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Theorem aannenlem2 22452

Description: Lemma for aannen 22454. (Contributed by Stefan O'Rear, 16-Nov-2014.)

**Hypothesis**
Ref	Expression
aannenlem.a	Poly $/\$ deg $/\$ coeff

**Assertion**
Ref	Expression
aannenlem2

Distinct variable group:

Allowed substitution hints:

(

)

Proof of Theorem aannenlem2 Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 993	. . . . . . . . . 10 Poly $\$ $/\$ $/\$
2		eldifi 3619	. . . . . . . . . . . . . 14 Poly $\$ Poly
3	2	adantr 465	. . . . . . . . . . . . 13 Poly $\$ $/\$ Poly
4	3	3adant2 1010	. . . . . . . . . . . 12 Poly $\$ $/\$ $/\$ Poly
5		eldifsni 4146	. . . . . . . . . . . . . . 15 Poly $\$
6	5	adantr 465	. . . . . . . . . . . . . 14 Poly $\$ $/\$
7		0nn0 10799	. . . . . . . . . . . . . . . . . 18
8		dgrcl 22358	. . . . . . . . . . . . . . . . . . 19 Poly deg
9	3, 8	syl 16	. . . . . . . . . . . . . . . . . 18 Poly $\$ $/\$ deg
10		prssi 4176	. . . . . . . . . . . . . . . . . 18 $/\$ deg deg
11	7, 9, 10	sylancr 663	. . . . . . . . . . . . . . . . 17 Poly $\$ $/\$ deg
12		ssrab2 3578	. . . . . . . . . . . . . . . . . 18 deg coeff
13	12	a1i 11	. . . . . . . . . . . . . . . . 17 Poly $\$ $/\$ deg coeff
14	11, 13	unssd 3673	. . . . . . . . . . . . . . . 16 Poly $\$ $/\$ deg deg coeff
15		nn0ssre 10788	. . . . . . . . . . . . . . . . 17
16		ressxr 9626	. . . . . . . . . . . . . . . . 17
17	15, 16	sstri 3506	. . . . . . . . . . . . . . . 16
18	14, 17	syl6ss 3509	. . . . . . . . . . . . . . 15 Poly $\$ $/\$ deg deg coeff
19		fvex 5867	. . . . . . . . . . . . . . . . 17 deg
20	19	prid2 4129	. . . . . . . . . . . . . . . 16 deg deg
21		elun1 3664	. . . . . . . . . . . . . . . 16 deg deg deg deg deg coeff
22	20, 21	ax-mp 5	. . . . . . . . . . . . . . 15 deg deg deg coeff
23		supxrub 11505	. . . . . . . . . . . . . . 15 deg deg coeff $/\$ deg deg deg coeff deg deg deg coeff
24	18, 22, 23	sylancl 662	. . . . . . . . . . . . . 14 Poly $\$ $/\$ deg deg deg coeff
25	18	adantr 465	. . . . . . . . . . . . . . . 16 Poly $\$ $/\$ $/\$ deg deg coeff
26		fveq2 5857	. . . . . . . . . . . . . . . . . . . 20 coeff coeff
27		abs0 13068	. . . . . . . . . . . . . . . . . . . 20
28	26, 27	syl6eq 2517	. . . . . . . . . . . . . . . . . . 19 coeff coeff
29		c0ex 9579	. . . . . . . . . . . . . . . . . . . . 21
30	29	prid1 4128	. . . . . . . . . . . . . . . . . . . 20 deg
31		elun1 3664	. . . . . . . . . . . . . . . . . . . 20 deg deg deg coeff
32	30, 31	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 deg deg coeff
33	28, 32	syl6eqel 2556	. . . . . . . . . . . . . . . . . 18 coeff coeff deg deg coeff
34	33	adantl 466	. . . . . . . . . . . . . . . . 17 Poly $\$ $/\$ $/\$ $/\$ coeff coeff deg deg coeff
35		0z 10864	. . . . . . . . . . . . . . . . . . . . . . 23
36		eqid 2460	. . . . . . . . . . . . . . . . . . . . . . . 24 coeff coeff
37	36	coef2 22356	. . . . . . . . . . . . . . . . . . . . . . 23 Poly $/\$ coeff
38	3, 35, 37	sylancl 662	. . . . . . . . . . . . . . . . . . . . . 22 Poly $\$ $/\$ coeff
39	38	ffvelrnda 6012	. . . . . . . . . . . . . . . . . . . . 21 Poly $\$ $/\$ $/\$ coeff
40		nn0abscl 13095	. . . . . . . . . . . . . . . . . . . . 21 coeff coeff
41	39, 40	syl 16	. . . . . . . . . . . . . . . . . . . 20 Poly $\$ $/\$ $/\$ coeff
42	41	adantr 465	. . . . . . . . . . . . . . . . . . 19 Poly $\$ $/\$ $/\$ $/\$ coeff coeff
43		simplr 754	. . . . . . . . . . . . . . . . . . . . 21 Poly $\$ $/\$ $/\$ $/\$ coeff
44	9	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . 21 Poly $\$ $/\$ $/\$ $/\$ coeff deg
45	3	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . 22 Poly $\$ $/\$ $/\$ $/\$ coeff Poly
46		simpr 461	. . . . . . . . . . . . . . . . . . . . . 22 Poly $\$ $/\$ $/\$ $/\$ coeff coeff
47		eqid 2460	. . . . . . . . . . . . . . . . . . . . . . 23 deg deg
48	36, 47	dgrub 22359	. . . . . . . . . . . . . . . . . . . . . 22 Poly $/\$ $/\$ coeff deg
49	45, 43, 46, 48	syl3anc 1223	. . . . . . . . . . . . . . . . . . . . 21 Poly $\$ $/\$ $/\$ $/\$ coeff deg
50		elfz2nn0 11757	. . . . . . . . . . . . . . . . . . . . 21 deg $/\$ deg $/\$ deg
51	43, 44, 49, 50	syl3anbrc 1175	. . . . . . . . . . . . . . . . . . . 20 Poly $\$ $/\$ $/\$ $/\$ coeff deg
52		eqid 2460	. . . . . . . . . . . . . . . . . . . 20 coeff coeff
53		fveq2 5857	. . . . . . . . . . . . . . . . . . . . . . 23 coeff coeff
54	53	fveq2d 5861	. . . . . . . . . . . . . . . . . . . . . 22 coeff coeff
55	54	eqeq2d 2474	. . . . . . . . . . . . . . . . . . . . 21 coeff coeff coeff coeff
56	55	rspcev 3207	. . . . . . . . . . . . . . . . . . . 20 deg $/\$ coeff coeff degcoeff coeff
57	51, 52, 56	sylancl 662	. . . . . . . . . . . . . . . . . . 19 Poly $\$ $/\$ $/\$ $/\$ coeff degcoeff coeff
58		eqeq1 2464	. . . . . . . . . . . . . . . . . . . . 21 coeff coeff coeff coeff
59	58	rexbidv 2966	. . . . . . . . . . . . . . . . . . . 20 coeff deg coeff degcoeff coeff
60	59	elrab 3254	. . . . . . . . . . . . . . . . . . 19 coeff deg coeff coeff $/\$ degcoeff coeff
61	42, 57, 60	sylanbrc 664	. . . . . . . . . . . . . . . . . 18 Poly $\$ $/\$ $/\$ $/\$ coeff coeff deg coeff
62		elun2 3665	. . . . . . . . . . . . . . . . . 18 coeff deg coeff coeff deg deg coeff
63	61, 62	syl 16	. . . . . . . . . . . . . . . . 17 Poly $\$ $/\$ $/\$ $/\$ coeff coeff deg deg coeff
64	34, 63	pm2.61dane 2778	. . . . . . . . . . . . . . . 16 Poly $\$ $/\$ $/\$ coeff deg deg coeff
65		supxrub 11505	. . . . . . . . . . . . . . . 16 deg deg coeff $/\$ coeff deg deg coeff coeff deg deg coeff
66	25, 64, 65	syl2anc 661	. . . . . . . . . . . . . . 15 Poly $\$ $/\$ $/\$ coeff deg deg coeff
67	66	ralrimiva 2871	. . . . . . . . . . . . . 14 Poly $\$ $/\$ coeff deg deg coeff
68	6, 24, 67	3jca 1171	. . . . . . . . . . . . 13 Poly $\$ $/\$ $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
69	68	3adant2 1010	. . . . . . . . . . . 12 Poly $\$ $/\$ $/\$ $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
70		neeq1 2741	. . . . . . . . . . . . . 14
71		fveq2 5857	. . . . . . . . . . . . . . 15 deg deg
72	71	breq1d 4450	. . . . . . . . . . . . . 14 deg deg deg coeff deg deg deg coeff
73		fveq2 5857	. . . . . . . . . . . . . . . . . 18 coeff coeff
74	73	fveq1d 5859	. . . . . . . . . . . . . . . . 17 coeff coeff
75	74	fveq2d 5861	. . . . . . . . . . . . . . . 16 coeff coeff
76	75	breq1d 4450	. . . . . . . . . . . . . . 15 coeff deg deg coeff coeff deg deg coeff
77	76	ralbidv 2896	. . . . . . . . . . . . . 14 coeff deg deg coeff coeff deg deg coeff
78	70, 72, 77	3anbi123d 1294	. . . . . . . . . . . . 13 $/\$ deg deg deg coeff $/\$ coeff deg deg coeff $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
79	78	elrab 3254	. . . . . . . . . . . 12 Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff Poly $/\$ $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
80	4, 69, 79	sylanbrc 664	. . . . . . . . . . 11 Poly $\$ $/\$ $/\$ Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
81		simp2 992	. . . . . . . . . . 11 Poly $\$ $/\$ $/\$
82		fveq1 5856	. . . . . . . . . . . . 13
83	82	eqeq1d 2462	. . . . . . . . . . . 12
84	83	rspcev 3207	. . . . . . . . . . 11 Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff $/\$ Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
85	80, 81, 84	syl2anc 661	. . . . . . . . . 10 Poly $\$ $/\$ $/\$ Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
86		fveq2 5857	. . . . . . . . . . . . 13
87	86	eqeq1d 2462	. . . . . . . . . . . 12
88	87	rexbidv 2966	. . . . . . . . . . 11 Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
89	88	elrab 3254	. . . . . . . . . 10 Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff $/\$ Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
90	1, 85, 89	sylanbrc 664	. . . . . . . . 9 Poly $\$ $/\$ $/\$ Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
91		prfi 7784	. . . . . . . . . . . . . . 15 deg
92		fzfi 12038	. . . . . . . . . . . . . . . . 17 deg
93		abrexfi 7809	. . . . . . . . . . . . . . . . 17 deg deg coeff
94	92, 93	ax-mp 5	. . . . . . . . . . . . . . . 16 deg coeff
95		rabssab 3580	. . . . . . . . . . . . . . . 16 deg coeff deg coeff
96		ssfi 7730	. . . . . . . . . . . . . . . 16 deg coeff $/\$ deg coeff deg coeff deg coeff
97	94, 95, 96	mp2an 672	. . . . . . . . . . . . . . 15 deg coeff
98		unfi 7776	. . . . . . . . . . . . . . 15 deg $/\$ deg coeff deg deg coeff
99	91, 97, 98	mp2an 672	. . . . . . . . . . . . . 14 deg deg coeff
100	99	a1i 11	. . . . . . . . . . . . 13 Poly $\$ $/\$ deg deg coeff
101		ne0i 3784	. . . . . . . . . . . . . . 15 deg deg deg coeff deg deg coeff
102	22, 101	ax-mp 5	. . . . . . . . . . . . . 14 deg deg coeff
103	102	a1i 11	. . . . . . . . . . . . 13 Poly $\$ $/\$ deg deg coeff
104		xrltso 11336	. . . . . . . . . . . . . 14
105		fisupcl 7916	. . . . . . . . . . . . . 14 $/\$ deg deg coeff $/\$ deg deg coeff $/\$ deg deg coeff deg deg coeff deg deg coeff
106	104, 105	mpan 670	. . . . . . . . . . . . 13 deg deg coeff $/\$ deg deg coeff $/\$ deg deg coeff deg deg coeff deg deg coeff
107	100, 103, 18, 106	syl3anc 1223	. . . . . . . . . . . 12 Poly $\$ $/\$ deg deg coeff deg deg coeff
108	14, 107	sseldd 3498	. . . . . . . . . . 11 Poly $\$ $/\$ deg deg coeff
109	108	3adant2 1010	. . . . . . . . . 10 Poly $\$ $/\$ $/\$ deg deg coeff
110		eqidd 2461	. . . . . . . . . 10 Poly $\$ $/\$ $/\$ Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
111		breq2 4444	. . . . . . . . . . . . . . . 16 deg deg coeff deg deg deg deg coeff
112		breq2 4444	. . . . . . . . . . . . . . . . 17 deg deg coeff coeff coeff deg deg coeff
113	112	ralbidv 2896	. . . . . . . . . . . . . . . 16 deg deg coeff coeff coeff deg deg coeff
114	111, 113	3anbi23d 1297	. . . . . . . . . . . . . . 15 deg deg coeff $/\$ deg $/\$ coeff $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
115	114	rabbidv 3098	. . . . . . . . . . . . . 14 deg deg coeff Poly $/\$ deg $/\$ coeff Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
116	115	rexeqdv 3058	. . . . . . . . . . . . 13 deg deg coeff Poly $/\$ deg $/\$ coeff Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
117	116	rabbidv 3098	. . . . . . . . . . . 12 deg deg coeff Poly $/\$ deg $/\$ coeff Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
118	117	eqeq2d 2474	. . . . . . . . . . 11 deg deg coeff Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff Poly $/\$ deg $/\$ coeff Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
119	118	rspcev 3207	. . . . . . . . . 10 deg deg coeff $/\$ Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff Poly $/\$ deg $/\$ coeff
120	109, 110, 119	syl2anc 661	. . . . . . . . 9 Poly $\$ $/\$ $/\$ Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff Poly $/\$ deg $/\$ coeff
121		cnex 9562	. . . . . . . . . . 11
122	121	rabex 4591	. . . . . . . . . 10 Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
123		eleq2 2533	. . . . . . . . . . 11 Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
124		eqeq1 2464	. . . . . . . . . . . 12 Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff Poly $/\$ deg $/\$ coeff Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff Poly $/\$ deg $/\$ coeff
125	124	rexbidv 2966	. . . . . . . . . . 11 Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff Poly $/\$ deg $/\$ coeff Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff Poly $/\$ deg $/\$ coeff
126	123, 125	anbi12d 710	. . . . . . . . . 10 Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff $/\$ Poly $/\$ deg $/\$ coeff Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff $/\$ Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff Poly $/\$ deg $/\$ coeff
127	122, 126	spcev 3198	. . . . . . . . 9 Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff $/\$ Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff Poly $/\$ deg $/\$ coeff $/\$ Poly $/\$ deg $/\$ coeff
128	90, 120, 127	syl2anc 661	. . . . . . . 8 Poly $\$ $/\$ $/\$ $/\$ Poly $/\$ deg $/\$ coeff
129	128	3exp 1190	. . . . . . 7 Poly $\$ $/\$ Poly $/\$ deg $/\$ coeff
130	129	rexlimiv 2942	. . . . . 6 Poly $\$ $/\$ Poly $/\$ deg $/\$ coeff
131	130	impcom 430	. . . . 5 $/\$ Poly $\$ $/\$ Poly $/\$ deg $/\$ coeff
132		eleq2 2533	. . . . . . . . 9 Poly $/\$ deg $/\$ coeff Poly $/\$ deg $/\$ coeff
133	87	rexbidv 2966	. . . . . . . . . . 11 Poly $/\$ deg $/\$ coeff Poly $/\$ deg $/\$ coeff
134	133	elrab 3254	. . . . . . . . . 10 Poly $/\$ deg $/\$ coeff $/\$ Poly $/\$ deg $/\$ coeff
135		simp1 991	. . . . . . . . . . . . . . 15 $/\$ deg $/\$ coeff
136	135	anim2i 569	. . . . . . . . . . . . . 14 Poly $/\$ $/\$ deg $/\$ coeff Poly $/\$
137	71	breq1d 4450	. . . . . . . . . . . . . . . 16 deg deg
138	75	breq1d 4450	. . . . . . . . . . . . . . . . 17 coeff coeff
139	138	ralbidv 2896	. . . . . . . . . . . . . . . 16 coeff coeff
140	70, 137, 139	3anbi123d 1294	. . . . . . . . . . . . . . 15 $/\$ deg $/\$ coeff $/\$ deg $/\$ coeff
141	140	elrab 3254	. . . . . . . . . . . . . 14 Poly $/\$ deg $/\$ coeff Poly $/\$ $/\$ deg $/\$ coeff
142		eldifsn 4145	. . . . . . . . . . . . . 14 Poly $\$ Poly $/\$
143	136, 141, 142	3imtr4i 266	. . . . . . . . . . . . 13 Poly $/\$ deg $/\$ coeff Poly $\$
144	143	ssriv 3501	. . . . . . . . . . . 12 Poly $/\$ deg $/\$ coeff Poly $\$
145		ssrexv 3558	. . . . . . . . . . . . 13 Poly $/\$ deg $/\$ coeff Poly $\$ Poly $/\$ deg $/\$ coeff Poly $\$
146	83	cbvrexv 3082	. . . . . . . . . . . . 13 Poly $\$ Poly $\$
147	145, 146	syl6ib 226	. . . . . . . . . . . 12 Poly $/\$ deg $/\$ coeff Poly $\$ Poly $/\$ deg $/\$ coeff Poly $\$
148	144, 147	ax-mp 5	. . . . . . . . . . 11 Poly $/\$ deg $/\$ coeff Poly $\$
149	148	anim2i 569	. . . . . . . . . 10 $/\$ Poly $/\$ deg $/\$ coeff $/\$ Poly $\$
150	134, 149	sylbi 195	. . . . . . . . 9 Poly $/\$ deg $/\$ coeff $/\$ Poly $\$
151	132, 150	syl6bi 228	. . . . . . . 8 Poly $/\$ deg $/\$ coeff $/\$ Poly $\$
152	151	rexlimivw 2945	. . . . . . 7 Poly $/\$ deg $/\$ coeff $/\$ Poly $\$
153	152	impcom 430	. . . . . 6 $/\$ Poly $/\$ deg $/\$ coeff $/\$ Poly $\$
154	153	exlimiv 1693	. . . . 5 $/\$ Poly $/\$ deg $/\$ coeff $/\$ Poly $\$
155	131, 154	impbii 188	. . . 4 $/\$ Poly $\$ $/\$ Poly $/\$ deg $/\$ coeff
156		elaa 22439	. . . 4 $/\$ Poly $\$
157		eluniab 4249	. . . 4 Poly $/\$ deg $/\$ coeff $/\$ Poly $/\$ deg $/\$ coeff
158	155, 156, 157	3bitr4i 277	. . 3 Poly $/\$ deg $/\$ coeff
159	158	eqriv 2456	. 2 Poly $/\$ deg $/\$ coeff
160		aannenlem.a	. . . 4 Poly $/\$ deg $/\$ coeff
161	160	rnmpt 5239	. . 3 Poly $/\$ deg $/\$ coeff
162	161	unieqi 4247	. 2 Poly $/\$ deg $/\$ coeff
163	159, 162	eqtr4i 2492	1

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 369 $/\$ w3a 968

wceq 1374

wex 1591

wcel 1762

cab 2445

wne 2655

wral 2807

wrex 2808

crab 2811 $\$ cdif 3466

cun 3467

wss 3469

c0 3778

csn 4020

cpr 4022

cuni 4238 class class class wbr 4440

cmpt 4498

wor 4792

crn 4993

wf 5575

cfv 5579 (class class class)co 6275

cfn 7506

csup 7889

cc 9479

cr 9480

cc0 9481

cxr 9616

clt 9617

cle 9618

cn0 10784

cz 10853

cfz 11661

cabs 13017

c0p 21804 Polycply 22309 coeffccoe 22311 degcdgr 22312

caa 22437

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1596 ax-4 1607 ax-5 1675 ax-6 1714 ax-7 1734 ax-8 1764 ax-9 1766 ax-10 1781 ax-11 1786 ax-12 1798 ax-13 1961 ax-ext 2438 ax-rep 4551 ax-sep 4561 ax-nul 4569 ax-pow 4618 ax-pr 4679 ax-un 6567 ax-inf2 8047 ax-cnex 9537 ax-resscn 9538 ax-1cn 9539 ax-icn 9540 ax-addcl 9541 ax-addrcl 9542 ax-mulcl 9543 ax-mulrcl 9544 ax-mulcom 9545 ax-addass 9546 ax-mulass 9547 ax-distr 9548 ax-i2m1 9549 ax-1ne0 9550 ax-1rid 9551 ax-rnegex 9552 ax-rrecex 9553 ax-cnre 9554 ax-pre-lttri 9555 ax-pre-lttrn 9556 ax-pre-ltadd 9557 ax-pre-mulgt0 9558 ax-pre-sup 9559 ax-addf 9560

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 969 df-3an 970 df-tru 1377 df-fal 1380 df-ex 1592 df-nf 1595 df-sb 1707 df-eu 2272 df-mo 2273 df-clab 2446 df-cleq 2452 df-clel 2455 df-nfc 2610 df-ne 2657 df-nel 2658 df-ral 2812 df-rex 2813 df-reu 2814 df-rmo 2815 df-rab 2816 df-v 3108 df-sbc 3325 df-csb 3429 df-dif 3472 df-un 3474 df-in 3476 df-ss 3483 df-pss 3485 df-nul 3779 df-if 3933 df-pw 4005 df-sn 4021 df-pr 4023 df-tp 4025 df-op 4027 df-uni 4239 df-int 4276 df-iun 4320 df-br 4441 df-opab 4499 df-mpt 4500 df-tr 4534 df-eprel 4784 df-id 4788 df-po 4793 df-so 4794 df-fr 4831 df-se 4832 df-we 4833 df-ord 4874 df-on 4875 df-lim 4876 df-suc 4877 df-xp 4998 df-rel 4999 df-cnv 5000 df-co 5001 df-dm 5002 df-rn 5003 df-res 5004 df-ima 5005 df-iota 5542 df-fun 5581 df-fn 5582 df-f 5583 df-f1 5584 df-fo 5585 df-f1o 5586 df-fv 5587 df-isom 5588 df-riota 6236 df-ov 6278 df-oprab 6279 df-mpt2 6280 df-of 6515 df-om 6672 df-1st 6774 df-2nd 6775 df-recs 7032 df-rdg 7066 df-1o 7120 df-oadd 7124 df-er 7301 df-map 7412 df-pm 7413 df-en 7507 df-dom 7508 df-sdom 7509 df-fin 7510 df-sup 7890 df-oi 7924 df-card 8309 df-pnf 9619 df-mnf 9620 df-xr 9621 df-ltxr 9622 df-le 9623 df-sub 9796 df-neg 9797 df-div 10196 df-nn 10526 df-2 10583 df-3 10584 df-n0 10785 df-z 10854 df-uz 11072 df-rp 11210 df-fz 11662 df-fzo 11782 df-fl 11886 df-seq 12064 df-exp 12123 df-hash 12361 df-cj 12882 df-re 12883 df-im 12884 df-sqr 13018 df-abs 13019 df-clim 13260 df-rlim 13261 df-sum 13458 df-0p 21805 df-ply 22313 df-coe 22315 df-dgr 22316 df-aa 22438

This theorem is referenced by: aannenlem3 22453

W3C validator