padicabv - Metamath Proof Explorer

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Theorem padicabv 25319

Description: The p-adic absolute value (with arbitrary base) is an absolute value. (Contributed by Mario Carneiro, 9-Sep-2014.)

**Hypotheses**
Ref	Expression
qrng.q	ℂ_fld ↾_s
qabsabv.a	AbsVal
padic.f

**Assertion**
Ref	Expression
padicabv	$/\$

Distinct variable groups:

Allowed substitution hint:

(

)

Proof of Theorem padicabv Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		qabsabv.a	. . 3 AbsVal
2	1	a1i 11	. 2 $/\$ AbsVal
3		qrng.q	. . . 4 ℂ_fld ↾_s
4	3	qrngbas 25308	. . 3
5	4	a1i 11	. 2 $/\$
6		qex 11800	. . 3
7		cnfldadd 19751	. . . 4 ℂ_fld
8	3, 7	ressplusg 15993	. . 3
9	6, 8	mp1i 13	. 2 $/\$
10		cnfldmul 19752	. . . 4 ℂ_fld
11	3, 10	ressmulr 16006	. . 3
12	6, 11	mp1i 13	. 2 $/\$
13	3	qrng0 25310	. . 3
14	13	a1i 11	. 2 $/\$
15	3	qdrng 25309	. . 3
16		drngring 18754	. . 3
17	15, 16	mp1i 13	. 2 $/\$
18		0red 10041	. . . 4 $/\$ $/\$ $/\$
19		ioossre 12235	. . . . . . 7
20		simpr 477	. . . . . . 7 $/\$
21	19, 20	sseldi 3601	. . . . . 6 $/\$
22	21	ad2antrr 762	. . . . 5 $/\$ $/\$ $/\$
23		eliooord 12233	. . . . . . . . . 10 $/\$
24	23	adantl 482	. . . . . . . . 9 $/\$ $/\$
25	24	simpld 475	. . . . . . . 8 $/\$
26	21, 25	elrpd 11869	. . . . . . 7 $/\$
27	26	rpne0d 11877	. . . . . 6 $/\$
28	27	ad2antrr 762	. . . . 5 $/\$ $/\$ $/\$
29		df-ne 2795	. . . . . 6
30		pcqcl 15561	. . . . . . . 8 $/\$ $/\$
31	30	adantlr 751	. . . . . . 7 $/\$ $/\$ $/\$
32	31	anassrs 680	. . . . . 6 $/\$ $/\$ $/\$
33	29, 32	sylan2br 493	. . . . 5 $/\$ $/\$ $/\$
34	22, 28, 33	reexpclzd 13034	. . . 4 $/\$ $/\$ $/\$
35	18, 34	ifclda 4120	. . 3 $/\$ $/\$
36		padic.f	. . 3
37	35, 36	fmptd 6385	. 2 $/\$
38		0z 11388	. . . 4
39		zq 11794	. . . 4
40	38, 39	ax-mp 5	. . 3
41		iftrue 4092	. . . 4
42		c0ex 10034	. . . 4
43	41, 36, 42	fvmpt 6282	. . 3
44	40, 43	mp1i 13	. 2 $/\$
45	21	3ad2ant1 1082	. . . 4 $/\$ $/\$ $/\$
46		pcqcl 15561	. . . . . 6 $/\$ $/\$
47	46	adantlr 751	. . . . 5 $/\$ $/\$ $/\$
48	47	3impb 1260	. . . 4 $/\$ $/\$ $/\$
49	25	3ad2ant1 1082	. . . 4 $/\$ $/\$ $/\$
50		expgt0 12893	. . . 4 $/\$ $/\$
51	45, 48, 49, 50	syl3anc 1326	. . 3 $/\$ $/\$ $/\$
52		eqeq1 2626	. . . . . . 7
53		oveq2 6658	. . . . . . . 8
54	53	oveq2d 6666	. . . . . . 7
55	52, 54	ifbieq2d 4111	. . . . . 6
56		ovex 6678	. . . . . . 7
57	42, 56	ifex 4156	. . . . . 6
58	55, 36, 57	fvmpt 6282	. . . . 5
59	58	3ad2ant2 1083	. . . 4 $/\$ $/\$ $/\$
60		simp3 1063	. . . . . 6 $/\$ $/\$ $/\$
61	60	neneqd 2799	. . . . 5 $/\$ $/\$ $/\$
62	61	iffalsed 4097	. . . 4 $/\$ $/\$ $/\$
63	59, 62	eqtrd 2656	. . 3 $/\$ $/\$ $/\$
64	51, 63	breqtrrd 4681	. 2 $/\$ $/\$ $/\$
65		pcqmul 15558	. . . . . 6 $/\$ $/\$ $/\$ $/\$
66	65	3adant1r 1319	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
67	66	oveq2d 6666	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
68	21	recnd 10068	. . . . . 6 $/\$
69	68	3ad2ant1 1082	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
70	27	3ad2ant1 1082	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
71	47	3adant3 1081	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
72		simp1l 1085	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
73		simp3l 1089	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
74		simp3r 1090	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
75		pcqcl 15561	. . . . . 6 $/\$ $/\$
76	72, 73, 74, 75	syl12anc 1324	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
77		expaddz 12904	. . . . 5 $/\$ $/\$ $/\$
78	69, 70, 71, 76, 77	syl22anc 1327	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
79	67, 78	eqtrd 2656	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
80		simp2l 1087	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
81		qmulcl 11806	. . . . . 6 $/\$
82	80, 73, 81	syl2anc 693	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
83		eqeq1 2626	. . . . . . 7
84		oveq2 6658	. . . . . . . 8
85	84	oveq2d 6666	. . . . . . 7
86	83, 85	ifbieq2d 4111	. . . . . 6
87		ovex 6678	. . . . . . 7
88	42, 87	ifex 4156	. . . . . 6
89	86, 36, 88	fvmpt 6282	. . . . 5
90	82, 89	syl 17	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
91		qcn 11802	. . . . . . . 8
92	80, 91	syl 17	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
93		qcn 11802	. . . . . . . 8
94	73, 93	syl 17	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
95		simp2r 1088	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
96	92, 94, 95, 74	mulne0d 10679	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
97	96	neneqd 2799	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
98	97	iffalsed 4097	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
99	90, 98	eqtrd 2656	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
100	63	3expb 1266	. . . . 5 $/\$ $/\$ $/\$
101	100	3adant3 1081	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
102		eqeq1 2626	. . . . . . . 8
103		oveq2 6658	. . . . . . . . 9
104	103	oveq2d 6666	. . . . . . . 8
105	102, 104	ifbieq2d 4111	. . . . . . 7
106		ovex 6678	. . . . . . . 8
107	42, 106	ifex 4156	. . . . . . 7
108	105, 36, 107	fvmpt 6282	. . . . . 6
109	73, 108	syl 17	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
110	74	neneqd 2799	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
111	110	iffalsed 4097	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
112	109, 111	eqtrd 2656	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
113	101, 112	oveq12d 6668	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
114	79, 99, 113	3eqtr4d 2666	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
115		iftrue 4092	. . . . 5
116	115	breq1d 4663	. . . 4
117		ifnefalse 4098	. . . . . 6
118	117	adantl 482	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
119	71	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
120	119	zred 11482	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
121	76	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
122	121	zred 11482	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
123	21	3ad2ant1 1082	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
124	123	ad2antrr 762	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
125	70	ad2antrr 762	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
126	72	adantr 481	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
127		qaddcl 11804	. . . . . . . . . . . 12 $/\$
128	80, 73, 127	syl2anc 693	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
129	128	adantr 481	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
130		simpr 477	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
131		pcqcl 15561	. . . . . . . . . 10 $/\$ $/\$
132	126, 129, 130, 131	syl12anc 1324	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
133	132	adantr 481	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
134	124, 125, 133	reexpclzd 13034	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
135	119	adantr 481	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
136	124, 125, 135	reexpclzd 13034	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
137		simpl1 1064	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
138	137, 21	syl 17	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
139	137, 27	syl 17	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
140	138, 139, 119	reexpclzd 13034	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
141	138, 139, 121	reexpclzd 13034	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
142	140, 141	readdcld 10069	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
143	142	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
144	126	adantr 481	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
145	80	ad2antrr 762	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
146	73	ad2antrr 762	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
147		simpr 477	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
148	144, 145, 146, 147	pcadd 15593	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
149	137, 26	syl 17	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
150	24	simprd 479	. . . . . . . . . . . . 13 $/\$
151	137, 150	syl 17	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
152	149, 119, 132, 151	ltexp2rd 13033	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
153	152	notbid 308	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
154	132	zred 11482	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
155	120, 154	lenltd 10183	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
156	138, 139, 132	reexpclzd 13034	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
157	156, 140	lenltd 10183	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
158	153, 155, 157	3bitr4d 300	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
159	158	biimpa 501	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
160	148, 159	syldan 487	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
161	26	3ad2ant1 1082	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
162	161, 76	rpexpcld 13032	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
163	162	adantr 481	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
164	163	rpge0d 11876	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
165	140, 141	addge01d 10615	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
166	164, 165	mpbid 222	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
167	166	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
168	134, 136, 143, 160, 167	letrd 10194	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
169	156	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
170	141	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
171	142	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
172	126	adantr 481	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
173	73	ad2antrr 762	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
174	80	ad2antrr 762	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
175		simpr 477	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
176	172, 173, 174, 175	pcadd 15593	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
177	92, 94	addcomd 10238	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
178	177	oveq2d 6666	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
179	178	ad2antrr 762	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
180	176, 179	breqtrrd 4681	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
181	149, 121, 132, 151	ltexp2rd 13033	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
182	181	notbid 308	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
183	122, 154	lenltd 10183	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
184	156, 141	lenltd 10183	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
185	182, 183, 184	3bitr4d 300	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
186	185	biimpa 501	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
187	180, 186	syldan 487	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
188	161, 71	rpexpcld 13032	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
189	188	adantr 481	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
190	189	rpge0d 11876	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
191	141, 140	addge02d 10616	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
192	190, 191	mpbid 222	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
193	192	adantr 481	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
194	169, 170, 171, 187, 193	letrd 10194	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
195	120, 122, 168, 194	lecasei 10143	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
196	118, 195	eqbrtrd 4675	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
197	188, 162	rpaddcld 11887	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
198	197	rpge0d 11876	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
199	116, 196, 198	pm2.61ne 2879	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
200		eqeq1 2626	. . . . . 6
201		oveq2 6658	. . . . . . 7
202	201	oveq2d 6666	. . . . . 6
203	200, 202	ifbieq2d 4111	. . . . 5
204		ovex 6678	. . . . . 6
205	42, 204	ifex 4156	. . . . 5
206	203, 36, 205	fvmpt 6282	. . . 4
207	128, 206	syl 17	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
208	101, 112	oveq12d 6668	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
209	199, 207, 208	3brtr4d 4685	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
210	2, 5, 9, 12, 14, 17, 37, 44, 64, 114, 209	isabvd 18820	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

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

wceq 1483

wcel 1990

wne 2794

cvv 3200

cif 4086 class class class wbr 4653

cmpt 4729

cfv 5888 (class class class)co 6650

cc 9934

cr 9935

cc0 9936

c1 9937

caddc 9939

cmul 9941

clt 10074

cle 10075

cz 11377

cq 11788

crp 11832

cioo 12175

cexp 12860

cprime 15385

cpc 15541

cbs 15857 ↾_s cress 15858

cplusg 15941

cmulr 15942

c0g 16100

crg 18547

cdr 18747 AbsValcabv 18816 ℂ_fldccnfld 19746

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-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 ax-mulf 10016

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-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-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 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-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-ioo 12179 df-ico 12181 df-fz 12327 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-dvds 14984 df-gcd 15217 df-prm 15386 df-pc 15542 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-subg 17591 df-cmn 18195 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-subrg 18778 df-abv 18817 df-cnfld 19747

This theorem is referenced by: padicabvf 25320 padicabvcxp 25321

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