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Theorem pceu 12249

Description: Uniqueness for the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)

**Hypotheses**
Ref	Expression
pcval.1
pcval.2

**Assertion**
Ref	Expression
pceu	$/\$ $/\$ $/\$

Distinct variable groups:

Allowed substitution hints:

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Proof of Theorem pceu Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simprl 526	. . . 4 $/\$ $/\$
2		elq 9581	. . . 4
3	1, 2	sylib 121	. . 3 $/\$ $/\$
4		simpr 109	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
5		simprr 527	. . . . . . . . . . . . 13 $/\$ $/\$
6	5	ad3antrrr 489	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
7	1	ad3antrrr 489	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
8		0z 9223	. . . . . . . . . . . . . 14
9		zq 9585	. . . . . . . . . . . . . 14
10	8, 9	mp1i 10	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
11		qapne 9598	. . . . . . . . . . . . 13 $/\$ #
12	7, 10, 11	syl2anc 409	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ #
13	6, 12	mpbird 166	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ #
14	4, 13	eqbrtrrd 4013	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ #
15		simpllr 529	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
16	15	zcnd 9335	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
17		nnz 9231	. . . . . . . . . . . . . 14
18	17	adantl 275	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
19	18	adantr 274	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
20	19	zcnd 9335	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
21		simplr 525	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
22	21	nnap0d 8924	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ #
23	16, 20, 22	divap0bd 8719	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ # #
24	14, 23	mpbird 166	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ #
25		0zd 9224	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
26		zapne 9286	. . . . . . . . . 10 $/\$ #
27	15, 25, 26	syl2anc 409	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ #
28	24, 27	mpbid 146	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
29	28	ex 114	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
30	29	adantrd 277	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
31	30	exlimdv 1812	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
32		prmuz2 12085	. . . . . . . . . . . . . . 15
33	32	ad3antrrr 489	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
34	33	adantr 274	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
35		simpllr 529	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
36		simpr 109	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
37		eqid 2170	. . . . . . . . . . . . . . 15
38		pcval.1	. . . . . . . . . . . . . . 15
39	37, 38	pcprecl 12243	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
40	39	simpld 111	. . . . . . . . . . . . 13 $/\$ $/\$
41	34, 35, 36, 40	syl12anc 1231	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
42	41	nn0zd 9332	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
43		nnne0 8906	. . . . . . . . . . . . . . . 16
44	43	adantl 275	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
45		eqid 2170	. . . . . . . . . . . . . . . 16
46		pcval.2	. . . . . . . . . . . . . . . 16
47	45, 46	pcprecl 12243	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
48	33, 18, 44, 47	syl12anc 1231	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
49	48	simpld 111	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
50	49	adantr 274	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
51	50	nn0zd 9332	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
52	42, 51	zsubcld 9339	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
53		biidd 171	. . . . . . . . . . 11
54	53	ceqsexgv 2859	. . . . . . . . . 10 $/\$
55	52, 54	syl 14	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56		exancom 1601	. . . . . . . . 9 $/\$ $/\$
57	55, 56	bitr3di 194	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58	57	ex 114	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
59	29, 31, 58	pm5.21ndd 700	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
60	59	rexbidva 2467	. . . . 5 $/\$ $/\$ $/\$ $/\$
61	60	rexbidva 2467	. . . 4 $/\$ $/\$ $/\$
62		rexcom4 2753	. . . . . 6 $/\$ $/\$
63	62	rexbii 2477	. . . . 5 $/\$ $/\$
64		rexcom4 2753	. . . . 5 $/\$ $/\$
65	63, 64	bitri 183	. . . 4 $/\$ $/\$
66	61, 65	bitrdi 195	. . 3 $/\$ $/\$ $/\$
67	3, 66	mpbid 146	. 2 $/\$ $/\$ $/\$
68		eqid 2170	. . . . . . . . . . 11
69		eqid 2170	. . . . . . . . . . 11
70		simp11l 1103	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
71		simp11r 1104	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
72		simp12 1023	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73		simp13l 1107	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
74		simp2 993	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
75		simp3l 1020	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
76	38, 46, 68, 69, 70, 71, 72, 73, 74, 75	pceulem 12248	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
77		simp13r 1108	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
78		simp3r 1021	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
79	76, 77, 78	3eqtr4d 2213	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
80	79	3exp 1197	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
81	80	rexlimdvv 2594	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
82	81	3exp 1197	. . . . . 6 $/\$ $/\$ $/\$ $/\$
83	82	adantrl 475	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
84	83	rexlimdvv 2594	. . . 4 $/\$ $/\$ $/\$ $/\$
85	84	impd 252	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
86	85	alrimivv 1868	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
87		eqeq1 2177	. . . . . 6
88	87	anbi2d 461	. . . . 5 $/\$ $/\$
89	88	2rexbidv 2495	. . . 4 $/\$ $/\$
90		oveq1 5860	. . . . . . . . 9
91	90	eqeq2d 2182	. . . . . . . 8
92		breq2 3993	. . . . . . . . . . . . 13
93	92	rabbidv 2719	. . . . . . . . . . . 12
94	93	supeq1d 6964	. . . . . . . . . . 11
95	38, 94	eqtrid 2215	. . . . . . . . . 10
96	95	oveq1d 5868	. . . . . . . . 9
97	96	eqeq2d 2182	. . . . . . . 8
98	91, 97	anbi12d 470	. . . . . . 7 $/\$ $/\$
99	98	rexbidv 2471	. . . . . 6 $/\$ $/\$
100		oveq2 5861	. . . . . . . . 9
101	100	eqeq2d 2182	. . . . . . . 8
102		breq2 3993	. . . . . . . . . . . . 13
103	102	rabbidv 2719	. . . . . . . . . . . 12
104	103	supeq1d 6964	. . . . . . . . . . 11
105	46, 104	eqtrid 2215	. . . . . . . . . 10
106	105	oveq2d 5869	. . . . . . . . 9
107	106	eqeq2d 2182	. . . . . . . 8
108	101, 107	anbi12d 470	. . . . . . 7 $/\$ $/\$
109	108	cbvrexvw 2701	. . . . . 6 $/\$ $/\$
110	99, 109	bitrdi 195	. . . . 5 $/\$ $/\$
111	110	cbvrexvw 2701	. . . 4 $/\$ $/\$
112	89, 111	bitrdi 195	. . 3 $/\$ $/\$
113	112	eu4 2081	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
114	67, 86, 113	sylanbrc 415	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $/\$ w3a 973

wal 1346

wceq 1348

wex 1485

weu 2019

wcel 2141

wne 2340

wrex 2449

crab 2452 class class class wbr 3989

cfv 5198 (class class class)co 5853

csup 6959

cr 7773

cc0 7774

clt 7954

cmin 8090 # cap 8500

cdiv 8589

cn 8878

c2 8929

cn0 9135

cz 9212

cuz 9487

cq 9578

cexp 10475

cdvds 11749

cprime 12061

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-1o 6395 df-2o 6396 df-er 6513 df-en 6719 df-sup 6961 df-inf 6962 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-fz 9966 df-fzo 10099 df-fl 10226 df-mod 10279 df-seqfrec 10402 df-exp 10476 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-dvds 11750 df-gcd 11898 df-prm 12062

This theorem is referenced by: pcval 12250 pczpre 12251 pcdiv 12256

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