aannenlem2 - Metamath Proof Explorer

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Theorem aannenlem2 24084

Description: Lemma for aannen 24086. (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 1063	. . . . . . . . . 10 Poly $\$ $/\$ $/\$
2		eldifi 3732	. . . . . . . . . . . . . 14 Poly $\$ Poly
3	2	adantr 481	. . . . . . . . . . . . 13 Poly $\$ $/\$ Poly
4	3	3adant2 1080	. . . . . . . . . . . 12 Poly $\$ $/\$ $/\$ Poly
5		eldifsni 4320	. . . . . . . . . . . . . . 15 Poly $\$
6	5	adantr 481	. . . . . . . . . . . . . 14 Poly $\$ $/\$
7		0nn0 11307	. . . . . . . . . . . . . . . . . 18
8		dgrcl 23989	. . . . . . . . . . . . . . . . . . 19 Poly deg
9	3, 8	syl 17	. . . . . . . . . . . . . . . . . 18 Poly $\$ $/\$ deg
10		prssi 4353	. . . . . . . . . . . . . . . . . 18 $/\$ deg deg
11	7, 9, 10	sylancr 695	. . . . . . . . . . . . . . . . 17 Poly $\$ $/\$ deg
12		ssrab2 3687	. . . . . . . . . . . . . . . . . 18 deg coeff
13	12	a1i 11	. . . . . . . . . . . . . . . . 17 Poly $\$ $/\$ deg coeff
14	11, 13	unssd 3789	. . . . . . . . . . . . . . . 16 Poly $\$ $/\$ deg deg coeff
15		nn0ssre 11296	. . . . . . . . . . . . . . . . 17
16		ressxr 10083	. . . . . . . . . . . . . . . . 17
17	15, 16	sstri 3612	. . . . . . . . . . . . . . . 16
18	14, 17	syl6ss 3615	. . . . . . . . . . . . . . 15 Poly $\$ $/\$ deg deg coeff
19		fvex 6201	. . . . . . . . . . . . . . . . 17 deg
20	19	prid2 4298	. . . . . . . . . . . . . . . 16 deg deg
21		elun1 3780	. . . . . . . . . . . . . . . 16 deg deg deg deg deg coeff
22	20, 21	ax-mp 5	. . . . . . . . . . . . . . 15 deg deg deg coeff
23		supxrub 12154	. . . . . . . . . . . . . . 15 deg deg coeff $/\$ deg deg deg coeff deg deg deg coeff
24	18, 22, 23	sylancl 694	. . . . . . . . . . . . . 14 Poly $\$ $/\$ deg deg deg coeff
25	18	adantr 481	. . . . . . . . . . . . . . . 16 Poly $\$ $/\$ $/\$ deg deg coeff
26		fveq2 6191	. . . . . . . . . . . . . . . . . . . 20 coeff coeff
27		abs0 14025	. . . . . . . . . . . . . . . . . . . 20
28	26, 27	syl6eq 2672	. . . . . . . . . . . . . . . . . . 19 coeff coeff
29		c0ex 10034	. . . . . . . . . . . . . . . . . . . . 21
30	29	prid1 4297	. . . . . . . . . . . . . . . . . . . 20 deg
31		elun1 3780	. . . . . . . . . . . . . . . . . . . 20 deg deg deg coeff
32	30, 31	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 deg deg coeff
33	28, 32	syl6eqel 2709	. . . . . . . . . . . . . . . . . 18 coeff coeff deg deg coeff
34	33	adantl 482	. . . . . . . . . . . . . . . . 17 Poly $\$ $/\$ $/\$ $/\$ coeff coeff deg deg coeff
35		0z 11388	. . . . . . . . . . . . . . . . . . . . . . 23
36		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . . 24 coeff coeff
37	36	coef2 23987	. . . . . . . . . . . . . . . . . . . . . . 23 Poly $/\$ coeff
38	3, 35, 37	sylancl 694	. . . . . . . . . . . . . . . . . . . . . 22 Poly $\$ $/\$ coeff
39	38	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . . . 21 Poly $\$ $/\$ $/\$ coeff
40		nn0abscl 14052	. . . . . . . . . . . . . . . . . . . . 21 coeff coeff
41	39, 40	syl 17	. . . . . . . . . . . . . . . . . . . 20 Poly $\$ $/\$ $/\$ coeff
42	41	adantr 481	. . . . . . . . . . . . . . . . . . 19 Poly $\$ $/\$ $/\$ $/\$ coeff coeff
43		simplr 792	. . . . . . . . . . . . . . . . . . . . 21 Poly $\$ $/\$ $/\$ $/\$ coeff
44	9	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . 21 Poly $\$ $/\$ $/\$ $/\$ coeff deg
45	3	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . 22 Poly $\$ $/\$ $/\$ $/\$ coeff Poly
46		simpr 477	. . . . . . . . . . . . . . . . . . . . . 22 Poly $\$ $/\$ $/\$ $/\$ coeff coeff
47		eqid 2622	. . . . . . . . . . . . . . . . . . . . . . 23 deg deg
48	36, 47	dgrub 23990	. . . . . . . . . . . . . . . . . . . . . 22 Poly $/\$ $/\$ coeff deg
49	45, 43, 46, 48	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . 21 Poly $\$ $/\$ $/\$ $/\$ coeff deg
50		elfz2nn0 12431	. . . . . . . . . . . . . . . . . . . . 21 deg $/\$ deg $/\$ deg
51	43, 44, 49, 50	syl3anbrc 1246	. . . . . . . . . . . . . . . . . . . 20 Poly $\$ $/\$ $/\$ $/\$ coeff deg
52		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 coeff coeff
53		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . 23 coeff coeff
54	53	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . 22 coeff coeff
55	54	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . 21 coeff coeff coeff coeff
56	55	rspcev 3309	. . . . . . . . . . . . . . . . . . . 20 deg $/\$ coeff coeff degcoeff coeff
57	51, 52, 56	sylancl 694	. . . . . . . . . . . . . . . . . . 19 Poly $\$ $/\$ $/\$ $/\$ coeff degcoeff coeff
58		eqeq1 2626	. . . . . . . . . . . . . . . . . . . . 21 coeff coeff coeff coeff
59	58	rexbidv 3052	. . . . . . . . . . . . . . . . . . . 20 coeff deg coeff degcoeff coeff
60	59	elrab 3363	. . . . . . . . . . . . . . . . . . 19 coeff deg coeff coeff $/\$ degcoeff coeff
61	42, 57, 60	sylanbrc 698	. . . . . . . . . . . . . . . . . 18 Poly $\$ $/\$ $/\$ $/\$ coeff coeff deg coeff
62		elun2 3781	. . . . . . . . . . . . . . . . . 18 coeff deg coeff coeff deg deg coeff
63	61, 62	syl 17	. . . . . . . . . . . . . . . . 17 Poly $\$ $/\$ $/\$ $/\$ coeff coeff deg deg coeff
64	34, 63	pm2.61dane 2881	. . . . . . . . . . . . . . . 16 Poly $\$ $/\$ $/\$ coeff deg deg coeff
65		supxrub 12154	. . . . . . . . . . . . . . . 16 deg deg coeff $/\$ coeff deg deg coeff coeff deg deg coeff
66	25, 64, 65	syl2anc 693	. . . . . . . . . . . . . . 15 Poly $\$ $/\$ $/\$ coeff deg deg coeff
67	66	ralrimiva 2966	. . . . . . . . . . . . . 14 Poly $\$ $/\$ coeff deg deg coeff
68	6, 24, 67	3jca 1242	. . . . . . . . . . . . 13 Poly $\$ $/\$ $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
69	68	3adant2 1080	. . . . . . . . . . . 12 Poly $\$ $/\$ $/\$ $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
70		neeq1 2856	. . . . . . . . . . . . . 14
71		fveq2 6191	. . . . . . . . . . . . . . 15 deg deg
72	71	breq1d 4663	. . . . . . . . . . . . . 14 deg deg deg coeff deg deg deg coeff
73		fveq2 6191	. . . . . . . . . . . . . . . . . 18 coeff coeff
74	73	fveq1d 6193	. . . . . . . . . . . . . . . . 17 coeff coeff
75	74	fveq2d 6195	. . . . . . . . . . . . . . . 16 coeff coeff
76	75	breq1d 4663	. . . . . . . . . . . . . . 15 coeff deg deg coeff coeff deg deg coeff
77	76	ralbidv 2986	. . . . . . . . . . . . . 14 coeff deg deg coeff coeff deg deg coeff
78	70, 72, 77	3anbi123d 1399	. . . . . . . . . . . . 13 $/\$ deg deg deg coeff $/\$ coeff deg deg coeff $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
79	78	elrab 3363	. . . . . . . . . . . 12 Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff Poly $/\$ $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
80	4, 69, 79	sylanbrc 698	. . . . . . . . . . 11 Poly $\$ $/\$ $/\$ Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
81		simp2 1062	. . . . . . . . . . 11 Poly $\$ $/\$ $/\$
82		fveq1 6190	. . . . . . . . . . . . 13
83	82	eqeq1d 2624	. . . . . . . . . . . 12
84	83	rspcev 3309	. . . . . . . . . . 11 Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff $/\$ Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
85	80, 81, 84	syl2anc 693	. . . . . . . . . 10 Poly $\$ $/\$ $/\$ Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
86		fveq2 6191	. . . . . . . . . . . . 13
87	86	eqeq1d 2624	. . . . . . . . . . . 12
88	87	rexbidv 3052	. . . . . . . . . . 11 Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
89	88	elrab 3363	. . . . . . . . . 10 Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff $/\$ Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
90	1, 85, 89	sylanbrc 698	. . . . . . . . 9 Poly $\$ $/\$ $/\$ Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
91		prfi 8235	. . . . . . . . . . . . . . 15 deg
92		fzfi 12771	. . . . . . . . . . . . . . . . 17 deg
93		abrexfi 8266	. . . . . . . . . . . . . . . . 17 deg deg coeff
94	92, 93	ax-mp 5	. . . . . . . . . . . . . . . 16 deg coeff
95		rabssab 3690	. . . . . . . . . . . . . . . 16 deg coeff deg coeff
96		ssfi 8180	. . . . . . . . . . . . . . . 16 deg coeff $/\$ deg coeff deg coeff deg coeff
97	94, 95, 96	mp2an 708	. . . . . . . . . . . . . . 15 deg coeff
98		unfi 8227	. . . . . . . . . . . . . . 15 deg $/\$ deg coeff deg deg coeff
99	91, 97, 98	mp2an 708	. . . . . . . . . . . . . 14 deg deg coeff
100	99	a1i 11	. . . . . . . . . . . . 13 Poly $\$ $/\$ deg deg coeff
101	22	ne0ii 3923	. . . . . . . . . . . . . 14 deg deg coeff
102	101	a1i 11	. . . . . . . . . . . . 13 Poly $\$ $/\$ deg deg coeff
103		xrltso 11974	. . . . . . . . . . . . . 14
104		fisupcl 8375	. . . . . . . . . . . . . 14 $/\$ deg deg coeff $/\$ deg deg coeff $/\$ deg deg coeff deg deg coeff deg deg coeff
105	103, 104	mpan 706	. . . . . . . . . . . . 13 deg deg coeff $/\$ deg deg coeff $/\$ deg deg coeff deg deg coeff deg deg coeff
106	100, 102, 18, 105	syl3anc 1326	. . . . . . . . . . . 12 Poly $\$ $/\$ deg deg coeff deg deg coeff
107	14, 106	sseldd 3604	. . . . . . . . . . 11 Poly $\$ $/\$ deg deg coeff
108	107	3adant2 1080	. . . . . . . . . 10 Poly $\$ $/\$ $/\$ deg deg coeff
109		eqidd 2623	. . . . . . . . . 10 Poly $\$ $/\$ $/\$ Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
110		breq2 4657	. . . . . . . . . . . . . . . 16 deg deg coeff deg deg deg deg coeff
111		breq2 4657	. . . . . . . . . . . . . . . . 17 deg deg coeff coeff coeff deg deg coeff
112	111	ralbidv 2986	. . . . . . . . . . . . . . . 16 deg deg coeff coeff coeff deg deg coeff
113	110, 112	3anbi23d 1402	. . . . . . . . . . . . . . 15 deg deg coeff $/\$ deg $/\$ coeff $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
114	113	rabbidv 3189	. . . . . . . . . . . . . 14 deg deg coeff Poly $/\$ deg $/\$ coeff Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
115	114	rexeqdv 3145	. . . . . . . . . . . . 13 deg deg coeff Poly $/\$ deg $/\$ coeff Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
116	115	rabbidv 3189	. . . . . . . . . . . 12 deg deg coeff Poly $/\$ deg $/\$ coeff Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
117	116	eqeq2d 2632	. . . . . . . . . . 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
118	117	rspcev 3309	. . . . . . . . . 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
119	108, 109, 118	syl2anc 693	. . . . . . . . 9 Poly $\$ $/\$ $/\$ Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff Poly $/\$ deg $/\$ coeff
120		cnex 10017	. . . . . . . . . . 11
121	120	rabex 4813	. . . . . . . . . 10 Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
122		eleq2 2690	. . . . . . . . . . 11 Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff Poly $/\$ deg deg deg coeff $/\$ coeff deg deg coeff
123		eqeq1 2626	. . . . . . . . . . . 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
124	123	rexbidv 3052	. . . . . . . . . . 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
125	122, 124	anbi12d 747	. . . . . . . . . 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
126	121, 125	spcev 3300	. . . . . . . . 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
127	90, 119, 126	syl2anc 693	. . . . . . . 8 Poly $\$ $/\$ $/\$ $/\$ Poly $/\$ deg $/\$ coeff
128	127	3exp 1264	. . . . . . 7 Poly $\$ $/\$ Poly $/\$ deg $/\$ coeff
129	128	rexlimiv 3027	. . . . . 6 Poly $\$ $/\$ Poly $/\$ deg $/\$ coeff
130	129	impcom 446	. . . . 5 $/\$ Poly $\$ $/\$ Poly $/\$ deg $/\$ coeff
131		eleq2 2690	. . . . . . . . 9 Poly $/\$ deg $/\$ coeff Poly $/\$ deg $/\$ coeff
132	87	rexbidv 3052	. . . . . . . . . . 11 Poly $/\$ deg $/\$ coeff Poly $/\$ deg $/\$ coeff
133	132	elrab 3363	. . . . . . . . . 10 Poly $/\$ deg $/\$ coeff $/\$ Poly $/\$ deg $/\$ coeff
134		simp1 1061	. . . . . . . . . . . . . . 15 $/\$ deg $/\$ coeff
135	134	anim2i 593	. . . . . . . . . . . . . 14 Poly $/\$ $/\$ deg $/\$ coeff Poly $/\$
136	71	breq1d 4663	. . . . . . . . . . . . . . . 16 deg deg
137	75	breq1d 4663	. . . . . . . . . . . . . . . . 17 coeff coeff
138	137	ralbidv 2986	. . . . . . . . . . . . . . . 16 coeff coeff
139	70, 136, 138	3anbi123d 1399	. . . . . . . . . . . . . . 15 $/\$ deg $/\$ coeff $/\$ deg $/\$ coeff
140	139	elrab 3363	. . . . . . . . . . . . . 14 Poly $/\$ deg $/\$ coeff Poly $/\$ $/\$ deg $/\$ coeff
141		eldifsn 4317	. . . . . . . . . . . . . 14 Poly $\$ Poly $/\$
142	135, 140, 141	3imtr4i 281	. . . . . . . . . . . . 13 Poly $/\$ deg $/\$ coeff Poly $\$
143	142	ssriv 3607	. . . . . . . . . . . 12 Poly $/\$ deg $/\$ coeff Poly $\$
144		ssrexv 3667	. . . . . . . . . . . . 13 Poly $/\$ deg $/\$ coeff Poly $\$ Poly $/\$ deg $/\$ coeff Poly $\$
145	83	cbvrexv 3172	. . . . . . . . . . . . 13 Poly $\$ Poly $\$
146	144, 145	syl6ib 241	. . . . . . . . . . . 12 Poly $/\$ deg $/\$ coeff Poly $\$ Poly $/\$ deg $/\$ coeff Poly $\$
147	143, 146	ax-mp 5	. . . . . . . . . . 11 Poly $/\$ deg $/\$ coeff Poly $\$
148	147	anim2i 593	. . . . . . . . . 10 $/\$ Poly $/\$ deg $/\$ coeff $/\$ Poly $\$
149	133, 148	sylbi 207	. . . . . . . . 9 Poly $/\$ deg $/\$ coeff $/\$ Poly $\$
150	131, 149	syl6bi 243	. . . . . . . 8 Poly $/\$ deg $/\$ coeff $/\$ Poly $\$
151	150	rexlimivw 3029	. . . . . . 7 Poly $/\$ deg $/\$ coeff $/\$ Poly $\$
152	151	impcom 446	. . . . . 6 $/\$ Poly $/\$ deg $/\$ coeff $/\$ Poly $\$
153	152	exlimiv 1858	. . . . 5 $/\$ Poly $/\$ deg $/\$ coeff $/\$ Poly $\$
154	130, 153	impbii 199	. . . 4 $/\$ Poly $\$ $/\$ Poly $/\$ deg $/\$ coeff
155		elaa 24071	. . . 4 $/\$ Poly $\$
156		eluniab 4447	. . . 4 Poly $/\$ deg $/\$ coeff $/\$ Poly $/\$ deg $/\$ coeff
157	154, 155, 156	3bitr4i 292	. . 3 Poly $/\$ deg $/\$ coeff
158	157	eqriv 2619	. 2 Poly $/\$ deg $/\$ coeff
159		aannenlem.a	. . . 4 Poly $/\$ deg $/\$ coeff
160	159	rnmpt 5371	. . 3 Poly $/\$ deg $/\$ coeff
161	160	unieqi 4445	. 2 Poly $/\$ deg $/\$ coeff
162	158, 161	eqtr4i 2647	1

Colors of variables: wff setvar class

Syntax hints:

wi 4 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wex 1704

wcel 1990

cab 2608

wne 2794

wral 2912

wrex 2913

crab 2916 $\$ cdif 3571

cun 3572

wss 3574

c0 3915

csn 4177

cpr 4179

cuni 4436 class class class wbr 4653

cmpt 4729

wor 5034

crn 5115

wf 5884

cfv 5888 (class class class)co 6650

cfn 7955

csup 8346

cc 9934

cr 9935

cc0 9936

cxr 10073

clt 10074

cle 10075

cn0 11292

cz 11377

cfz 12326

cabs 13974

c0p 23436 Polycply 23940 coeffccoe 23942 degcdgr 23943

caa 24069

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-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-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-coe 23946 df-dgr 23947 df-aa 24070

This theorem is referenced by: aannenlem3 24085

W3C validator