pcqmul - Metamath Proof Explorer

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Theorem pcqmul 15558

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 1087	. . 3 $/\$ $/\$ $/\$ $/\$
2		elq 11790	. . 3
3	1, 2	sylib 208	. 2 $/\$ $/\$ $/\$ $/\$
4		simp3l 1089	. . 3 $/\$ $/\$ $/\$ $/\$
5		elq 11790	. . 3
6	4, 5	sylib 208	. 2 $/\$ $/\$ $/\$ $/\$
7		reeanv 3107	. . 3 $/\$ $/\$
8		reeanv 3107	. . . . 5 $/\$ $/\$
9		simp2r 1088	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
10		simp3r 1090	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
11	9, 10	jca 554	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
12	11	ad2antrr 762	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
13		simp1 1061	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
14		simprl 794	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
15	14	nncnd 11036	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
16	14	nnne0d 11065	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
17	15, 16	div0d 10800	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
18		oveq1 6657	. . . . . . . . . . . . 13
19	18	eqeq1d 2624	. . . . . . . . . . . 12
20	17, 19	syl5ibrcom 237	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
21	20	necon3d 2815	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
22		simprr 796	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
23	22	nncnd 11036	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
24	22	nnne0d 11065	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
25	23, 24	div0d 10800	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
26		oveq1 6657	. . . . . . . . . . . . 13
27	26	eqeq1d 2624	. . . . . . . . . . . 12
28	25, 27	syl5ibrcom 237	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
29	28	necon3d 2815	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
30		simpll 790	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31		simplrl 800	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		simplrr 801	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	31, 32	zmulcld 11488	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	31	zcnd 11483	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	32	zcnd 11483	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36		simprrl 804	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		simprrr 805	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38	34, 35, 36, 37	mulne0d 10679	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	14	adantrr 753	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
40	22	adantrr 753	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	39, 40	nnmulcld 11068	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		pcdiv 15557	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
43	30, 33, 38, 41, 42	syl121anc 1331	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
44		pcmul 15556	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
45	30, 31, 36, 32, 37, 44	syl122anc 1335	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
46	39	nnzd 11481	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
47	16	adantrr 753	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
48	40	nnzd 11481	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49	24	adantrr 753	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
50		pcmul 15556	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
51	30, 46, 47, 48, 49, 50	syl122anc 1335	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	45, 51	oveq12d 6668	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53		pczcl 15553	. . . . . . . . . . . . . . . 16 $/\$ $/\$
54	30, 31, 36, 53	syl12anc 1324	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	54	nn0cnd 11353	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56		pczcl 15553	. . . . . . . . . . . . . . . 16 $/\$ $/\$
57	30, 32, 37, 56	syl12anc 1324	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58	57	nn0cnd 11353	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
59	30, 39	pccld 15555	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60	59	nn0cnd 11353	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
61	30, 40	pccld 15555	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62	61	nn0cnd 11353	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63	55, 58, 60, 62	addsub4d 10439	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
64	43, 52, 63	3eqtrd 2660	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65	15	adantrr 753	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
66	23	adantrr 753	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
67	34, 65, 35, 66, 47, 49	divmuldivd 10842	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
68	67	oveq2d 6666	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
69		pcdiv 15557	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
70	30, 31, 36, 39, 69	syl121anc 1331	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
71		pcdiv 15557	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
72	30, 32, 37, 40, 71	syl121anc 1331	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73	70, 72	oveq12d 6668	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
74	64, 68, 73	3eqtr4d 2666	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
75	74	expr 643	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
76	21, 29, 75	syl2and 500	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
77		neeq1 2856	. . . . . . . . . . 11
78		neeq1 2856	. . . . . . . . . . 11
79	77, 78	bi2anan9 917	. . . . . . . . . 10 $/\$ $/\$ $/\$
80		oveq12 6659	. . . . . . . . . . . 12 $/\$
81	80	oveq2d 6666	. . . . . . . . . . 11 $/\$
82		oveq2 6658	. . . . . . . . . . . 12
83		oveq2 6658	. . . . . . . . . . . 12
84	82, 83	oveqan12d 6669	. . . . . . . . . . 11 $/\$
85	81, 84	eqeq12d 2637	. . . . . . . . . 10 $/\$
86	79, 85	imbi12d 334	. . . . . . . . 9 $/\$ $/\$ $/\$
87	76, 86	syl5ibrcom 237	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
88	13, 87	sylanl1 682	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
89	12, 88	mpid 44	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
90	89	rexlimdvva 3038	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
91	8, 90	syl5bir 233	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
92	91	rexlimdvva 3038	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
93	7, 92	syl5bir 233	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
94	3, 6, 93	mp2and 715	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

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

wceq 1483

wcel 1990

wne 2794

wrex 2913 (class class class)co 6650

cc 9934

cc0 9936

caddc 9939

cmul 9941

cmin 10266

cdiv 10684

cn 11020

cn0 11292

cz 11377

cq 11788

cprime 15385

cpc 15541

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-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

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-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-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-er 7742 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-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 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

This theorem is referenced by: pcqdiv 15562 pcexp 15564 pcaddlem 15592 sylow1lem1 18013 padicabv 25319

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