Theorem plyexmo 24068

Description: An infinite set of values can be extended to a polynomial in at most one way. (Contributed by Stefan O'Rear, 14-Nov-2014.)

Assertion

Ref	Expression
plyexmo	|- ( ( D C_ CC /\ -. D e. Fin ) -> E* p ( p e. (Poly ` S ) /\ ( p |` D ) = F ) )

Distinct variable groups:   S, p   F, p   D, p

Proof of Theorem plyexmo

Dummy variables a b are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 792	. . . . . . . . 9 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> -. D e. Fin )
2		simpll 790	. . . . . . . . . . . . . 14 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> D C_ CC )
3	2	sseld 3602	. . . . . . . . . . . . 13 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> ( b e. D -> b e. CC ) )
4		simprll 802	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> p e. (Poly ` CC ) )
5		plyf 23954	. . . . . . . . . . . . . . . . . . 19 |- ( p e. (Poly ` CC ) -> p : CC --> CC )
6	4, 5	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> p : CC --> CC )
7		ffn 6045	. . . . . . . . . . . . . . . . . 18 |- ( p : CC --> CC -> p Fn CC )
8	6, 7	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> p Fn CC )
9	8	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) /\ b e. D ) -> p Fn CC )
10		simprrl 804	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> a e. (Poly ` CC ) )
11		plyf 23954	. . . . . . . . . . . . . . . . . . 19 |- ( a e. (Poly ` CC ) -> a : CC --> CC )
12	10, 11	syl 17	. . . . . . . . . . . . . . . . . 18 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> a : CC --> CC )
13		ffn 6045	. . . . . . . . . . . . . . . . . 18 |- ( a : CC --> CC -> a Fn CC )
14	12, 13	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> a Fn CC )
15	14	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) /\ b e. D ) -> a Fn CC )
16		cnex 10017	. . . . . . . . . . . . . . . . 17 |- CC e. _V
17	16	a1i 11	. . . . . . . . . . . . . . . 16 |- ( ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) /\ b e. D ) -> CC e. _V )
18	2	sselda 3603	. . . . . . . . . . . . . . . 16 |- ( ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) /\ b e. D ) -> b e. CC )
19		fnfvof 6911	. . . . . . . . . . . . . . . 16 |- ( ( ( p Fn CC /\ a Fn CC ) /\ ( CC e. _V /\ b e. CC ) ) -> ( ( p oF - a ) ` b ) = ( ( p ` b ) - ( a ` b ) ) )
20	9, 15, 17, 18, 19	syl22anc 1327	. . . . . . . . . . . . . . 15 |- ( ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) /\ b e. D ) -> ( ( p oF - a ) ` b ) = ( ( p ` b ) - ( a ` b ) ) )
21	6	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) /\ b e. D ) -> p : CC --> CC )
22	21, 18	ffvelrnd 6360	. . . . . . . . . . . . . . . 16 |- ( ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) /\ b e. D ) -> ( p ` b ) e. CC )
23		simprlr 803	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> ( p |` D ) = F )
24		simprrr 805	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> ( a |` D ) = F )
25	23, 24	eqtr4d 2659	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> ( p |` D ) = ( a |` D ) )
26	25	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) /\ b e. D ) -> ( p |` D ) = ( a |` D ) )
27	26	fveq1d 6193	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) /\ b e. D ) -> ( ( p |` D ) ` b ) = ( ( a |` D ) ` b ) )
28		fvres 6207	. . . . . . . . . . . . . . . . . 18 |- ( b e. D -> ( ( p |` D ) ` b ) = ( p ` b ) )
29	28	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) /\ b e. D ) -> ( ( p |` D ) ` b ) = ( p ` b ) )
30		fvres 6207	. . . . . . . . . . . . . . . . . 18 |- ( b e. D -> ( ( a |` D ) ` b ) = ( a ` b ) )
31	30	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) /\ b e. D ) -> ( ( a |` D ) ` b ) = ( a ` b ) )
32	27, 29, 31	3eqtr3d 2664	. . . . . . . . . . . . . . . 16 |- ( ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) /\ b e. D ) -> ( p ` b ) = ( a ` b ) )
33	22, 32	subeq0bd 10456	. . . . . . . . . . . . . . 15 |- ( ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) /\ b e. D ) -> ( ( p ` b ) - ( a ` b ) ) = 0 )
34	20, 33	eqtrd 2656	. . . . . . . . . . . . . 14 |- ( ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) /\ b e. D ) -> ( ( p oF - a ) ` b ) = 0 )
35	34	ex 450	. . . . . . . . . . . . 13 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> ( b e. D -> ( ( p oF - a ) ` b ) = 0 ) )
36	3, 35	jcad 555	. . . . . . . . . . . 12 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> ( b e. D -> ( b e. CC /\ ( ( p oF - a ) ` b ) = 0 ) ) )
37		plysubcl 23978	. . . . . . . . . . . . . 14 |- ( ( p e. (Poly ` CC ) /\ a e. (Poly ` CC ) ) -> ( p oF - a ) e. (Poly ` CC ) )
38	4, 10, 37	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> ( p oF - a ) e. (Poly ` CC ) )
39		plyf 23954	. . . . . . . . . . . . 13 |- ( ( p oF - a ) e. (Poly ` CC ) -> ( p oF - a ) : CC --> CC )
40		ffn 6045	. . . . . . . . . . . . 13 |- ( ( p oF - a ) : CC --> CC -> ( p oF - a ) Fn CC )
41		fniniseg 6338	. . . . . . . . . . . . 13 |- ( ( p oF - a ) Fn CC -> ( b e. ( `' ( p oF - a ) " { 0 } ) <-> ( b e. CC /\ ( ( p oF - a ) ` b ) = 0 ) ) )
42	38, 39, 40, 41	4syl 19	. . . . . . . . . . . 12 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> ( b e. ( `' ( p oF - a ) " { 0 } ) <-> ( b e. CC /\ ( ( p oF - a ) ` b ) = 0 ) ) )
43	36, 42	sylibrd 249	. . . . . . . . . . 11 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> ( b e. D -> b e. ( `' ( p oF - a ) " { 0 } ) ) )
44	43	ssrdv 3609	. . . . . . . . . 10 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> D C_ ( `' ( p oF - a ) " { 0 } ) )
45		ssfi 8180	. . . . . . . . . . 11 |- ( ( ( `' ( p oF - a ) " { 0 } ) e. Fin /\ D C_ ( `' ( p oF - a ) " { 0 } ) ) -> D e. Fin )
46	45	expcom 451	. . . . . . . . . 10 |- ( D C_ ( `' ( p oF - a ) " { 0 } ) -> ( ( `' ( p oF - a ) " { 0 } ) e. Fin -> D e. Fin ) )
47	44, 46	syl 17	. . . . . . . . 9 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> ( ( `' ( p oF - a ) " { 0 } ) e. Fin -> D e. Fin ) )
48	1, 47	mtod 189	. . . . . . . 8 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> -. ( `' ( p oF - a ) " { 0 } ) e. Fin )
49		df-ne 2795	. . . . . . . . . . . 12 |- ( ( p oF - a ) =/= 0p <-> -. ( p oF - a ) = 0p )
50	49	biimpri 218	. . . . . . . . . . 11 |- ( -. ( p oF - a ) = 0p -> ( p oF - a ) =/= 0p )
51		eqid 2622	. . . . . . . . . . . 12 |- ( `' ( p oF - a ) " { 0 } ) = ( `' ( p oF - a ) " { 0 } )
52	51	fta1 24063	. . . . . . . . . . 11 |- ( ( ( p oF - a ) e. (Poly ` CC ) /\ ( p oF - a ) =/= 0p ) -> ( ( `' ( p oF - a ) " { 0 } ) e. Fin /\ ( # ` ( `' ( p oF - a ) " { 0 } ) ) <_ (deg ` ( p oF - a ) ) ) )
53	38, 50, 52	syl2an 494	. . . . . . . . . 10 |- ( ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) /\ -. ( p oF - a ) = 0p ) -> ( ( `' ( p oF - a ) " { 0 } ) e. Fin /\ ( # ` ( `' ( p oF - a ) " { 0 } ) ) <_ (deg ` ( p oF - a ) ) ) )
54	53	simpld 475	. . . . . . . . 9 |- ( ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) /\ -. ( p oF - a ) = 0p ) -> ( `' ( p oF - a ) " { 0 } ) e. Fin )
55	54	ex 450	. . . . . . . 8 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> ( -. ( p oF - a ) = 0p -> ( `' ( p oF - a ) " { 0 } ) e. Fin ) )
56	48, 55	mt3d 140	. . . . . . 7 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> ( p oF - a ) = 0p )
57		df-0p 23437	. . . . . . 7 |- 0p = ( CC X. { 0 } )
58	56, 57	syl6eq 2672	. . . . . 6 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> ( p oF - a ) = ( CC X. { 0 } ) )
59	16	a1i 11	. . . . . . 7 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> CC e. _V )
60		ofsubeq0 11017	. . . . . . 7 |- ( ( CC e. _V /\ p : CC --> CC /\ a : CC --> CC ) -> ( ( p oF - a ) = ( CC X. { 0 } ) <-> p = a ) )
61	59, 6, 12, 60	syl3anc 1326	. . . . . 6 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> ( ( p oF - a ) = ( CC X. { 0 } ) <-> p = a ) )
62	58, 61	mpbid 222	. . . . 5 |- ( ( ( D C_ CC /\ -. D e. Fin ) /\ ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) ) -> p = a )
63	62	ex 450	. . . 4 |- ( ( D C_ CC /\ -. D e. Fin ) -> ( ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) -> p = a ) )
64	63	alrimivv 1856	. . 3 |- ( ( D C_ CC /\ -. D e. Fin ) -> A. p A. a ( ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) -> p = a ) )
65		eleq1 2689	. . . . 5 |- ( p = a -> ( p e. (Poly ` CC ) <-> a e. (Poly ` CC ) ) )
66		reseq1 5390	. . . . . 6 |- ( p = a -> ( p |` D ) = ( a |` D ) )
67	66	eqeq1d 2624	. . . . 5 |- ( p = a -> ( ( p |` D ) = F <-> ( a |` D ) = F ) )
68	65, 67	anbi12d 747	. . . 4 |- ( p = a -> ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) <-> ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) )
69	68	mo4 2517	. . 3 |- ( E* p ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) <-> A. p A. a ( ( ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) /\ ( a e. (Poly ` CC ) /\ ( a |` D ) = F ) ) -> p = a ) )
70	64, 69	sylibr 224	. 2 |- ( ( D C_ CC /\ -. D e. Fin ) -> E* p ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) )
71		plyssc 23956	. . . . 5 |- (Poly ` S ) C_ (Poly ` CC )
72	71	sseli 3599	. . . 4 |- ( p e. (Poly ` S ) -> p e. (Poly ` CC ) )
73	72	anim1i 592	. . 3 |- ( ( p e. (Poly ` S ) /\ ( p |` D ) = F ) -> ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) )
74	73	moimi 2520	. 2 |- ( E* p ( p e. (Poly ` CC ) /\ ( p |` D ) = F ) -> E* p ( p e. (Poly ` S ) /\ ( p |` D ) = F ) )
75	70, 74	syl 17	1 |- ( ( D C_ CC /\ -. D e. Fin ) -> E* p ( p e. (Poly ` S ) /\ ( p |` D ) = F ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   E*wmo 2471   =/= wne 2794   _Vcvv 3200   C_ wss 3574   {csn 4177   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   |` cres 5116   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   Fincfn 7955   CCcc 9934   0cc0 9936   <_ cle 10075   - cmin 10266   #chash 13117   0pc0p 23436  Polycply 23940  degcdgr 23943

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  ax-pre-sup 10014  ax-addf 10015

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-rlim 14220  df-sum 14417  df-0p 23437  df-ply 23944  df-idp 23945  df-coe 23946  df-dgr 23947  df-quot 24046

This theorem is referenced by: (None)