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Theorem pcqmul 12257

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 1018	. . 3 $/\$ $/\$ $/\$ $/\$
2		elq 9581	. . 3
3	1, 2	sylib 121	. 2 $/\$ $/\$ $/\$ $/\$
4		simp3l 1020	. . 3 $/\$ $/\$ $/\$ $/\$
5		elq 9581	. . 3
6	4, 5	sylib 121	. 2 $/\$ $/\$ $/\$ $/\$
7		reeanv 2639	. . 3 $/\$ $/\$
8		reeanv 2639	. . . . 5 $/\$ $/\$
9		simp2r 1019	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
10		simp3r 1021	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
11	9, 10	jca 304	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
12	11	ad2antrr 485	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
13		simp1 992	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
14		simprl 526	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
15	14	nncnd 8892	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
16	14	nnap0d 8924	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ #
17	15, 16	div0apd 8704	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
18		oveq1 5860	. . . . . . . . . . . . 13
19	18	eqeq1d 2179	. . . . . . . . . . . 12
20	17, 19	syl5ibrcom 156	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
21	20	necon3d 2384	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
22		simprr 527	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
23	22	nncnd 8892	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
24	22	nnap0d 8924	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ #
25	23, 24	div0apd 8704	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
26		oveq1 5860	. . . . . . . . . . . . 13
27	26	eqeq1d 2179	. . . . . . . . . . . 12
28	25, 27	syl5ibrcom 156	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
29	28	necon3d 2384	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
30		simpll 524	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
31		simplrl 530	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32		simplrr 531	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	31, 32	zmulcld 9340	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	31	zcnd 9335	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35	32	zcnd 9335	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
36		simprrl 534	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37		0zd 9224	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
38		zapne 9286	. . . . . . . . . . . . . . . . . 18 $/\$ #
39	31, 37, 38	syl2anc 409	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
40	36, 39	mpbird 166	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
41		simprrr 535	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		zapne 9286	. . . . . . . . . . . . . . . . . 18 $/\$ #
43	32, 37, 42	syl2anc 409	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
44	41, 43	mpbird 166	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
45	34, 35, 40, 44	mulap0d 8576	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
46		zapne 9286	. . . . . . . . . . . . . . . 16 $/\$ #
47	33, 37, 46	syl2anc 409	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
48	45, 47	mpbid 146	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
49	14	adantrr 476	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
50	22	adantrr 476	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	49, 50	nnmulcld 8927	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52		pcdiv 12256	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
53	30, 33, 48, 51, 52	syl121anc 1238	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54		pcmul 12255	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
55	30, 31, 36, 32, 41, 54	syl122anc 1242	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	49	nnzd 9333	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57	14	nnne0d 8923	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
58	57	adantrr 476	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
59	50	nnzd 9333	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60	22	nnne0d 8923	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
61	60	adantrr 476	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
62		pcmul 12255	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
63	30, 56, 58, 59, 61, 62	syl122anc 1242	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
64	55, 63	oveq12d 5871	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
65		pczcl 12252	. . . . . . . . . . . . . . . 16 $/\$ $/\$
66	30, 31, 36, 65	syl12anc 1231	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
67	66	nn0cnd 9190	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
68		pczcl 12252	. . . . . . . . . . . . . . . 16 $/\$ $/\$
69	30, 32, 41, 68	syl12anc 1231	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
70	69	nn0cnd 9190	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
71	30, 49	pccld 12254	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
72	71	nn0cnd 9190	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
73	30, 50	pccld 12254	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
74	73	nn0cnd 9190	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
75	67, 70, 72, 74	addsub4d 8277	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
76	53, 64, 75	3eqtrd 2207	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
77	15	adantrr 476	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
78	23	adantrr 476	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
79	16	adantrr 476	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
80	24	adantrr 476	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ #
81	34, 77, 35, 78, 79, 80	divmuldivapd 8749	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
82	81	oveq2d 5869	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
83		pcdiv 12256	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
84	30, 31, 36, 49, 83	syl121anc 1238	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
85		pcdiv 12256	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
86	30, 32, 41, 50, 85	syl121anc 1238	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
87	84, 86	oveq12d 5871	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
88	76, 82, 87	3eqtr4d 2213	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
89	88	expr 373	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
90	21, 29, 89	syl2and 293	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
91		neeq1 2353	. . . . . . . . . . 11
92		neeq1 2353	. . . . . . . . . . 11
93	91, 92	bi2anan9 601	. . . . . . . . . 10 $/\$ $/\$ $/\$
94		oveq12 5862	. . . . . . . . . . . 12 $/\$
95	94	oveq2d 5869	. . . . . . . . . . 11 $/\$
96		oveq2 5861	. . . . . . . . . . . 12
97		oveq2 5861	. . . . . . . . . . . 12
98	96, 97	oveqan12d 5872	. . . . . . . . . . 11 $/\$
99	95, 98	eqeq12d 2185	. . . . . . . . . 10 $/\$
100	93, 99	imbi12d 233	. . . . . . . . 9 $/\$ $/\$ $/\$
101	90, 100	syl5ibrcom 156	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
102	13, 101	sylanl1 400	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
103	12, 102	mpid 42	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
104	103	rexlimdvva 2595	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
105	8, 104	syl5bir 152	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
106	105	rexlimdvva 2595	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
107	7, 106	syl5bir 152	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
108	3, 6, 107	mp2and 431	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $/\$ w3a 973

wceq 1348

wcel 2141

wne 2340

wrex 2449 class class class wbr 3989 (class class class)co 5853

cc 7772

cc0 7774

caddc 7777

cmul 7779

cmin 8090 # cap 8500

cdiv 8589

cn 8878

cn0 9135

cz 9212

cq 9578

cprime 12061

cpc 12238

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 df-pc 12239

This theorem is referenced by: pcqdiv 12261 pcexp 12263 pcaddlem 12292

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