Theorem fourierdlem52 32187

Description: d16:d17,d18:jca |- ( ph -> ( ( S 0 ) <_ A /\ A <_ ( S 0 ) ) ) (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fourierdlem52.tf	|- ( ph -> T e. Fin )
fourierdlem52.n	|- N = ( ( # ` T ) - 1 )
fourierdlem52.s	|- S = ( iota f f Isom < , < ( ( 0 ... N ) , T ) )
fourierdlem52.a	|- ( ph -> A e. RR )
fourierdlem52.b	|- ( ph -> B e. RR )
fourierdlem52.t	|- ( ph -> T C_ ( A [,] B ) )
fourierdlem52.at	|- ( ph -> A e. T )
fourierdlem52.bt	|- ( ph -> B e. T )

Assertion

Ref	Expression
fourierdlem52	|- ( ph -> ( ( S : ( 0 ... N ) --> ( A [,] B ) /\ ( S ` 0 ) = A ) /\ ( S ` N ) = B ) )

Distinct variable groups:   f, N   S, f   T, f   ph, f

Allowed substitution hints:   A( f)   B( f)

Proof of Theorem fourierdlem52

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fourierdlem52.tf	. . . . 5 |- ( ph -> T e. Fin )
2		fourierdlem52.t	. . . . . 6 |- ( ph -> T C_ ( A [,] B ) )
3		fourierdlem52.a	. . . . . . 7 |- ( ph -> A e. RR )
4		fourierdlem52.b	. . . . . . 7 |- ( ph -> B e. RR )
5	3, 4	iccssred 31785	. . . . . 6 |- ( ph -> ( A [,] B ) C_ RR )
6	2, 5	sstrd 3509	. . . . 5 |- ( ph -> T C_ RR )
7		fourierdlem52.s	. . . . 5 |- S = ( iota f f Isom < , < ( ( 0 ... N ) , T ) )
8		fourierdlem52.n	. . . . 5 |- N = ( ( # ` T ) - 1 )
9	1, 6, 7, 8	fourierdlem36 32171	. . . 4 |- ( ph -> S Isom < , < ( ( 0 ... N ) , T ) )
10		isof1o 6222	. . . 4 |- ( S Isom < , < ( ( 0 ... N ) , T ) -> S : ( 0 ... N ) -1-1-onto-> T )
11		f1of 5822	. . . 4 |- ( S : ( 0 ... N ) -1-1-onto-> T -> S : ( 0 ... N ) --> T )
12	9, 10, 11	3syl 20	. . 3 |- ( ph -> S : ( 0 ... N ) --> T )
13	12, 2	fssd 5746	. 2 |- ( ph -> S : ( 0 ... N ) --> ( A [,] B ) )
14		f1ofo 5829	. . . . . 6 |- ( S : ( 0 ... N ) -1-1-onto-> T -> S : ( 0 ... N ) -onto-> T )
15	9, 10, 14	3syl 20	. . . . 5 |- ( ph -> S : ( 0 ... N ) -onto-> T )
16		fourierdlem52.at	. . . . 5 |- ( ph -> A e. T )
17		foelrn 6051	. . . . 5 |- ( ( S : ( 0 ... N ) -onto-> T /\ A e. T ) -> E. j e. ( 0 ... N ) A = ( S ` j ) )
18	15, 16, 17	syl2anc 661	. . . 4 |- ( ph -> E. j e. ( 0 ... N ) A = ( S ` j ) )
19		elfzle1 11714	. . . . . . . . 9 |- ( j e. ( 0 ... N ) -> 0 <_ j )
20	19	adantl 466	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) ) -> 0 <_ j )
21	9	adantr 465	. . . . . . . . 9 |- ( ( ph /\ j e. ( 0 ... N ) ) -> S Isom < , < ( ( 0 ... N ) , T ) )
22		ressxr 9654	. . . . . . . . . . . 12 |- RR C_ RR*
23	6, 22	syl6ss 3511	. . . . . . . . . . 11 |- ( ph -> T C_ RR* )
24	23	adantr 465	. . . . . . . . . 10 |- ( ( ph /\ j e. ( 0 ... N ) ) -> T C_ RR* )
25		fzssz 31712	. . . . . . . . . . 11 |- ( 0 ... N ) C_ ZZ
26		zssre 10892	. . . . . . . . . . . 12 |- ZZ C_ RR
27	26, 22	sstri 3508	. . . . . . . . . . 11 |- ZZ C_ RR*
28	25, 27	sstri 3508	. . . . . . . . . 10 |- ( 0 ... N ) C_ RR*
29	24, 28	jctil 537	. . . . . . . . 9 |- ( ( ph /\ j e. ( 0 ... N ) ) -> ( ( 0 ... N ) C_ RR* /\ T C_ RR* ) )
30		hashcl 12431	. . . . . . . . . . . . . . . 16 |- ( T e. Fin -> ( # ` T ) e. NN0 )
31	1, 30	syl 16	. . . . . . . . . . . . . . 15 |- ( ph -> ( # ` T ) e. NN0 )
32		ne0i 3799	. . . . . . . . . . . . . . . . 17 |- ( A e. T -> T =/= (/) )
33	16, 32	syl 16	. . . . . . . . . . . . . . . 16 |- ( ph -> T =/= (/) )
34		hashge1 12460	. . . . . . . . . . . . . . . 16 |- ( ( T e. Fin /\ T =/= (/) ) -> 1 <_ ( # ` T ) )
35	1, 33, 34	syl2anc 661	. . . . . . . . . . . . . . 15 |- ( ph -> 1 <_ ( # ` T ) )
36		elnnnn0c 10862	. . . . . . . . . . . . . . 15 |- ( ( # ` T ) e. NN <-> ( ( # ` T ) e. NN0 /\ 1 <_ ( # ` T ) ) )
37	31, 35, 36	sylanbrc 664	. . . . . . . . . . . . . 14 |- ( ph -> ( # ` T ) e. NN )
38		nnm1nn0 10858	. . . . . . . . . . . . . 14 |- ( ( # ` T ) e. NN -> ( ( # ` T ) - 1 ) e. NN0 )
39	37, 38	syl 16	. . . . . . . . . . . . 13 |- ( ph -> ( ( # ` T ) - 1 ) e. NN0 )
40	8, 39	syl5eqel 2549	. . . . . . . . . . . 12 |- ( ph -> N e. NN0 )
41		nn0uz 11140	. . . . . . . . . . . 12 |- NN0 = ( ZZ>= ` 0 )
42	40, 41	syl6eleq 2555	. . . . . . . . . . 11 |- ( ph -> N e. ( ZZ>= ` 0 ) )
43		eluzfz1 11718	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... N ) )
44	42, 43	syl 16	. . . . . . . . . 10 |- ( ph -> 0 e. ( 0 ... N ) )
45	44	anim1i 568	. . . . . . . . 9 |- ( ( ph /\ j e. ( 0 ... N ) ) -> ( 0 e. ( 0 ... N ) /\ j e. ( 0 ... N ) ) )
46		leisorel 12513	. . . . . . . . 9 |- ( ( S Isom < , < ( ( 0 ... N ) , T ) /\ ( ( 0 ... N ) C_ RR* /\ T C_ RR* ) /\ ( 0 e. ( 0 ... N ) /\ j e. ( 0 ... N ) ) ) -> ( 0 <_ j <-> ( S ` 0 ) <_ ( S ` j ) ) )
47	21, 29, 45, 46	syl3anc 1228	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) ) -> ( 0 <_ j <-> ( S ` 0 ) <_ ( S ` j ) ) )
48	20, 47	mpbid 210	. . . . . . 7 |- ( ( ph /\ j e. ( 0 ... N ) ) -> ( S ` 0 ) <_ ( S ` j ) )
49	48	3adant3 1016	. . . . . 6 |- ( ( ph /\ j e. ( 0 ... N ) /\ A = ( S ` j ) ) -> ( S ` 0 ) <_ ( S ` j ) )
50		eqcom 2466	. . . . . . . 8 |- ( A = ( S ` j ) <-> ( S ` j ) = A )
51	50	biimpi 194	. . . . . . 7 |- ( A = ( S ` j ) -> ( S ` j ) = A )
52	51	3ad2ant3 1019	. . . . . 6 |- ( ( ph /\ j e. ( 0 ... N ) /\ A = ( S ` j ) ) -> ( S ` j ) = A )
53	49, 52	breqtrd 4480	. . . . 5 |- ( ( ph /\ j e. ( 0 ... N ) /\ A = ( S ` j ) ) -> ( S ` 0 ) <_ A )
54	53	rexlimdv3a 2951	. . . 4 |- ( ph -> ( E. j e. ( 0 ... N ) A = ( S ` j ) -> ( S ` 0 ) <_ A ) )
55	18, 54	mpd 15	. . 3 |- ( ph -> ( S ` 0 ) <_ A )
56	3	rexrd 9660	. . . 4 |- ( ph -> A e. RR* )
57	4	rexrd 9660	. . . 4 |- ( ph -> B e. RR* )
58	13, 44	ffvelrnd 6033	. . . 4 |- ( ph -> ( S ` 0 ) e. ( A [,] B ) )
59		iccgelb 11606	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ ( S ` 0 ) e. ( A [,] B ) ) -> A <_ ( S ` 0 ) )
60	56, 57, 58, 59	syl3anc 1228	. . 3 |- ( ph -> A <_ ( S ` 0 ) )
61	5, 58	sseldd 3500	. . . 4 |- ( ph -> ( S ` 0 ) e. RR )
62	61, 3	letri3d 9744	. . 3 |- ( ph -> ( ( S ` 0 ) = A <-> ( ( S ` 0 ) <_ A /\ A <_ ( S ` 0 ) ) ) )
63	55, 60, 62	mpbir2and 922	. 2 |- ( ph -> ( S ` 0 ) = A )
64		eluzfz2 11719	. . . . . 6 |- ( N e. ( ZZ>= ` 0 ) -> N e. ( 0 ... N ) )
65	42, 64	syl 16	. . . . 5 |- ( ph -> N e. ( 0 ... N ) )
66	13, 65	ffvelrnd 6033	. . . 4 |- ( ph -> ( S ` N ) e. ( A [,] B ) )
67		iccleub 11605	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ ( S ` N ) e. ( A [,] B ) ) -> ( S ` N ) <_ B )
68	56, 57, 66, 67	syl3anc 1228	. . 3 |- ( ph -> ( S ` N ) <_ B )
69		fourierdlem52.bt	. . . . 5 |- ( ph -> B e. T )
70		foelrn 6051	. . . . 5 |- ( ( S : ( 0 ... N ) -onto-> T /\ B e. T ) -> E. j e. ( 0 ... N ) B = ( S ` j ) )
71	15, 69, 70	syl2anc 661	. . . 4 |- ( ph -> E. j e. ( 0 ... N ) B = ( S ` j ) )
72		simp3 998	. . . . . 6 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> B = ( S ` j ) )
73		elfzle2 11715	. . . . . . . 8 |- ( j e. ( 0 ... N ) -> j <_ N )
74	73	3ad2ant2 1018	. . . . . . 7 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> j <_ N )
75	9	3ad2ant1 1017	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> S Isom < , < ( ( 0 ... N ) , T ) )
76	29	3adant3 1016	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> ( ( 0 ... N ) C_ RR* /\ T C_ RR* ) )
77		simp2 997	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> j e. ( 0 ... N ) )
78	65	3ad2ant1 1017	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> N e. ( 0 ... N ) )
79		leisorel 12513	. . . . . . . 8 |- ( ( S Isom < , < ( ( 0 ... N ) , T ) /\ ( ( 0 ... N ) C_ RR* /\ T C_ RR* ) /\ ( j e. ( 0 ... N ) /\ N e. ( 0 ... N ) ) ) -> ( j <_ N <-> ( S ` j ) <_ ( S ` N ) ) )
80	75, 76, 77, 78, 79	syl112anc 1232	. . . . . . 7 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> ( j <_ N <-> ( S ` j ) <_ ( S ` N ) ) )
81	74, 80	mpbid 210	. . . . . 6 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> ( S ` j ) <_ ( S ` N ) )
82	72, 81	eqbrtrd 4476	. . . . 5 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> B <_ ( S ` N ) )
83	82	rexlimdv3a 2951	. . . 4 |- ( ph -> ( E. j e. ( 0 ... N ) B = ( S ` j ) -> B <_ ( S ` N ) ) )
84	71, 83	mpd 15	. . 3 |- ( ph -> B <_ ( S ` N ) )
85	5, 66	sseldd 3500	. . . 4 |- ( ph -> ( S ` N ) e. RR )
86	85, 4	letri3d 9744	. . 3 |- ( ph -> ( ( S ` N ) = B <-> ( ( S ` N ) <_ B /\ B <_ ( S ` N ) ) ) )
87	68, 84, 86	mpbir2and 922	. 2 |- ( ph -> ( S ` N ) = B )
88	13, 63, 87	jca31 534	1 |- ( ph -> ( ( S : ( 0 ... N ) --> ( A [,] B ) /\ ( S ` 0 ) = A ) /\ ( S ` N ) = B ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   = wceq 1395   e. wcel 1819   =/= wne 2652   E.wrex 2808   C_ wss 3471   (/)c0 3793   class class class wbr 4456   iotacio 5555   -->wf 5590   -onto->wfo 5592   -1-1-onto->wf1o 5593   ` cfv 5594   Isom wiso 5595  (class class class)co 6296   Fincfn 7535   RRcr 9508   0cc0 9509   1c1 9510   RR*cxr 9644   < clt 9645   <_ cle 9646   - cmin 9824   NNcn 10556   NN0cn0 10816   ZZcz 10885   ZZ>=cuz 11106   [,]cicc 11557   ...cfz 11697   #chash 12408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-se 4848  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-oi 7953  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-icc 11561  df-fz 11698  df-hash 12409

This theorem is referenced by:  fourierdlem103  32238  fourierdlem104  32239