pceu - Intuitionistic Logic Explorer

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Theorem pceu 13052

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 535	. . . 4 $/\$ $/\$
2		elq 10001	. . . 4
3	1, 2	sylib 122	. . 3 $/\$ $/\$
4		simpr 110	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
5		simprr 537	. . . . . . . . . . . . 13 $/\$ $/\$
6	5	ad3antrrr 496	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
7	1	ad3antrrr 496	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
8		0z 9634	. . . . . . . . . . . . . 14
9		zq 10005	. . . . . . . . . . . . . 14
10	8, 9	mp1i 10	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
11		qapne 10018	. . . . . . . . . . . . 13 $/\$ #
12	7, 10, 11	syl2anc 415	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ #
13	6, 12	mpbird 167	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ #
14	4, 13	eqbrtrrd 4149	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ #
15		simpllr 540	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
16	15	zcnd 9748	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
17		nnz 9642	. . . . . . . . . . . . . 14
18	17	adantl 277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
19	18	adantr 276	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
20	19	zcnd 9748	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
21		simplr 533	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
22	21	nnap0d 9329	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ #
23	16, 20, 22	divap0bd 9122	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ # #
24	14, 23	mpbird 167	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ #
25		0zd 9635	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
26		zapne 9698	. . . . . . . . . 10 $/\$ #
27	15, 25, 26	syl2anc 415	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ #
28	24, 27	mpbid 147	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
29	28	ex 115	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
30	29	adantrd 279	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
31	30	exlimdv 1872	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
32		prmuz2 12887	. . . . . . . . . . . . . . 15
33	32	ad3antrrr 496	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
34	33	adantr 276	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
35		simpllr 540	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
36		simpr 110	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$
37		eqid 2238	. . . . . . . . . . . . . . 15
38		pcval.1	. . . . . . . . . . . . . . 15
39	37, 38	pcprecl 13046	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
40	39	simpld 112	. . . . . . . . . . . . 13 $/\$ $/\$
41	34, 35, 36, 40	syl12anc 1276	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
42	41	nn0zd 9745	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
43		nnne0 9311	. . . . . . . . . . . . . . . 16
44	43	adantl 277	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
45		eqid 2238	. . . . . . . . . . . . . . . 16
46		pcval.2	. . . . . . . . . . . . . . . 16
47	45, 46	pcprecl 13046	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$
48	33, 18, 44, 47	syl12anc 1276	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$
49	48	simpld 112	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
50	49	adantr 276	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
51	50	nn0zd 9745	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
52	42, 51	zsubcld 9752	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
53		biidd 172	. . . . . . . . . . 11
54	53	ceqsexgv 2955	. . . . . . . . . 10 $/\$
55	52, 54	syl 14	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56		exancom 1661	. . . . . . . . 9 $/\$ $/\$
57	55, 56	bitr3di 195	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58	57	ex 115	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
59	29, 31, 58	pm5.21ndd 717	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
60	59	rexbidva 2547	. . . . 5 $/\$ $/\$ $/\$ $/\$
61	60	rexbidva 2547	. . . 4 $/\$ $/\$ $/\$
62		rexcom4 2845	. . . . . 6 $/\$ $/\$
63	62	rexbii 2557	. . . . 5 $/\$ $/\$
64		rexcom4 2845	. . . . 5 $/\$ $/\$
65	63, 64	bitri 184	. . . 4 $/\$ $/\$
66	61, 65	bitrdi 196	. . 3 $/\$ $/\$ $/\$
67	3, 66	mpbid 147	. 2 $/\$ $/\$ $/\$
68		eqid 2238	. . . . . . . . . . 11
69		eqid 2238	. . . . . . . . . . 11
70		simp11l 1139	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
71		simp11r 1140	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
72		simp12 1059	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73		simp13l 1143	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
74		simp2 1029	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
75		simp3l 1056	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
76	38, 46, 68, 69, 70, 71, 72, 73, 74, 75	pceulem 13051	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
77		simp13r 1144	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
78		simp3r 1057	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
79	76, 77, 78	3eqtr4d 2281	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
80	79	3exp 1233	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
81	80	rexlimdvv 2675	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
82	81	3exp 1233	. . . . . 6 $/\$ $/\$ $/\$ $/\$
83	82	adantrl 482	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
84	83	rexlimdvv 2675	. . . 4 $/\$ $/\$ $/\$ $/\$
85	84	impd 254	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
86	85	alrimivv 1928	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
87		eqeq1 2245	. . . . . 6
88	87	anbi2d 468	. . . . 5 $/\$ $/\$
89	88	2rexbidv 2575	. . . 4 $/\$ $/\$
90		oveq1 6082	. . . . . . . . 9
91	90	eqeq2d 2250	. . . . . . . 8
92		breq2 4129	. . . . . . . . . . . . 13
93	92	rabbidv 2810	. . . . . . . . . . . 12
94	93	supeq1d 7317	. . . . . . . . . . 11
95	38, 94	eqtrid 2283	. . . . . . . . . 10
96	95	oveq1d 6090	. . . . . . . . 9
97	96	eqeq2d 2250	. . . . . . . 8
98	91, 97	anbi12d 477	. . . . . . 7 $/\$ $/\$
99	98	rexbidv 2551	. . . . . 6 $/\$ $/\$
100		oveq2 6083	. . . . . . . . 9
101	100	eqeq2d 2250	. . . . . . . 8
102		breq2 4129	. . . . . . . . . . . . 13
103	102	rabbidv 2810	. . . . . . . . . . . 12
104	103	supeq1d 7317	. . . . . . . . . . 11
105	46, 104	eqtrid 2283	. . . . . . . . . 10
106	105	oveq2d 6091	. . . . . . . . 9
107	106	eqeq2d 2250	. . . . . . . 8
108	101, 107	anbi12d 477	. . . . . . 7 $/\$ $/\$
109	108	cbvrexvw 2791	. . . . . 6 $/\$ $/\$
110	99, 109	bitrdi 196	. . . . 5 $/\$ $/\$
111	110	cbvrexvw 2791	. . . 4 $/\$ $/\$
112	89, 111	bitrdi 196	. . 3 $/\$ $/\$
113	112	eu4 2149	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
114	67, 86, 113	sylanbrc 421	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1009

wal 1400

wceq 1402

wex 1545

weu 2086

wcel 2209

wne 2420

wrex 2529

crab 2532 class class class wbr 4125

cfv 5372 (class class class)co 6075

csup 7312

cr 8168

cc0 8169

clt 8350

cmin 8487 # cap 8899

cdiv 8992

cn 9283

c2 9334

cn0 9542

cz 9623

cuz 9900

cq 9998

cexp 10953

cdvds 12532

cprime 12863

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287 ax-arch 8288 ax-caucvg 8289

This theorem depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-isom 5381 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-frec 6652 df-1o 6677 df-2o 6678 df-er 6797 df-en 7013 df-sup 7314 df-inf 7315 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-inn 9284 df-2 9342 df-3 9343 df-4 9344 df-n0 9543 df-z 9624 df-uz 9901 df-q 9999 df-rp 10034 df-fz 10391 df-fzo 10528 df-fl 10683 df-mod 10738 df-seqfrec 10863 df-exp 10954 df-cj 11585 df-re 11586 df-im 11587 df-rsqrt 11742 df-abs 11743 df-dvds 12533 df-gcd 12709 df-prm 12864

This theorem is referenced by: pcval 13053 pczpre 13054 pcdiv 13059

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