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Theorem pcqmul 13060

Description: Multiplication property of the prime power function. (Contributed by Mario Carneiro, 9-Sep-2014.)

**Assertion**
Ref	Expression
pcqmul	$/\$ $/\$ $/\$ $/\$

Proof of Theorem pcqmul Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2l 1054	. . 3 $/\$ $/\$ $/\$ $/\$
2		elq 10001	. . 3
3	1, 2	sylib 122	. 2 $/\$ $/\$ $/\$ $/\$
4		simp3l 1056	. . 3 $/\$ $/\$ $/\$ $/\$
5		elq 10001	. . 3
6	4, 5	sylib 122	. 2 $/\$ $/\$ $/\$ $/\$
7		reeanv 2721	. . 3 $/\$ $/\$
8		reeanv 2721	. . . . 5 $/\$ $/\$
9		simp2r 1055	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
10		simp3r 1057	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
11	9, 10	jca 306	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
12	11	ad2antrr 492	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
13		simp1 1028	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
14		simprl 535	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
15	14	nncnd 9297	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
16	14	nnap0d 9329	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ #
17	15, 16	div0apd 9107	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
18		oveq1 6082	. . . . . . . . . . . . 13
19	18	eqeq1d 2247	. . . . . . . . . . . 12
20	17, 19	syl5ibrcom 157	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
21	20	necon3d 2464	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
22		simprr 537	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
23	22	nncnd 9297	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
24	22	nnap0d 9329	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ #
25	23, 24	div0apd 9107	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
26		oveq1 6082	. . . . . . . . . . . . 13
27	26	eqeq1d 2247	. . . . . . . . . . . 12
28	25, 27	syl5ibrcom 157	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
29	28	necon3d 2464	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
30		simpll 531	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31		simplrl 541	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		simplrr 542	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	31, 32	zmulcld 9753	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	31	zcnd 9748	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	32	zcnd 9748	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36		simprrl 545	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		0zd 9635	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38		zapne 9698	. . . . . . . . . . . . . . . . . 18 $/\$ #
39	31, 37, 38	syl2anc 415	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
40	36, 39	mpbird 167	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
41		simprrr 546	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		zapne 9698	. . . . . . . . . . . . . . . . . 18 $/\$ #
43	32, 37, 42	syl2anc 415	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
44	41, 43	mpbird 167	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
45	34, 35, 40, 44	mulap0d 8976	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
46		zapne 9698	. . . . . . . . . . . . . . . 16 $/\$ #
47	33, 37, 46	syl2anc 415	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
48	45, 47	mpbid 147	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49	14	adantrr 483	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
50	22	adantrr 483	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	49, 50	nnmulcld 9332	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52		pcdiv 13059	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
53	30, 33, 48, 51, 52	syl121anc 1283	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54		pcmul 13058	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
55	30, 31, 36, 32, 41, 54	syl122anc 1287	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	49	nnzd 9746	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57	14	nnne0d 9328	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
58	57	adantrr 483	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
59	50	nnzd 9746	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60	22	nnne0d 9328	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
61	60	adantrr 483	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62		pcmul 13058	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
63	30, 56, 58, 59, 61, 62	syl122anc 1287	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
64	55, 63	oveq12d 6093	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65		pczcl 13055	. . . . . . . . . . . . . . . 16 $/\$ $/\$
66	30, 31, 36, 65	syl12anc 1276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
67	66	nn0cnd 9601	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
68		pczcl 13055	. . . . . . . . . . . . . . . 16 $/\$ $/\$
69	30, 32, 41, 68	syl12anc 1276	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70	69	nn0cnd 9601	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
71	30, 49	pccld 13057	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
72	71	nn0cnd 9601	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73	30, 50	pccld 13057	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
74	73	nn0cnd 9601	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
75	67, 70, 72, 74	addsub4d 8674	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
76	53, 64, 75	3eqtrd 2275	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
77	15	adantrr 483	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
78	23	adantrr 483	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
79	16	adantrr 483	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
80	24	adantrr 483	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
81	34, 77, 35, 78, 79, 80	divmuldivapd 9152	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
82	81	oveq2d 6091	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
83		pcdiv 13059	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
84	30, 31, 36, 49, 83	syl121anc 1283	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
85		pcdiv 13059	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
86	30, 32, 41, 50, 85	syl121anc 1283	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
87	84, 86	oveq12d 6093	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
88	76, 82, 87	3eqtr4d 2281	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
89	88	expr 375	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
90	21, 29, 89	syl2and 295	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
91		neeq1 2433	. . . . . . . . . . 11
92		neeq1 2433	. . . . . . . . . . 11
93	91, 92	bi2anan9 614	. . . . . . . . . 10 $/\$ $/\$ $/\$
94		oveq12 6084	. . . . . . . . . . . 12 $/\$
95	94	oveq2d 6091	. . . . . . . . . . 11 $/\$
96		oveq2 6083	. . . . . . . . . . . 12
97		oveq2 6083	. . . . . . . . . . . 12
98	96, 97	oveqan12d 6094	. . . . . . . . . . 11 $/\$
99	95, 98	eqeq12d 2253	. . . . . . . . . 10 $/\$
100	93, 99	imbi12d 234	. . . . . . . . 9 $/\$ $/\$ $/\$
101	90, 100	syl5ibrcom 157	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
102	13, 101	sylanl1 406	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
103	12, 102	mpid 42	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
104	103	rexlimdvva 2676	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
105	8, 104	biimtrrid 153	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
106	105	rexlimdvva 2676	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
107	7, 106	biimtrrid 153	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
108	3, 6, 107	mp2and 437	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1009

wceq 1402

wcel 2209

wne 2420

wrex 2529 class class class wbr 4125 (class class class)co 6075

cc 8167

cc0 8169

caddc 8172

cmul 8174

cmin 8487 # cap 8899

cdiv 8992

cn 9283

cn0 9542

cz 9623

cq 9998

cprime 12863

cpc 13041

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 df-pc 13042

This theorem is referenced by: pcqdiv 13064 pcexp 13066 pcaddlem 13096

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