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Theorem pcqmul 12472

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 1025	. . 3 $/\$ $/\$ $/\$ $/\$
2		elq 9696	. . 3
3	1, 2	sylib 122	. 2 $/\$ $/\$ $/\$ $/\$
4		simp3l 1027	. . 3 $/\$ $/\$ $/\$ $/\$
5		elq 9696	. . 3
6	4, 5	sylib 122	. 2 $/\$ $/\$ $/\$ $/\$
7		reeanv 2667	. . 3 $/\$ $/\$
8		reeanv 2667	. . . . 5 $/\$ $/\$
9		simp2r 1026	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
10		simp3r 1028	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
11	9, 10	jca 306	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
12	11	ad2antrr 488	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
13		simp1 999	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
14		simprl 529	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
15	14	nncnd 9004	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
16	14	nnap0d 9036	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ #
17	15, 16	div0apd 8814	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
18		oveq1 5929	. . . . . . . . . . . . 13
19	18	eqeq1d 2205	. . . . . . . . . . . 12
20	17, 19	syl5ibrcom 157	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
21	20	necon3d 2411	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
22		simprr 531	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
23	22	nncnd 9004	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
24	22	nnap0d 9036	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ #
25	23, 24	div0apd 8814	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
26		oveq1 5929	. . . . . . . . . . . . 13
27	26	eqeq1d 2205	. . . . . . . . . . . 12
28	25, 27	syl5ibrcom 157	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
29	28	necon3d 2411	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
30		simpll 527	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31		simplrl 535	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		simplrr 536	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	31, 32	zmulcld 9454	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	31	zcnd 9449	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	32	zcnd 9449	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36		simprrl 539	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		0zd 9338	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38		zapne 9400	. . . . . . . . . . . . . . . . . 18 $/\$ #
39	31, 37, 38	syl2anc 411	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
40	36, 39	mpbird 167	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
41		simprrr 540	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		zapne 9400	. . . . . . . . . . . . . . . . . 18 $/\$ #
43	32, 37, 42	syl2anc 411	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
44	41, 43	mpbird 167	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
45	34, 35, 40, 44	mulap0d 8685	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
46		zapne 9400	. . . . . . . . . . . . . . . 16 $/\$ #
47	33, 37, 46	syl2anc 411	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
48	45, 47	mpbid 147	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49	14	adantrr 479	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
50	22	adantrr 479	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	49, 50	nnmulcld 9039	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52		pcdiv 12471	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
53	30, 33, 48, 51, 52	syl121anc 1254	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54		pcmul 12470	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
55	30, 31, 36, 32, 41, 54	syl122anc 1258	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	49	nnzd 9447	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57	14	nnne0d 9035	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
58	57	adantrr 479	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
59	50	nnzd 9447	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60	22	nnne0d 9035	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
61	60	adantrr 479	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62		pcmul 12470	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
63	30, 56, 58, 59, 61, 62	syl122anc 1258	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
64	55, 63	oveq12d 5940	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65		pczcl 12467	. . . . . . . . . . . . . . . 16 $/\$ $/\$
66	30, 31, 36, 65	syl12anc 1247	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
67	66	nn0cnd 9304	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
68		pczcl 12467	. . . . . . . . . . . . . . . 16 $/\$ $/\$
69	30, 32, 41, 68	syl12anc 1247	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70	69	nn0cnd 9304	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
71	30, 49	pccld 12469	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
72	71	nn0cnd 9304	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73	30, 50	pccld 12469	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
74	73	nn0cnd 9304	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
75	67, 70, 72, 74	addsub4d 8384	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
76	53, 64, 75	3eqtrd 2233	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
77	15	adantrr 479	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
78	23	adantrr 479	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
79	16	adantrr 479	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
80	24	adantrr 479	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
81	34, 77, 35, 78, 79, 80	divmuldivapd 8859	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
82	81	oveq2d 5938	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
83		pcdiv 12471	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
84	30, 31, 36, 49, 83	syl121anc 1254	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
85		pcdiv 12471	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
86	30, 32, 41, 50, 85	syl121anc 1254	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
87	84, 86	oveq12d 5940	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
88	76, 82, 87	3eqtr4d 2239	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
89	88	expr 375	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
90	21, 29, 89	syl2and 295	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
91		neeq1 2380	. . . . . . . . . . 11
92		neeq1 2380	. . . . . . . . . . 11
93	91, 92	bi2anan9 606	. . . . . . . . . 10 $/\$ $/\$ $/\$
94		oveq12 5931	. . . . . . . . . . . 12 $/\$
95	94	oveq2d 5938	. . . . . . . . . . 11 $/\$
96		oveq2 5930	. . . . . . . . . . . 12
97		oveq2 5930	. . . . . . . . . . . 12
98	96, 97	oveqan12d 5941	. . . . . . . . . . 11 $/\$
99	95, 98	eqeq12d 2211	. . . . . . . . . 10 $/\$
100	93, 99	imbi12d 234	. . . . . . . . 9 $/\$ $/\$ $/\$
101	90, 100	syl5ibrcom 157	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
102	13, 101	sylanl1 402	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
103	12, 102	mpid 42	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
104	103	rexlimdvva 2622	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
105	8, 104	biimtrrid 153	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
106	105	rexlimdvva 2622	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
107	7, 106	biimtrrid 153	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
108	3, 6, 107	mp2and 433	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 980

wceq 1364

wcel 2167

wne 2367

wrex 2476 class class class wbr 4033 (class class class)co 5922

cc 7877

cc0 7879

caddc 7882

cmul 7884

cmin 8197 # cap 8608

cdiv 8699

cn 8990

cn0 9249

cz 9326

cq 9693

cprime 12275

cpc 12453

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-isom 5267 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-frec 6449 df-1o 6474 df-2o 6475 df-er 6592 df-en 6800 df-sup 7050 df-inf 7051 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-fz 10084 df-fzo 10218 df-fl 10360 df-mod 10415 df-seqfrec 10540 df-exp 10631 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-dvds 11953 df-gcd 12121 df-prm 12276 df-pc 12454

This theorem is referenced by: pcqdiv 12476 pcexp 12478 pcaddlem 12508

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