Theorem fourierdlem52 37848

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 37439	. . . . . 6 |- ( ph -> ( A [,] B ) C_ RR )
6	2, 5	sstrd 3475	. . . . 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 37832	. . . 4 |- ( ph -> S Isom < , < ( ( 0 ... N ) , T ) )
10		isof1o 6229	. . . 4 |- ( S Isom < , < ( ( 0 ... N ) , T ) -> S : ( 0 ... N ) -1-1-onto-> T )
11		f1of 5829	. . . 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 5753	. 2 |- ( ph -> S : ( 0 ... N ) --> ( A [,] B ) )
14		f1ofo 5836	. . . . . 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 6054	. . . . 5 |- ( ( S : ( 0 ... N ) -onto-> T /\ A e. T ) -> E. j e. ( 0 ... N ) A = ( S ` j ) )
18	15, 16, 17	syl2anc 666	. . . 4 |- ( ph -> E. j e. ( 0 ... N ) A = ( S ` j ) )
19		elfzle1 11804	. . . . . . . . 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 9686	. . . . . . . . . . . 12 |- RR C_ RR*
23	6, 22	syl6ss 3477	. . . . . . . . . . 11 |- ( ph -> T C_ RR* )
24	23	adantr 467	. . . . . . . . . 10 |- ( ( ph /\ j e. ( 0 ... N ) ) -> T C_ RR* )
25		fzssz 11803	. . . . . . . . . . 11 |- ( 0 ... N ) C_ ZZ
26		zssre 10946	. . . . . . . . . . . 12 |- ZZ C_ RR
27	26, 22	sstri 3474	. . . . . . . . . . 11 |- ZZ C_ RR*
28	25, 27	sstri 3474	. . . . . . . . . 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 12539	. . . . . . . . . . . . . . . 16 |- ( T e. Fin -> ( # ` T ) e. NN0 )
31	1, 30	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> ( # ` T ) e. NN0 )
32		ne0i 3768	. . . . . . . . . . . . . . . . 17 |- ( A e. T -> T =/= (/) )
33	16, 32	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> T =/= (/) )
34		hashge1 12569	. . . . . . . . . . . . . . . 16 |- ( ( T e. Fin /\ T =/= (/) ) -> 1 <_ ( # ` T ) )
35	1, 33, 34	syl2anc 666	. . . . . . . . . . . . . . 15 |- ( ph -> 1 <_ ( # ` T ) )
36		elnnnn0c 10917	. . . . . . . . . . . . . . 15 |- ( ( # ` T ) e. NN <-> ( ( # ` T ) e. NN0 /\ 1 <_ ( # ` T ) ) )
37	31, 35, 36	sylanbrc 669	. . . . . . . . . . . . . 14 |- ( ph -> ( # ` T ) e. NN )
38		nnm1nn0 10913	. . . . . . . . . . . . . 14 |- ( ( # ` T ) e. NN -> ( ( # ` T ) - 1 ) e. NN0 )
39	37, 38	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( ( # ` T ) - 1 ) e. NN0 )
40	8, 39	syl5eqel 2515	. . . . . . . . . . . 12 |- ( ph -> N e. NN0 )
41		nn0uz 11195	. . . . . . . . . . . 12 |- NN0 = ( ZZ>= ` 0 )
42	40, 41	syl6eleq 2521	. . . . . . . . . . 11 |- ( ph -> N e. ( ZZ>= ` 0 ) )
43		eluzfz1 11808	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... N ) )
44	42, 43	syl 17	. . . . . . . . . 10 |- ( ph -> 0 e. ( 0 ... N ) )
45	44	anim1i 571	. . . . . . . . 9 |- ( ( ph /\ j e. ( 0 ... N ) ) -> ( 0 e. ( 0 ... N ) /\ j e. ( 0 ... N ) ) )
46		leisorel 12622	. . . . . . . . 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 1265	. . . . . . . 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 1026	. . . . . 6 |- ( ( ph /\ j e. ( 0 ... N ) /\ A = ( S ` j ) ) -> ( S ` 0 ) <_ ( S ` j ) )
50		eqcom 2432	. . . . . . . 8 |- ( A = ( S ` j ) <-> ( S ` j ) = A )
51	50	biimpi 198	. . . . . . 7 |- ( A = ( S ` j ) -> ( S ` j ) = A )
52	51	3ad2ant3 1029	. . . . . 6 |- ( ( ph /\ j e. ( 0 ... N ) /\ A = ( S ` j ) ) -> ( S ` j ) = A )
53	49, 52	breqtrd 4446	. . . . 5 |- ( ( ph /\ j e. ( 0 ... N ) /\ A = ( S ` j ) ) -> ( S ` 0 ) <_ A )
54	53	rexlimdv3a 2920	. . . 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 9692	. . . 4 |- ( ph -> A e. RR* )
57	4	rexrd 9692	. . . 4 |- ( ph -> B e. RR* )
58	13, 44	ffvelrnd 6036	. . . 4 |- ( ph -> ( S ` 0 ) e. ( A [,] B ) )
59		iccgelb 11693	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ ( S ` 0 ) e. ( A [,] B ) ) -> A <_ ( S ` 0 ) )
60	56, 57, 58, 59	syl3anc 1265	. . 3 |- ( ph -> A <_ ( S ` 0 ) )
61	5, 58	sseldd 3466	. . . 4 |- ( ph -> ( S ` 0 ) e. RR )
62	61, 3	letri3d 9779	. . 3 |- ( ph -> ( ( S ` 0 ) = A <-> ( ( S ` 0 ) <_ A /\ A <_ ( S ` 0 ) ) ) )
63	55, 60, 62	mpbir2and 931	. 2 |- ( ph -> ( S ` 0 ) = A )
64		eluzfz2 11809	. . . . . 6 |- ( N e. ( ZZ>= ` 0 ) -> N e. ( 0 ... N ) )
65	42, 64	syl 17	. . . . 5 |- ( ph -> N e. ( 0 ... N ) )
66	13, 65	ffvelrnd 6036	. . . 4 |- ( ph -> ( S ` N ) e. ( A [,] B ) )
67		iccleub 11692	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ ( S ` N ) e. ( A [,] B ) ) -> ( S ` N ) <_ B )
68	56, 57, 66, 67	syl3anc 1265	. . 3 |- ( ph -> ( S ` N ) <_ B )
69		fourierdlem52.bt	. . . . 5 |- ( ph -> B e. T )
70		foelrn 6054	. . . . 5 |- ( ( S : ( 0 ... N ) -onto-> T /\ B e. T ) -> E. j e. ( 0 ... N ) B = ( S ` j ) )
71	15, 69, 70	syl2anc 666	. . . 4 |- ( ph -> E. j e. ( 0 ... N ) B = ( S ` j ) )
72		simp3 1008	. . . . . 6 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> B = ( S ` j ) )
73		elfzle2 11805	. . . . . . . 8 |- ( j e. ( 0 ... N ) -> j <_ N )
74	73	3ad2ant2 1028	. . . . . . 7 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> j <_ N )
75	9	3ad2ant1 1027	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> S Isom < , < ( ( 0 ... N ) , T ) )
76	29	3adant3 1026	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> ( ( 0 ... N ) C_ RR* /\ T C_ RR* ) )
77		simp2 1007	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> j e. ( 0 ... N ) )
78	65	3ad2ant1 1027	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> N e. ( 0 ... N ) )
79		leisorel 12622	. . . . . . . 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 1269	. . . . . . 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 4442	. . . . 5 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> B <_ ( S ` N ) )
83	82	rexlimdv3a 2920	. . . 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 3466	. . . 4 |- ( ph -> ( S ` N ) e. RR )
86	85, 4	letri3d 9779	. . 3 |- ( ph -> ( ( S ` N ) = B <-> ( ( S ` N ) <_ B /\ B <_ ( S ` N ) ) ) )
87	68, 84, 86	mpbir2and 931	. 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 983   = wceq 1438   e. wcel 1869   =/= wne 2619   E.wrex 2777   C_ wss 3437   (/)c0 3762   class class class wbr 4421   iotacio 5561   -->wf 5595   -onto->wfo 5597   -1-1-onto->wf1o 5598   ` cfv 5599   Isom wiso 5600  (class class class)co 6303   Fincfn 7575   RRcr 9540   0cc0 9541   1c1 9542   RR*cxr 9676   < clt 9677   <_ cle 9678   - cmin 9862   NNcn 10611   NN0cn0 10871   ZZcz 10939   ZZ>=cuz 11161   [,]cicc 11640   ...cfz 11786   #chash 12516

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-inf2 8150  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-int 4254  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-se 4811  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-isom 5608  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-1o 7188  df-er 7369  df-en 7576  df-dom 7577  df-sdom 7578  df-fin 7579  df-oi 8029  df-card 8376  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-n0 10872  df-z 10940  df-uz 11162  df-icc 11644  df-fz 11787  df-hash 12517

This theorem is referenced by:  fourierdlem103  37899  fourierdlem104  37900