Theorem fourierdlem52 40375

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 39727	. . . . . 6 |- ( ph -> ( A [,] B ) C_ RR )
6	2, 5	sstrd 3613	. . . . 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 40360	. . . 4 |- ( ph -> S Isom < , < ( ( 0 ... N ) , T ) )
10		isof1o 6573	. . . 4 |- ( S Isom < , < ( ( 0 ... N ) , T ) -> S : ( 0 ... N ) -1-1-onto-> T )
11		f1of 6137	. . . 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 6057	. 2 |- ( ph -> S : ( 0 ... N ) --> ( A [,] B ) )
14		f1ofo 6144	. . . . . 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 6378	. . . . 5 |- ( ( S : ( 0 ... N ) -onto-> T /\ A e. T ) -> E. j e. ( 0 ... N ) A = ( S ` j ) )
18	15, 16, 17	syl2anc 693	. . . 4 |- ( ph -> E. j e. ( 0 ... N ) A = ( S ` j ) )
19		elfzle1 12344	. . . . . . . . 9 |- ( j e. ( 0 ... N ) -> 0 <_ j )
20	19	adantl 482	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) ) -> 0 <_ j )
21	9	adantr 481	. . . . . . . . 9 |- ( ( ph /\ j e. ( 0 ... N ) ) -> S Isom < , < ( ( 0 ... N ) , T ) )
22		ressxr 10083	. . . . . . . . . . . 12 |- RR C_ RR*
23	6, 22	syl6ss 3615	. . . . . . . . . . 11 |- ( ph -> T C_ RR* )
24	23	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ j e. ( 0 ... N ) ) -> T C_ RR* )
25		fzssz 12343	. . . . . . . . . . 11 |- ( 0 ... N ) C_ ZZ
26		zssre 11384	. . . . . . . . . . . 12 |- ZZ C_ RR
27	26, 22	sstri 3612	. . . . . . . . . . 11 |- ZZ C_ RR*
28	25, 27	sstri 3612	. . . . . . . . . 10 |- ( 0 ... N ) C_ RR*
29	24, 28	jctil 560	. . . . . . . . 9 |- ( ( ph /\ j e. ( 0 ... N ) ) -> ( ( 0 ... N ) C_ RR* /\ T C_ RR* ) )
30		hashcl 13147	. . . . . . . . . . . . . . . 16 |- ( T e. Fin -> ( # ` T ) e. NN0 )
31	1, 30	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> ( # ` T ) e. NN0 )
32		ne0i 3921	. . . . . . . . . . . . . . . . 17 |- ( A e. T -> T =/= (/) )
33	16, 32	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> T =/= (/) )
34		hashge1 13178	. . . . . . . . . . . . . . . 16 |- ( ( T e. Fin /\ T =/= (/) ) -> 1 <_ ( # ` T ) )
35	1, 33, 34	syl2anc 693	. . . . . . . . . . . . . . 15 |- ( ph -> 1 <_ ( # ` T ) )
36		elnnnn0c 11338	. . . . . . . . . . . . . . 15 |- ( ( # ` T ) e. NN <-> ( ( # ` T ) e. NN0 /\ 1 <_ ( # ` T ) ) )
37	31, 35, 36	sylanbrc 698	. . . . . . . . . . . . . 14 |- ( ph -> ( # ` T ) e. NN )
38		nnm1nn0 11334	. . . . . . . . . . . . . 14 |- ( ( # ` T ) e. NN -> ( ( # ` T ) - 1 ) e. NN0 )
39	37, 38	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( ( # ` T ) - 1 ) e. NN0 )
40	8, 39	syl5eqel 2705	. . . . . . . . . . . 12 |- ( ph -> N e. NN0 )
41		nn0uz 11722	. . . . . . . . . . . 12 |- NN0 = ( ZZ>= ` 0 )
42	40, 41	syl6eleq 2711	. . . . . . . . . . 11 |- ( ph -> N e. ( ZZ>= ` 0 ) )
43		eluzfz1 12348	. . . . . . . . . . 11 |- ( N e. ( ZZ>= ` 0 ) -> 0 e. ( 0 ... N ) )
44	42, 43	syl 17	. . . . . . . . . 10 |- ( ph -> 0 e. ( 0 ... N ) )
45	44	anim1i 592	. . . . . . . . 9 |- ( ( ph /\ j e. ( 0 ... N ) ) -> ( 0 e. ( 0 ... N ) /\ j e. ( 0 ... N ) ) )
46		leisorel 13244	. . . . . . . . 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 1326	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) ) -> ( 0 <_ j <-> ( S ` 0 ) <_ ( S ` j ) ) )
48	20, 47	mpbid 222	. . . . . . 7 |- ( ( ph /\ j e. ( 0 ... N ) ) -> ( S ` 0 ) <_ ( S ` j ) )
49	48	3adant3 1081	. . . . . 6 |- ( ( ph /\ j e. ( 0 ... N ) /\ A = ( S ` j ) ) -> ( S ` 0 ) <_ ( S ` j ) )
50		eqcom 2629	. . . . . . . 8 |- ( A = ( S ` j ) <-> ( S ` j ) = A )
51	50	biimpi 206	. . . . . . 7 |- ( A = ( S ` j ) -> ( S ` j ) = A )
52	51	3ad2ant3 1084	. . . . . 6 |- ( ( ph /\ j e. ( 0 ... N ) /\ A = ( S ` j ) ) -> ( S ` j ) = A )
53	49, 52	breqtrd 4679	. . . . 5 |- ( ( ph /\ j e. ( 0 ... N ) /\ A = ( S ` j ) ) -> ( S ` 0 ) <_ A )
54	53	rexlimdv3a 3033	. . . 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 10089	. . . 4 |- ( ph -> A e. RR* )
57	4	rexrd 10089	. . . 4 |- ( ph -> B e. RR* )
58	13, 44	ffvelrnd 6360	. . . 4 |- ( ph -> ( S ` 0 ) e. ( A [,] B ) )
59		iccgelb 12230	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ ( S ` 0 ) e. ( A [,] B ) ) -> A <_ ( S ` 0 ) )
60	56, 57, 58, 59	syl3anc 1326	. . 3 |- ( ph -> A <_ ( S ` 0 ) )
61	5, 58	sseldd 3604	. . . 4 |- ( ph -> ( S ` 0 ) e. RR )
62	61, 3	letri3d 10179	. . 3 |- ( ph -> ( ( S ` 0 ) = A <-> ( ( S ` 0 ) <_ A /\ A <_ ( S ` 0 ) ) ) )
63	55, 60, 62	mpbir2and 957	. 2 |- ( ph -> ( S ` 0 ) = A )
64		eluzfz2 12349	. . . . . 6 |- ( N e. ( ZZ>= ` 0 ) -> N e. ( 0 ... N ) )
65	42, 64	syl 17	. . . . 5 |- ( ph -> N e. ( 0 ... N ) )
66	13, 65	ffvelrnd 6360	. . . 4 |- ( ph -> ( S ` N ) e. ( A [,] B ) )
67		iccleub 12229	. . . 4 |- ( ( A e. RR* /\ B e. RR* /\ ( S ` N ) e. ( A [,] B ) ) -> ( S ` N ) <_ B )
68	56, 57, 66, 67	syl3anc 1326	. . 3 |- ( ph -> ( S ` N ) <_ B )
69		fourierdlem52.bt	. . . . 5 |- ( ph -> B e. T )
70		foelrn 6378	. . . . 5 |- ( ( S : ( 0 ... N ) -onto-> T /\ B e. T ) -> E. j e. ( 0 ... N ) B = ( S ` j ) )
71	15, 69, 70	syl2anc 693	. . . 4 |- ( ph -> E. j e. ( 0 ... N ) B = ( S ` j ) )
72		simp3 1063	. . . . . 6 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> B = ( S ` j ) )
73		elfzle2 12345	. . . . . . . 8 |- ( j e. ( 0 ... N ) -> j <_ N )
74	73	3ad2ant2 1083	. . . . . . 7 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> j <_ N )
75	9	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> S Isom < , < ( ( 0 ... N ) , T ) )
76	29	3adant3 1081	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> ( ( 0 ... N ) C_ RR* /\ T C_ RR* ) )
77		simp2 1062	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> j e. ( 0 ... N ) )
78	65	3ad2ant1 1082	. . . . . . . 8 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> N e. ( 0 ... N ) )
79		leisorel 13244	. . . . . . . 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 1330	. . . . . . 7 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> ( j <_ N <-> ( S ` j ) <_ ( S ` N ) ) )
81	74, 80	mpbid 222	. . . . . 6 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> ( S ` j ) <_ ( S ` N ) )
82	72, 81	eqbrtrd 4675	. . . . 5 |- ( ( ph /\ j e. ( 0 ... N ) /\ B = ( S ` j ) ) -> B <_ ( S ` N ) )
83	82	rexlimdv3a 3033	. . . 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 3604	. . . 4 |- ( ph -> ( S ` N ) e. RR )
86	85, 4	letri3d 10179	. . 3 |- ( ph -> ( ( S ` N ) = B <-> ( ( S ` N ) <_ B /\ B <_ ( S ` N ) ) ) )
87	68, 84, 86	mpbir2and 957	. 2 |- ( ph -> ( S ` N ) = B )
88	13, 63, 87	jca31 557	1 |- ( ph -> ( ( S : ( 0 ... N ) --> ( A [,] B ) /\ ( S ` 0 ) = A ) /\ ( S ` N ) = B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   C_ wss 3574   (/)c0 3915   class class class wbr 4653   iotacio 5849   -->wf 5884   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   Fincfn 7955   RRcr 9935   0cc0 9936   1c1 9937   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   [,]cicc 12178   ...cfz 12326   #chash 13117

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  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-int 4476  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-icc 12182  df-fz 12327  df-hash 13118

This theorem is referenced by:  fourierdlem103  40426  fourierdlem104  40427