Theorem fourierdlem52 38022

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 37602	. . . . . 6 |- ( ph -> ( A [,] B ) C_ RR )
6	2, 5	sstrd 3442	. . . . 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 38006	. . . 4 |- ( ph -> S Isom < , < ( ( 0 ... N ) , T ) )
10		isof1o 6216	. . . 4 |- ( S Isom < , < ( ( 0 ... N ) , T ) -> S : ( 0 ... N ) -1-1-onto-> T )
11		f1of 5814	. . . 4 |- ( S : ( 0 ... N ) -1-1-onto-> T -> S : ( 0 ... N ) --> T )
12	9, 10, 11	3syl 18	. . 3 |- ( ph -> S : ( 0 ... N ) --> T )
13	12, 2	fssd 5738	. 2 |- ( ph -> S : ( 0 ... N ) --> ( A [,] B ) )
14		f1ofo 5821	. . . . . 6 |- ( S : ( 0 ... N ) -1-1-onto-> T -> S : ( 0 ... N ) -onto-> T )
15	9, 10, 14	3syl 18	. . . . 5 |- ( ph -> S : ( 0 ... N ) -onto-> T )
16		fourierdlem52.at	. . . . 5 |- ( ph -> A e. T )
17		foelrn 6041	. . . . 5 |- ( ( S : ( 0 ... N ) -onto-> T /\ A e. T ) -> E. j e. ( 0 ... N ) A = ( S ` j ) )
18	15, 16, 17	syl2anc 667	. . . 4 |- ( ph -> E. j e. ( 0 ... N ) A = ( S ` j ) )
19		elfzle1 11802	. . . . . . . . 9 |- ( j e. ( 0 ... N ) -> 0 <_ j )
20	19	adantl 468	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) ) -> 0 <_ j )
21	9	adantr 467	. . . . . . . . 9 |- ( ( ph /\ j e. ( 0 ... N ) ) -> S Isom < , < ( ( 0 ... N ) , T ) )
22		ressxr 9684	. . . . . . . . . . . 12 |- RR C_ RR*
23	6, 22	syl6ss 3444	. . . . . . . . . . 11 |- ( ph -> T C_ RR* )
24	23	adantr 467	. . . . . . . . . 10 |- ( ( ph /\ j e. ( 0 ... N ) ) -> T C_ RR* )
25		fzssz 11801	. . . . . . . . . . 11 |- ( 0 ... N ) C_ ZZ
26		zssre 10944	. . . . . . . . . . . 12 |- ZZ C_ RR
27	26, 22	sstri 3441	. . . . . . . . . . 11 |- ZZ C_ RR*
28	25, 27	sstri 3441	. . . . . . . . . 10 |- ( 0 ... N ) C_ RR*
29	24, 28	jctil 540	. . . . . . . . 9 |- ( ( ph /\ j e. ( 0 ... N ) ) -> ( ( 0 ... N ) C_ RR* /\ T C_ RR* ) )
30		hashcl 12538	. . . . . . . . . . . . . . . 16 |- ( T e. Fin -> ( # ` T ) e. NN0 )
31	1, 30	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> ( # ` T ) e. NN0 )
32		ne0i 3737	. . . . . . . . . . . . . . . . 17 |- ( A e. T -> T =/= (/) )
33	16, 32	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> T =/= (/) )
34		hashge1 12568	. . . . . . . . . . . . . . . 16 |- ( ( T e. Fin /\ T =/= (/) ) -> 1 <_ ( # ` T ) )
35	1, 33, 34	syl2anc 667	. . . . . . . . . . . . . . 15 |- ( ph -> 1 <_ ( # ` T ) )
36		elnnnn0c 10915	. . . . . . . . . . . . . . 15 |- ( ( # ` T ) e. NN <-> ( ( # ` T ) e. NN0 /\ 1 <_ ( # ` T ) ) )
37	31, 35, 36	sylanbrc 670	. . . . . . . . . . . . . 14 |- ( ph -> ( # ` T ) e. NN )
38		nnm1nn0 10911	. . . . . . . . . . . . . 14 |- ( ( # ` T ) e. NN -> ( ( # ` T ) - 1 ) e. NN0 )
39	37, 38	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( ( # ` T ) - 1 ) e. NN0 )
40	8, 39	syl5eqel 2533	. . . . . . . . . . . 12 |- ( ph -> N e. NN0 )
41		nn0uz 11193	. . . . . . . . . . . 12 |- NN0 = ( ZZ>= ` 0 )
42	40, 41	syl6eleq 2539	. . . . . . . . . . 11 |- ( ph -> N e. ( ZZ>= ` 0 ) )
43		eluzfz1 11806	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... N ) )
44	42, 43	syl 17	. . . . . . . . . 10 |- ( ph -> 0 e. ( 0 ... N ) )
45	44	anim1i 572	. . . . . . . . 9 |- ( ( ph /\ j e. ( 0 ... N ) ) -> ( 0 e. ( 0 ... N ) /\ j e. ( 0 ... N ) ) )
46		leisorel 12623	. . . . . . . . 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 1268	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) ) -> ( 0 <_ j <-> ( S ` 0 ) <_ ( S ` j ) ) )
48	20, 47	mpbid 214	. . . . . . 7 |- ( ( ph /\ j e. ( 0 ... N ) ) -> ( S ` 0 ) <_ ( S ` j ) )
49	48	3adant3 1028	. . . . . 6 |- ( ( ph /\ j e. ( 0 ... N ) /\ A = ( S ` j ) ) -> ( S ` 0 ) <_ ( S ` j ) )
50		eqcom 2458	. . . . . . . 8 |- ( A = ( S ` j ) <-> ( S ` j ) = A )
51	50	biimpi 198	. . . . . . 7 |- ( A = ( S ` j ) -> ( S ` j ) = A )
52	51	3ad2ant3 1031	. . . . . 6 |- ( ( ph /\ j e. ( 0 ... N ) /\ A = ( S ` j ) ) -> ( S ` j ) = A )
53	49, 52	breqtrd 4427	. . . . 5 |- ( ( ph /\ j e. ( 0 ... N ) /\ A = ( S ` j ) ) -> ( S ` 0 ) <_ A )
54	53	rexlimdv3a 2881	. . . 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 9690	. . . 4 |- ( ph -> A e. RR* )
57	4	rexrd 9690	. . . 4 |- ( ph -> B e. RR* )
58	13, 44	ffvelrnd 6023	. . . 4 |- ( ph -> ( S ` 0 ) e. ( A [,] B ) )
59		iccgelb 11691	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ ( S ` 0 ) e. ( A [,] B ) ) -> A <_ ( S ` 0 ) )
60	56, 57, 58, 59	syl3anc 1268	. . 3 |- ( ph -> A <_ ( S ` 0 ) )
61	5, 58	sseldd 3433	. . . 4 |- ( ph -> ( S ` 0 ) e. RR )
62	61, 3	letri3d 9777	. . 3 |- ( ph -> ( ( S ` 0 ) = A <-> ( ( S ` 0 ) <_ A /\ A <_ ( S ` 0 ) ) ) )
63	55, 60, 62	mpbir2and 933	. 2 |- ( ph -> ( S ` 0 ) = A )
64		eluzfz2 11807	. . . . . 6 |- ( N e. ( ZZ>= ` 0 ) -> N e. ( 0 ... N ) )
65	42, 64	syl 17	. . . . 5 |- ( ph -> N e. ( 0 ... N ) )
66	13, 65	ffvelrnd 6023	. . . 4 |- ( ph -> ( S ` N ) e. ( A [,] B ) )
67		iccleub 11690	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ ( S ` N ) e. ( A [,] B ) ) -> ( S ` N ) <_ B )
68	56, 57, 66, 67	syl3anc 1268	. . 3 |- ( ph -> ( S ` N ) <_ B )
69		fourierdlem52.bt	. . . . 5 |- ( ph -> B e. T )
70		foelrn 6041	. . . . 5 |- ( ( S : ( 0 ... N ) -onto-> T /\ B e. T ) -> E. j e. ( 0 ... N ) B = ( S ` j ) )
71	15, 69, 70	syl2anc 667	. . . 4 |- ( ph -> E. j e. ( 0 ... N ) B = ( S ` j ) )
72		simp3 1010	. . . . . 6 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> B = ( S ` j ) )
73		elfzle2 11803	. . . . . . . 8 |- ( j e. ( 0 ... N ) -> j <_ N )
74	73	3ad2ant2 1030	. . . . . . 7 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> j <_ N )
75	9	3ad2ant1 1029	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> S Isom < , < ( ( 0 ... N ) , T ) )
76	29	3adant3 1028	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> ( ( 0 ... N ) C_ RR* /\ T C_ RR* ) )
77		simp2 1009	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> j e. ( 0 ... N ) )
78	65	3ad2ant1 1029	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> N e. ( 0 ... N ) )
79		leisorel 12623	. . . . . . . 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 1272	. . . . . . 7 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> ( j <_ N <-> ( S ` j ) <_ ( S ` N ) ) )
81	74, 80	mpbid 214	. . . . . 6 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> ( S ` j ) <_ ( S ` N ) )
82	72, 81	eqbrtrd 4423	. . . . 5 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> B <_ ( S ` N ) )
83	82	rexlimdv3a 2881	. . . 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 3433	. . . 4 |- ( ph -> ( S ` N ) e. RR )
86	85, 4	letri3d 9777	. . 3 |- ( ph -> ( ( S ` N ) = B <-> ( ( S ` N ) <_ B /\ B <_ ( S ` N ) ) ) )
87	68, 84, 86	mpbir2and 933	. 2 |- ( ph -> ( S ` N ) = B )
88	13, 63, 87	jca31 537	1 |- ( ph -> ( ( S : ( 0 ... N ) --> ( A [,] B ) /\ ( S ` 0 ) = A ) /\ ( S ` N ) = B ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   E.wrex 2738   C_ wss 3404   (/)c0 3731   class class class wbr 4402   iotacio 5544   -->wf 5578   -onto->wfo 5580   -1-1-onto->wf1o 5581   ` cfv 5582   Isom wiso 5583  (class class class)co 6290   Fincfn 7569   RRcr 9538   0cc0 9539   1c1 9540   RR*cxr 9674   < clt 9675   <_ cle 9676   - cmin 9860   NNcn 10609   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   [,]cicc 11638   ...cfz 11784   #chash 12515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-inf2 8146  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-int 4235  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-se 4794  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-isom 5591  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-1o 7182  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-fin 7573  df-oi 8025  df-card 8373  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-icc 11642  df-fz 11785  df-hash 12516

This theorem is referenced by:  fourierdlem103  38073  fourierdlem104  38074