Theorem fourierdlem36 40360

Description: F is an isomorphism. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fourierdlem36.a	|- ( ph -> A e. Fin )
fourierdlem36.assr	|- ( ph -> A C_ RR )
fourierdlem36.f	|- F = ( iota f f Isom < , < ( ( 0 ... N ) , A ) )
fourierdlem36.n	|- N = ( ( # ` A ) - 1 )

Assertion

Ref	Expression
fourierdlem36	|- ( ph -> F Isom < , < ( ( 0 ... N ) , A ) )

Distinct variable groups:   A, f   f, F   f, N   ph, f

Proof of Theorem fourierdlem36

Step	Hyp	Ref	Expression
1		fourierdlem36.f	. . 3 |- F = ( iota f f Isom < , < ( ( 0 ... N ) , A ) )
2		fourierdlem36.a	. . . . . 6 |- ( ph -> A e. Fin )
3		fourierdlem36.assr	. . . . . . 7 |- ( ph -> A C_ RR )
4		ltso 10118	. . . . . . 7 |- < Or RR
5		soss 5053	. . . . . . 7 |- ( A C_ RR -> ( < Or RR -> < Or A ) )
6	3, 4, 5	mpisyl 21	. . . . . 6 |- ( ph -> < Or A )
7		0zd 11389	. . . . . 6 |- ( ph -> 0 e. ZZ )
8		eqid 2622	. . . . . 6 |- ( ( # ` A ) + ( 0 - 1 ) ) = ( ( # ` A ) + ( 0 - 1 ) )
9	2, 6, 7, 8	fzisoeu 39514	. . . . 5 |- ( ph -> E! f f Isom < , < ( ( 0 ... ( ( # ` A ) + ( 0 - 1 ) ) ) , A ) )
10		hashcl 13147	. . . . . . . . . . . 12 |- ( A e. Fin -> ( # ` A ) e. NN0 )
11	2, 10	syl 17	. . . . . . . . . . 11 |- ( ph -> ( # ` A ) e. NN0 )
12	11	nn0cnd 11353	. . . . . . . . . 10 |- ( ph -> ( # ` A ) e. CC )
13		1cnd 10056	. . . . . . . . . 10 |- ( ph -> 1 e. CC )
14	12, 13	negsubd 10398	. . . . . . . . 9 |- ( ph -> ( ( # ` A ) + -u 1 ) = ( ( # ` A ) - 1 ) )
15		df-neg 10269	. . . . . . . . . . 11 |- -u 1 = ( 0 - 1 )
16	15	eqcomi 2631	. . . . . . . . . 10 |- ( 0 - 1 ) = -u 1
17	16	oveq2i 6661	. . . . . . . . 9 |- ( ( # ` A ) + ( 0 - 1 ) ) = ( ( # ` A ) + -u 1 )
18		fourierdlem36.n	. . . . . . . . 9 |- N = ( ( # ` A ) - 1 )
19	14, 17, 18	3eqtr4g 2681	. . . . . . . 8 |- ( ph -> ( ( # ` A ) + ( 0 - 1 ) ) = N )
20	19	oveq2d 6666	. . . . . . 7 |- ( ph -> ( 0 ... ( ( # ` A ) + ( 0 - 1 ) ) ) = ( 0 ... N ) )
21		isoeq4 6570	. . . . . . 7 |- ( ( 0 ... ( ( # ` A ) + ( 0 - 1 ) ) ) = ( 0 ... N ) -> ( f Isom < , < ( ( 0 ... ( ( # ` A ) + ( 0 - 1 ) ) ) , A ) <-> f Isom < , < ( ( 0 ... N ) , A ) ) )
22	20, 21	syl 17	. . . . . 6 |- ( ph -> ( f Isom < , < ( ( 0 ... ( ( # ` A ) + ( 0 - 1 ) ) ) , A ) <-> f Isom < , < ( ( 0 ... N ) , A ) ) )
23	22	eubidv 2490	. . . . 5 |- ( ph -> ( E! f f Isom < , < ( ( 0 ... ( ( # ` A ) + ( 0 - 1 ) ) ) , A ) <-> E! f f Isom < , < ( ( 0 ... N ) , A ) ) )
24	9, 23	mpbid 222	. . . 4 |- ( ph -> E! f f Isom < , < ( ( 0 ... N ) , A ) )
25		iotacl 5874	. . . 4 |- ( E! f f Isom < , < ( ( 0 ... N ) , A ) -> ( iota f f Isom < , < ( ( 0 ... N ) , A ) ) e. { f | f Isom < , < ( ( 0 ... N ) , A ) } )
26	24, 25	syl 17	. . 3 |- ( ph -> ( iota f f Isom < , < ( ( 0 ... N ) , A ) ) e. { f | f Isom < , < ( ( 0 ... N ) , A ) } )
27	1, 26	syl5eqel 2705	. 2 |- ( ph -> F e. { f | f Isom < , < ( ( 0 ... N ) , A ) } )
28		iotaex 5868	. . . 4 |- ( iota f f Isom < , < ( ( 0 ... N ) , A ) ) e. _V
29	1, 28	eqeltri 2697	. . 3 |- F e. _V
30		isoeq1 6567	. . 3 |- ( f = F -> ( f Isom < , < ( ( 0 ... N ) , A ) <-> F Isom < , < ( ( 0 ... N ) , A ) ) )
31	29, 30	elab 3350	. 2 |- ( F e. { f | f Isom < , < ( ( 0 ... N ) , A ) } <-> F Isom < , < ( ( 0 ... N ) , A ) )
32	27, 31	sylib 208	1 |- ( ph -> F Isom < , < ( ( 0 ... N ) , A ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   E!weu 2470   {cab 2608   _Vcvv 3200   C_ wss 3574   Or wor 5034   iotacio 5849   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   Fincfn 7955   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   - cmin 10266   -ucneg 10267   NN0cn0 11292   ...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-fz 12327  df-hash 13118

This theorem is referenced by:  fourierdlem50  40373  fourierdlem51  40374  fourierdlem52  40375  fourierdlem54  40377  fourierdlem76  40399  fourierdlem102  40425  fourierdlem103  40426  fourierdlem104  40427  fourierdlem114  40437