Theorem fzisoeu 39514

Description: A finite ordered set has a unique order isomorphism to a generic finite sequence of integers. This theorem generalizes fz1iso 13246 for the base index and also states the uniqueness condition. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fzisoeu.h	|- ( ph -> H e. Fin )
fzisoeu.or	|- ( ph -> < Or H )
fzisoeu.m	|- ( ph -> M e. ZZ )
fzisoeu.4	|- N = ( ( # ` H ) + ( M - 1 ) )

Assertion

Ref	Expression
fzisoeu	|- ( ph -> E! f f Isom < , < ( ( M ... N ) , H ) )

Distinct variable groups:   f, H   f, M   f, N

Allowed substitution hint:   ph( f)

Proof of Theorem fzisoeu

Dummy variables g h are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzssz 12343	. . . . . . . . 9 |- ( M ... N ) C_ ZZ
2		zssre 11384	. . . . . . . . 9 |- ZZ C_ RR
3	1, 2	sstri 3612	. . . . . . . 8 |- ( M ... N ) C_ RR
4		ltso 10118	. . . . . . . 8 |- < Or RR
5		soss 5053	. . . . . . . 8 |- ( ( M ... N ) C_ RR -> ( < Or RR -> < Or ( M ... N ) ) )
6	3, 4, 5	mp2 9	. . . . . . 7 |- < Or ( M ... N )
7		fzfi 12771	. . . . . . 7 |- ( M ... N ) e. Fin
8		fz1iso 13246	. . . . . . 7 |- ( ( < Or ( M ... N ) /\ ( M ... N ) e. Fin ) -> E. h h Isom < , < ( ( 1 ... ( # ` ( M ... N ) ) ) , ( M ... N ) ) )
9	6, 7, 8	mp2an 708	. . . . . 6 |- E. h h Isom < , < ( ( 1 ... ( # ` ( M ... N ) ) ) , ( M ... N ) )
10		fzisoeu.4	. . . . . . . . . . . . . . . 16 |- N = ( ( # ` H ) + ( M - 1 ) )
11		fveq2 6191	. . . . . . . . . . . . . . . . . 18 |- ( H = (/) -> ( # ` H ) = ( # ` (/) ) )
12		hash0 13158	. . . . . . . . . . . . . . . . . 18 |- ( # ` (/) ) = 0
13	11, 12	syl6eq 2672	. . . . . . . . . . . . . . . . 17 |- ( H = (/) -> ( # ` H ) = 0 )
14	13	oveq1d 6665	. . . . . . . . . . . . . . . 16 |- ( H = (/) -> ( ( # ` H ) + ( M - 1 ) ) = ( 0 + ( M - 1 ) ) )
15	10, 14	syl5eq 2668	. . . . . . . . . . . . . . 15 |- ( H = (/) -> N = ( 0 + ( M - 1 ) ) )
16	15	oveq2d 6666	. . . . . . . . . . . . . 14 |- ( H = (/) -> ( M ... N ) = ( M ... ( 0 + ( M - 1 ) ) ) )
17	16	adantl 482	. . . . . . . . . . . . 13 |- ( ( ph /\ H = (/) ) -> ( M ... N ) = ( M ... ( 0 + ( M - 1 ) ) ) )
18		fzisoeu.m	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> M e. ZZ )
19	18	zcnd 11483	. . . . . . . . . . . . . . . . . 18 |- ( ph -> M e. CC )
20		1cnd 10056	. . . . . . . . . . . . . . . . . 18 |- ( ph -> 1 e. CC )
21	19, 20	subcld 10392	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( M - 1 ) e. CC )
22	21	addid2d 10237	. . . . . . . . . . . . . . . 16 |- ( ph -> ( 0 + ( M - 1 ) ) = ( M - 1 ) )
23	22	oveq2d 6666	. . . . . . . . . . . . . . 15 |- ( ph -> ( M ... ( 0 + ( M - 1 ) ) ) = ( M ... ( M - 1 ) ) )
24	18	zred 11482	. . . . . . . . . . . . . . . . 17 |- ( ph -> M e. RR )
25	24	ltm1d 10956	. . . . . . . . . . . . . . . 16 |- ( ph -> ( M - 1 ) < M )
26		peano2zm 11420	. . . . . . . . . . . . . . . . . 18 |- ( M e. ZZ -> ( M - 1 ) e. ZZ )
27	18, 26	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( M - 1 ) e. ZZ )
28		fzn 12357	. . . . . . . . . . . . . . . . 17 |- ( ( M e. ZZ /\ ( M - 1 ) e. ZZ ) -> ( ( M - 1 ) < M <-> ( M ... ( M - 1 ) ) = (/) ) )
29	18, 27, 28	syl2anc 693	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( M - 1 ) < M <-> ( M ... ( M - 1 ) ) = (/) ) )
30	25, 29	mpbid 222	. . . . . . . . . . . . . . 15 |- ( ph -> ( M ... ( M - 1 ) ) = (/) )
31	23, 30	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ph -> ( M ... ( 0 + ( M - 1 ) ) ) = (/) )
32	31	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ H = (/) ) -> ( M ... ( 0 + ( M - 1 ) ) ) = (/) )
33		eqcom 2629	. . . . . . . . . . . . . . 15 |- ( H = (/) <-> (/) = H )
34	33	biimpi 206	. . . . . . . . . . . . . 14 |- ( H = (/) -> (/) = H )
35	34	adantl 482	. . . . . . . . . . . . 13 |- ( ( ph /\ H = (/) ) -> (/) = H )
36	17, 32, 35	3eqtrd 2660	. . . . . . . . . . . 12 |- ( ( ph /\ H = (/) ) -> ( M ... N ) = H )
37	36	fveq2d 6195	. . . . . . . . . . 11 |- ( ( ph /\ H = (/) ) -> ( # ` ( M ... N ) ) = ( # ` H ) )
38	20, 19	pncan3d 10395	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( 1 + ( M - 1 ) ) = M )
39	38	eqcomd 2628	. . . . . . . . . . . . . . . 16 |- ( ph -> M = ( 1 + ( M - 1 ) ) )
40	39	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ -. H = (/) ) -> M = ( 1 + ( M - 1 ) ) )
41		1red 10055	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ -. H = (/) ) -> 1 e. RR )
42		neqne 2802	. . . . . . . . . . . . . . . . . . . 20 |- ( -. H = (/) -> H =/= (/) )
43	42	adantl 482	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ -. H = (/) ) -> H =/= (/) )
44		fzisoeu.h	. . . . . . . . . . . . . . . . . . . . 21 |- ( ph -> H e. Fin )
45	44	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ph /\ -. H = (/) ) -> H e. Fin )
46		hashnncl 13157	. . . . . . . . . . . . . . . . . . . 20 |- ( H e. Fin -> ( ( # ` H ) e. NN <-> H =/= (/) ) )
47	45, 46	syl 17	. . . . . . . . . . . . . . . . . . 19 |- ( ( ph /\ -. H = (/) ) -> ( ( # ` H ) e. NN <-> H =/= (/) ) )
48	43, 47	mpbird 247	. . . . . . . . . . . . . . . . . 18 |- ( ( ph /\ -. H = (/) ) -> ( # ` H ) e. NN )
49	48	nnred 11035	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ -. H = (/) ) -> ( # ` H ) e. RR )
50	27	zred 11482	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( M - 1 ) e. RR )
51	50	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ -. H = (/) ) -> ( M - 1 ) e. RR )
52	48	nnge1d 11063	. . . . . . . . . . . . . . . . 17 |- ( ( ph /\ -. H = (/) ) -> 1 <_ ( # ` H ) )
53	41, 49, 51, 52	leadd1dd 10641	. . . . . . . . . . . . . . . 16 |- ( ( ph /\ -. H = (/) ) -> ( 1 + ( M - 1 ) ) <_ ( ( # ` H ) + ( M - 1 ) ) )
54	53, 10	syl6breqr 4695	. . . . . . . . . . . . . . 15 |- ( ( ph /\ -. H = (/) ) -> ( 1 + ( M - 1 ) ) <_ N )
55	40, 54	eqbrtrd 4675	. . . . . . . . . . . . . 14 |- ( ( ph /\ -. H = (/) ) -> M <_ N )
56	18	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ -. H = (/) ) -> M e. ZZ )
57		hashcl 13147	. . . . . . . . . . . . . . . . . . 19 |- ( H e. Fin -> ( # ` H ) e. NN0 )
58		nn0z 11400	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` H ) e. NN0 -> ( # ` H ) e. ZZ )
59	44, 57, 58	3syl 18	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( # ` H ) e. ZZ )
60	59, 27	zaddcld 11486	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( ( # ` H ) + ( M - 1 ) ) e. ZZ )
61	10, 60	syl5eqel 2705	. . . . . . . . . . . . . . . 16 |- ( ph -> N e. ZZ )
62	61	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ -. H = (/) ) -> N e. ZZ )
63		eluz 11701	. . . . . . . . . . . . . . 15 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( N e. ( ZZ>= ` M ) <-> M <_ N ) )
64	56, 62, 63	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ph /\ -. H = (/) ) -> ( N e. ( ZZ>= ` M ) <-> M <_ N ) )
65	55, 64	mpbird 247	. . . . . . . . . . . . 13 |- ( ( ph /\ -. H = (/) ) -> N e. ( ZZ>= ` M ) )
66		hashfz 13214	. . . . . . . . . . . . 13 |- ( N e. ( ZZ>= ` M ) -> ( # ` ( M ... N ) ) = ( ( N - M ) + 1 ) )
67	65, 66	syl 17	. . . . . . . . . . . 12 |- ( ( ph /\ -. H = (/) ) -> ( # ` ( M ... N ) ) = ( ( N - M ) + 1 ) )
68	10	oveq1i 6660	. . . . . . . . . . . . . . . 16 |- ( N - M ) = ( ( ( # ` H ) + ( M - 1 ) ) - M )
69	44, 57	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( # ` H ) e. NN0 )
70	69	nn0cnd 11353	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( # ` H ) e. CC )
71	70, 21, 19	addsubassd 10412	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( ( # ` H ) + ( M - 1 ) ) - M ) = ( ( # ` H ) + ( ( M - 1 ) - M ) ) )
72	68, 71	syl5eq 2668	. . . . . . . . . . . . . . 15 |- ( ph -> ( N - M ) = ( ( # ` H ) + ( ( M - 1 ) - M ) ) )
73	19, 20	negsubd 10398	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> ( M + -u 1 ) = ( M - 1 ) )
74	73	eqcomd 2628	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( M - 1 ) = ( M + -u 1 ) )
75	74	oveq1d 6665	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( ( M - 1 ) - M ) = ( ( M + -u 1 ) - M ) )
76	20	negcld 10379	. . . . . . . . . . . . . . . . . 18 |- ( ph -> -u 1 e. CC )
77	19, 76	pncan2d 10394	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( ( M + -u 1 ) - M ) = -u 1 )
78	75, 77	eqtrd 2656	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( M - 1 ) - M ) = -u 1 )
79	78	oveq2d 6666	. . . . . . . . . . . . . . 15 |- ( ph -> ( ( # ` H ) + ( ( M - 1 ) - M ) ) = ( ( # ` H ) + -u 1 ) )
80	72, 79	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ph -> ( N - M ) = ( ( # ` H ) + -u 1 ) )
81	80	oveq1d 6665	. . . . . . . . . . . . 13 |- ( ph -> ( ( N - M ) + 1 ) = ( ( ( # ` H ) + -u 1 ) + 1 ) )
82	81	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ -. H = (/) ) -> ( ( N - M ) + 1 ) = ( ( ( # ` H ) + -u 1 ) + 1 ) )
83	70, 20	negsubd 10398	. . . . . . . . . . . . . . 15 |- ( ph -> ( ( # ` H ) + -u 1 ) = ( ( # ` H ) - 1 ) )
84	83	oveq1d 6665	. . . . . . . . . . . . . 14 |- ( ph -> ( ( ( # ` H ) + -u 1 ) + 1 ) = ( ( ( # ` H ) - 1 ) + 1 ) )
85	70, 20	npcand 10396	. . . . . . . . . . . . . 14 |- ( ph -> ( ( ( # ` H ) - 1 ) + 1 ) = ( # ` H ) )
86	84, 85	eqtrd 2656	. . . . . . . . . . . . 13 |- ( ph -> ( ( ( # ` H ) + -u 1 ) + 1 ) = ( # ` H ) )
87	86	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ -. H = (/) ) -> ( ( ( # ` H ) + -u 1 ) + 1 ) = ( # ` H ) )
88	67, 82, 87	3eqtrd 2660	. . . . . . . . . . 11 |- ( ( ph /\ -. H = (/) ) -> ( # ` ( M ... N ) ) = ( # ` H ) )
89	37, 88	pm2.61dan 832	. . . . . . . . . 10 |- ( ph -> ( # ` ( M ... N ) ) = ( # ` H ) )
90	89	oveq2d 6666	. . . . . . . . 9 |- ( ph -> ( 1 ... ( # ` ( M ... N ) ) ) = ( 1 ... ( # ` H ) ) )
91		isoeq4 6570	. . . . . . . . 9 |- ( ( 1 ... ( # ` ( M ... N ) ) ) = ( 1 ... ( # ` H ) ) -> ( h Isom < , < ( ( 1 ... ( # ` ( M ... N ) ) ) , ( M ... N ) ) <-> h Isom < , < ( ( 1 ... ( # ` H ) ) , ( M ... N ) ) ) )
92	90, 91	syl 17	. . . . . . . 8 |- ( ph -> ( h Isom < , < ( ( 1 ... ( # ` ( M ... N ) ) ) , ( M ... N ) ) <-> h Isom < , < ( ( 1 ... ( # ` H ) ) , ( M ... N ) ) ) )
93	92	biimpd 219	. . . . . . 7 |- ( ph -> ( h Isom < , < ( ( 1 ... ( # ` ( M ... N ) ) ) , ( M ... N ) ) -> h Isom < , < ( ( 1 ... ( # ` H ) ) , ( M ... N ) ) ) )
94	93	eximdv 1846	. . . . . 6 |- ( ph -> ( E. h h Isom < , < ( ( 1 ... ( # ` ( M ... N ) ) ) , ( M ... N ) ) -> E. h h Isom < , < ( ( 1 ... ( # ` H ) ) , ( M ... N ) ) ) )
95	9, 94	mpi 20	. . . . 5 |- ( ph -> E. h h Isom < , < ( ( 1 ... ( # ` H ) ) , ( M ... N ) ) )
96		fzisoeu.or	. . . . . 6 |- ( ph -> < Or H )
97		fz1iso 13246	. . . . . 6 |- ( ( < Or H /\ H e. Fin ) -> E. g g Isom < , < ( ( 1 ... ( # ` H ) ) , H ) )
98	96, 44, 97	syl2anc 693	. . . . 5 |- ( ph -> E. g g Isom < , < ( ( 1 ... ( # ` H ) ) , H ) )
99		eeanv 2182	. . . . 5 |- ( E. h E. g ( h Isom < , < ( ( 1 ... ( # ` H ) ) , ( M ... N ) ) /\ g Isom < , < ( ( 1 ... ( # ` H ) ) , H ) ) <-> ( E. h h Isom < , < ( ( 1 ... ( # ` H ) ) , ( M ... N ) ) /\ E. g g Isom < , < ( ( 1 ... ( # ` H ) ) , H ) ) )
100	95, 98, 99	sylanbrc 698	. . . 4 |- ( ph -> E. h E. g ( h Isom < , < ( ( 1 ... ( # ` H ) ) , ( M ... N ) ) /\ g Isom < , < ( ( 1 ... ( # ` H ) ) , H ) ) )
101		isocnv 6580	. . . . . . . 8 |- ( h Isom < , < ( ( 1 ... ( # ` H ) ) , ( M ... N ) ) -> `' h Isom < , < ( ( M ... N ) , ( 1 ... ( # ` H ) ) ) )
102	101	ad2antrl 764	. . . . . . 7 |- ( ( ph /\ ( h Isom < , < ( ( 1 ... ( # ` H ) ) , ( M ... N ) ) /\ g Isom < , < ( ( 1 ... ( # ` H ) ) , H ) ) ) -> `' h Isom < , < ( ( M ... N ) , ( 1 ... ( # ` H ) ) ) )
103		simprr 796	. . . . . . 7 |- ( ( ph /\ ( h Isom < , < ( ( 1 ... ( # ` H ) ) , ( M ... N ) ) /\ g Isom < , < ( ( 1 ... ( # ` H ) ) , H ) ) ) -> g Isom < , < ( ( 1 ... ( # ` H ) ) , H ) )
104		isotr 6586	. . . . . . 7 |- ( ( `' h Isom < , < ( ( M ... N ) , ( 1 ... ( # ` H ) ) ) /\ g Isom < , < ( ( 1 ... ( # ` H ) ) , H ) ) -> ( g o. `' h ) Isom < , < ( ( M ... N ) , H ) )
105	102, 103, 104	syl2anc 693	. . . . . 6 |- ( ( ph /\ ( h Isom < , < ( ( 1 ... ( # ` H ) ) , ( M ... N ) ) /\ g Isom < , < ( ( 1 ... ( # ` H ) ) , H ) ) ) -> ( g o. `' h ) Isom < , < ( ( M ... N ) , H ) )
106	105	ex 450	. . . . 5 |- ( ph -> ( ( h Isom < , < ( ( 1 ... ( # ` H ) ) , ( M ... N ) ) /\ g Isom < , < ( ( 1 ... ( # ` H ) ) , H ) ) -> ( g o. `' h ) Isom < , < ( ( M ... N ) , H ) ) )
107	106	2eximdv 1848	. . . 4 |- ( ph -> ( E. h E. g ( h Isom < , < ( ( 1 ... ( # ` H ) ) , ( M ... N ) ) /\ g Isom < , < ( ( 1 ... ( # ` H ) ) , H ) ) -> E. h E. g ( g o. `' h ) Isom < , < ( ( M ... N ) , H ) ) )
108	100, 107	mpd 15	. . 3 |- ( ph -> E. h E. g ( g o. `' h ) Isom < , < ( ( M ... N ) , H ) )
109		vex 3203	. . . . . . 7 |- g e. _V
110		vex 3203	. . . . . . . 8 |- h e. _V
111	110	cnvex 7113	. . . . . . 7 |- `' h e. _V
112	109, 111	coex 7118	. . . . . 6 |- ( g o. `' h ) e. _V
113		isoeq1 6567	. . . . . 6 |- ( f = ( g o. `' h ) -> ( f Isom < , < ( ( M ... N ) , H ) <-> ( g o. `' h ) Isom < , < ( ( M ... N ) , H ) ) )
114	112, 113	spcev 3300	. . . . 5 |- ( ( g o. `' h ) Isom < , < ( ( M ... N ) , H ) -> E. f f Isom < , < ( ( M ... N ) , H ) )
115	114	a1i 11	. . . 4 |- ( ph -> ( ( g o. `' h ) Isom < , < ( ( M ... N ) , H ) -> E. f f Isom < , < ( ( M ... N ) , H ) ) )
116	115	exlimdvv 1862	. . 3 |- ( ph -> ( E. h E. g ( g o. `' h ) Isom < , < ( ( M ... N ) , H ) -> E. f f Isom < , < ( ( M ... N ) , H ) ) )
117	108, 116	mpd 15	. 2 |- ( ph -> E. f f Isom < , < ( ( M ... N ) , H ) )
118		ltwefz 12762	. . 3 |- < We ( M ... N )
119		wemoiso 7153	. . 3 |- ( < We ( M ... N ) -> E* f f Isom < , < ( ( M ... N ) , H ) )
120	118, 119	mp1i 13	. 2 |- ( ph -> E* f f Isom < , < ( ( M ... N ) , H ) )
121		eu5 2496	. 2 |- ( E! f f Isom < , < ( ( M ... N ) , H ) <-> ( E. f f Isom < , < ( ( M ... N ) , H ) /\ E* f f Isom < , < ( ( M ... N ) , H ) ) )
122	117, 120, 121	sylanbrc 698	1 |- ( ph -> E! f f Isom < , < ( ( M ... N ) , H ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   E!weu 2470   E*wmo 2471   =/= wne 2794   C_ wss 3574   (/)c0 3915   class class class wbr 4653   Or wor 5034   We wwe 5072   `'ccnv 5113   o. ccom 5118   ` cfv 5888   Isom wiso 5889  (class class class)co 6650   Fincfn 7955   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   -ucneg 10267   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687   ...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:  fourierdlem36  40360