |- ( J = ( I + 1 ) -> ( J x. T ) = ( ( I + 1 ) x. T ) )
84	83	eqcomd 2628	. . . . . . . . . . 11 |- ( J = ( I + 1 ) -> ( ( I + 1 ) x. T ) = ( J x. T ) )
85	84	oveq2d 6666	. . . . . . . . . 10 |- ( J = ( I + 1 ) -> ( X + ( ( I + 1 ) x. T ) ) = ( X + ( J x. T ) ) )
86	85	adantl 482	. . . . . . . . 9 |- ( ( ph /\ J = ( I + 1 ) ) -> ( X + ( ( I + 1 ) x. T ) ) = ( X + ( J x. T ) ) )
87	77, 82, 86	3eqtrrd 2661	. . . . . . . 8 |- ( ( ph /\ J = ( I + 1 ) ) -> ( X + ( J x. T ) ) = ( ( X + ( I x. T ) ) + T ) )
88	65, 73, 87	3brtr4d 4685	. . . . . . 7 |- ( ( ph /\ J = ( I + 1 ) ) -> B < ( X + ( J x. T ) ) )
89	4	adantr 481	. . . . . . . 8 |- ( ( ph /\ J = ( I + 1 ) ) -> B e. RR )
90	27, 57	remulcld 10070	. . . . . . . . . 10 |- ( ph -> ( J x. T ) e. RR )
91	9, 90	readdcld 10069	. . . . . . . . 9 |- ( ph -> ( X + ( J x. T ) ) e. RR )
92	91	adantr 481	. . . . . . . 8 |- ( ( ph /\ J = ( I + 1 ) ) -> ( X + ( J x. T ) ) e. RR )
93	89, 92	ltnled 10184	. . . . . . 7 |- ( ( ph /\ J = ( I + 1 ) ) -> ( B < ( X + ( J x. T ) ) <-> -. ( X + ( J x. T ) ) <_ B ) )
94	88, 93	mpbid 222	. . . . . 6 |- ( ( ph /\ J = ( I + 1 ) ) -> -. ( X + ( J x. T ) ) <_ B )
95	54, 94	pm2.65da 600	. . . . 5 |- ( ph -> -. J = ( I + 1 ) )
96		iocleub 39725	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* /\ ( X + ( I x. T ) ) e. ( A (,] B ) ) -> ( X + ( I x. T ) ) <_ B )
97	50, 51, 17, 96	syl3anc 1326	. . . . . . 7 |- ( ph -> ( X + ( I x. T ) ) <_ B )
98	97	adantr 481	. . . . . 6 |- ( ( ph /\ I = ( J + 1 ) ) -> ( X + ( I x. T ) ) <_ B )
99	2	adantr 481	. . . . . . . . 9 |- ( ( ph /\ I = ( J + 1 ) ) -> A e. RR )
100	91	adantr 481	. . . . . . . . 9 |- ( ( ph /\ I = ( J + 1 ) ) -> ( X + ( J x. T ) ) e. RR )
101	57	adantr 481	. . . . . . . . 9 |- ( ( ph /\ I = ( J + 1 ) ) -> T e. RR )
102		iocgtlb 39724	. . . . . . . . . . 11 |- ( ( A e. RR* /\ B e. RR* /\ ( X + ( J x. T ) ) e. ( A (,] B ) ) -> A < ( X + ( J x. T ) ) )
103	50, 51, 20, 102	syl3anc 1326	. . . . . . . . . 10 |- ( ph -> A < ( X + ( J x. T ) ) )
104	103	adantr 481	. . . . . . . . 9 |- ( ( ph /\ I = ( J + 1 ) ) -> A < ( X + ( J x. T ) ) )
105	99, 100, 101, 104	ltadd1dd 10638	. . . . . . . 8 |- ( ( ph /\ I = ( J + 1 ) ) -> ( A + T ) < ( ( X + ( J x. T ) ) + T ) )
106	72	adantr 481	. . . . . . . 8 |- ( ( ph /\ I = ( J + 1 ) ) -> B = ( A + T ) )
107	90	recnd 10068	. . . . . . . . . . 11 |- ( ph -> ( J x. T ) e. CC )
108	74, 107, 69	addassd 10062	. . . . . . . . . 10 |- ( ph -> ( ( X + ( J x. T ) ) + T ) = ( X + ( ( J x. T ) + T ) ) )
109	108	adantr 481	. . . . . . . . 9 |- ( ( ph /\ I = ( J + 1 ) ) -> ( ( X + ( J x. T ) ) + T ) = ( X + ( ( J x. T ) + T ) ) )
110	27	recnd 10068	. . . . . . . . . . . . 13 |- ( ph -> J e. CC )
111	110, 69	adddirp1d 10066	. . . . . . . . . . . 12 |- ( ph -> ( ( J + 1 ) x. T ) = ( ( J x. T ) + T ) )
112	111	eqcomd 2628	. . . . . . . . . . 11 |- ( ph -> ( ( J x. T ) + T ) = ( ( J + 1 ) x. T ) )
113	112	oveq2d 6666	. . . . . . . . . 10 |- ( ph -> ( X + ( ( J x. T ) + T ) ) = ( X + ( ( J + 1 ) x. T ) ) )
114	113	adantr 481	. . . . . . . . 9 |- ( ( ph /\ I = ( J + 1 ) ) -> ( X + ( ( J x. T ) + T ) ) = ( X + ( ( J + 1 ) x. T ) ) )
115		oveq1 6657	. . . . . . . . . . . 12 |- ( I = ( J + 1 ) -> ( I x. T ) = ( ( J + 1 ) x. T ) )
116	115	eqcomd 2628	. . . . . . . . . . 11 |- ( I = ( J + 1 ) -> ( ( J + 1 ) x. T ) = ( I x. T ) )
117	116	oveq2d 6666	. . . . . . . . . 10 |- ( I = ( J + 1 ) -> ( X + ( ( J + 1 ) x. T ) ) = ( X + ( I x. T ) ) )
118	117	adantl 482	. . . . . . . . 9 |- ( ( ph /\ I = ( J + 1 ) ) -> ( X + ( ( J + 1 ) x. T ) ) = ( X + ( I x. T ) ) )
119	109, 114, 118	3eqtrrd 2661	. . . . . . . 8 |- ( ( ph /\ I = ( J + 1 ) ) -> ( X + ( I x. T ) ) = ( ( X + ( J x. T ) ) + T ) )
120	105, 106, 119	3brtr4d 4685	. . . . . . 7 |- ( ( ph /\ I = ( J + 1 ) ) -> B < ( X + ( I x. T ) ) )
121	4	adantr 481	. . . . . . . 8 |- ( ( ph /\ I = ( J + 1 ) ) -> B e. RR )
122	59	adantr 481	. . . . . . . 8 |- ( ( ph /\ I = ( J + 1 ) ) -> ( X + ( I x. T ) ) e. RR )
123	121, 122	ltnled 10184	. . . . . . 7 |- ( ( ph /\ I = ( J + 1 ) ) -> ( B < ( X + ( I x. T ) ) <-> -. ( X + ( I x. T ) ) <_ B ) )
124	120, 123	mpbid 222	. . . . . 6 |- ( ( ph /\ I = ( J + 1 ) ) -> -. ( X + ( I x. T ) ) <_ B )
125	98, 124	pm2.65da 600	. . . . 5 |- ( ph -> -. I = ( J + 1 ) )
126	95, 125	jca 554	. . . 4 |- ( ph -> ( -. J = ( I + 1 ) /\ -. I = ( J + 1 ) ) )
127	126	adantr 481	. . 3 |- ( ( ph /\ -. I = J ) -> ( -. J = ( I + 1 ) /\ -. I = ( J + 1 ) ) )
128		pm4.56 516	. . 3 |- ( ( -. J = ( I + 1 ) /\ -. I = ( J + 1 ) ) <-> -. ( J = ( I + 1 ) \/ I = ( J + 1 ) ) )
129	127, 128	sylib 208	. 2 |- ( ( ph /\ -. I = J ) -> -. ( J = ( I + 1 ) \/ I = ( J + 1 ) ) )
130	49, 129	condan 835	1 |- ( ph -> I = J )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653  (class class class)co 6650   RRcr 9935   1c1 9937   + caddc 9939   x. cmul 9941   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   ZZcz 11377   (,]cioc 12176   [,]cicc 12178

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-rmo 2920  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-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-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-rp 11833  df-ioc 12180  df-icc 12182

This theorem is referenced by:  fourierdlem51  40374