Theorem mumullem2 24186

Description: Lemma for mumul 24187. The product of two coprime squarefree numbers is squarefree. (Contributed by Mario Carneiro, 3-Oct-2014.)

Assertion

Ref	Expression
mumullem2	|- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> ( mmu ` ( A x. B ) ) =/= 0 )

Proof of Theorem mumullem2

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.26 2904	. . . 4 |- ( A. p e. Prime ( ( p pCnt A ) <_ 1 /\ ( p pCnt B ) <_ 1 ) <-> ( A. p e. Prime ( p pCnt A ) <_ 1 /\ A. p e. Prime ( p pCnt B ) <_ 1 ) )
2		simpr 468	. . . . . . . . . 10 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> p e. Prime )
3		simpl1 1033	. . . . . . . . . 10 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> A e. NN )
4	2, 3	pccld 14879	. . . . . . . . 9 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( p pCnt A ) e. NN0 )
5	4	nn0red 10950	. . . . . . . 8 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( p pCnt A ) e. RR )
6		simpl2 1034	. . . . . . . . . 10 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> B e. NN )
7	2, 6	pccld 14879	. . . . . . . . 9 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( p pCnt B ) e. NN0 )
8	7	nn0red 10950	. . . . . . . 8 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( p pCnt B ) e. RR )
9		1red 9676	. . . . . . . 8 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> 1 e. RR )
10		le2add 10117	. . . . . . . 8 |- ( ( ( ( p pCnt A ) e. RR /\ ( p pCnt B ) e. RR ) /\ ( 1 e. RR /\ 1 e. RR ) ) -> ( ( ( p pCnt A ) <_ 1 /\ ( p pCnt B ) <_ 1 ) -> ( ( p pCnt A ) + ( p pCnt B ) ) <_ ( 1 + 1 ) ) )
11	5, 8, 9, 9, 10	syl22anc 1293	. . . . . . 7 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( ( p pCnt A ) <_ 1 /\ ( p pCnt B ) <_ 1 ) -> ( ( p pCnt A ) + ( p pCnt B ) ) <_ ( 1 + 1 ) ) )
12		ax-1ne0 9626	. . . . . . . . . . . 12 |- 1 =/= 0
13		simpl3 1035	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( A gcd B ) = 1 )
14	13	oveq2d 6324	. . . . . . . . . . . . . . 15 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( p pCnt ( A gcd B ) ) = ( p pCnt 1 ) )
15	3	nnzd 11062	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> A e. ZZ )
16	6	nnzd 11062	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> B e. ZZ )
17		pcgcd 14906	. . . . . . . . . . . . . . . 16 |- ( ( p e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( p pCnt ( A gcd B ) ) = if ( ( p pCnt A ) <_ ( p pCnt B ) , ( p pCnt A ) , ( p pCnt B ) ) )
18	2, 15, 16, 17	syl3anc 1292	. . . . . . . . . . . . . . 15 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( p pCnt ( A gcd B ) ) = if ( ( p pCnt A ) <_ ( p pCnt B ) , ( p pCnt A ) , ( p pCnt B ) ) )
19		pc1 14884	. . . . . . . . . . . . . . . 16 |- ( p e. Prime -> ( p pCnt 1 ) = 0 )
20	19	adantl 473	. . . . . . . . . . . . . . 15 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( p pCnt 1 ) = 0 )
21	14, 18, 20	3eqtr3d 2513	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> if ( ( p pCnt A ) <_ ( p pCnt B ) , ( p pCnt A ) , ( p pCnt B ) ) = 0 )
22		ifid 3909	. . . . . . . . . . . . . . . 16 |- if ( ( p pCnt A ) <_ ( p pCnt B ) , 1 , 1 ) = 1
23		ifeq12 3889	. . . . . . . . . . . . . . . 16 |- ( ( 1 = ( p pCnt A ) /\ 1 = ( p pCnt B ) ) -> if ( ( p pCnt A ) <_ ( p pCnt B ) , 1 , 1 ) = if ( ( p pCnt A ) <_ ( p pCnt B ) , ( p pCnt A ) , ( p pCnt B ) ) )
24	22, 23	syl5eqr 2519	. . . . . . . . . . . . . . 15 |- ( ( 1 = ( p pCnt A ) /\ 1 = ( p pCnt B ) ) -> 1 = if ( ( p pCnt A ) <_ ( p pCnt B ) , ( p pCnt A ) , ( p pCnt B ) ) )
25	24	eqeq1d 2473	. . . . . . . . . . . . . 14 |- ( ( 1 = ( p pCnt A ) /\ 1 = ( p pCnt B ) ) -> ( 1 = 0 <-> if ( ( p pCnt A ) <_ ( p pCnt B ) , ( p pCnt A ) , ( p pCnt B ) ) = 0 ) )
26	21, 25	syl5ibrcom 230	. . . . . . . . . . . . 13 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( 1 = ( p pCnt A ) /\ 1 = ( p pCnt B ) ) -> 1 = 0 ) )
27	26	necon3ad 2656	. . . . . . . . . . . 12 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( 1 =/= 0 -> -. ( 1 = ( p pCnt A ) /\ 1 = ( p pCnt B ) ) ) )
28	12, 27	mpi 20	. . . . . . . . . . 11 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> -. ( 1 = ( p pCnt A ) /\ 1 = ( p pCnt B ) ) )
29		ax-1cn 9615	. . . . . . . . . . . . 13 |- 1 e. CC
30	5	recnd 9687	. . . . . . . . . . . . 13 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( p pCnt A ) e. CC )
31		subeq0 9920	. . . . . . . . . . . . 13 |- ( ( 1 e. CC /\ ( p pCnt A ) e. CC ) -> ( ( 1 - ( p pCnt A ) ) = 0 <-> 1 = ( p pCnt A ) ) )
32	29, 30, 31	sylancr 676	. . . . . . . . . . . 12 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( 1 - ( p pCnt A ) ) = 0 <-> 1 = ( p pCnt A ) ) )
33	8	recnd 9687	. . . . . . . . . . . . 13 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( p pCnt B ) e. CC )
34		subeq0 9920	. . . . . . . . . . . . 13 |- ( ( 1 e. CC /\ ( p pCnt B ) e. CC ) -> ( ( 1 - ( p pCnt B ) ) = 0 <-> 1 = ( p pCnt B ) ) )
35	29, 33, 34	sylancr 676	. . . . . . . . . . . 12 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( 1 - ( p pCnt B ) ) = 0 <-> 1 = ( p pCnt B ) ) )
36	32, 35	anbi12d 725	. . . . . . . . . . 11 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( ( 1 - ( p pCnt A ) ) = 0 /\ ( 1 - ( p pCnt B ) ) = 0 ) <-> ( 1 = ( p pCnt A ) /\ 1 = ( p pCnt B ) ) ) )
37	28, 36	mtbird 308	. . . . . . . . . 10 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> -. ( ( 1 - ( p pCnt A ) ) = 0 /\ ( 1 - ( p pCnt B ) ) = 0 ) )
38	37	adantr 472	. . . . . . . . 9 |- ( ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) /\ ( ( p pCnt A ) <_ 1 /\ ( p pCnt B ) <_ 1 ) ) -> -. ( ( 1 - ( p pCnt A ) ) = 0 /\ ( 1 - ( p pCnt B ) ) = 0 ) )
39		eqcom 2478	. . . . . . . . . . 11 |- ( ( 1 + 1 ) = ( ( p pCnt A ) + ( p pCnt B ) ) <-> ( ( p pCnt A ) + ( p pCnt B ) ) = ( 1 + 1 ) )
40		1re 9660	. . . . . . . . . . . . . . . . . 18 |- 1 e. RR
41	40, 40	readdcli 9674	. . . . . . . . . . . . . . . . 17 |- ( 1 + 1 ) e. RR
42	41	recni 9673	. . . . . . . . . . . . . . . 16 |- ( 1 + 1 ) e. CC
43	4, 7	nn0addcld 10953	. . . . . . . . . . . . . . . . . 18 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( p pCnt A ) + ( p pCnt B ) ) e. NN0 )
44	43	nn0red 10950	. . . . . . . . . . . . . . . . 17 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( p pCnt A ) + ( p pCnt B ) ) e. RR )
45	44	recnd 9687	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( p pCnt A ) + ( p pCnt B ) ) e. CC )
46		subeq0 9920	. . . . . . . . . . . . . . . 16 |- ( ( ( 1 + 1 ) e. CC /\ ( ( p pCnt A ) + ( p pCnt B ) ) e. CC ) -> ( ( ( 1 + 1 ) - ( ( p pCnt A ) + ( p pCnt B ) ) ) = 0 <-> ( 1 + 1 ) = ( ( p pCnt A ) + ( p pCnt B ) ) ) )
47	42, 45, 46	sylancr 676	. . . . . . . . . . . . . . 15 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( ( 1 + 1 ) - ( ( p pCnt A ) + ( p pCnt B ) ) ) = 0 <-> ( 1 + 1 ) = ( ( p pCnt A ) + ( p pCnt B ) ) ) )
48	47, 39	syl6bb 269	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( ( 1 + 1 ) - ( ( p pCnt A ) + ( p pCnt B ) ) ) = 0 <-> ( ( p pCnt A ) + ( p pCnt B ) ) = ( 1 + 1 ) ) )
49	9	recnd 9687	. . . . . . . . . . . . . . . 16 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> 1 e. CC )
50	49, 49, 30, 33	addsub4d 10052	. . . . . . . . . . . . . . 15 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( 1 + 1 ) - ( ( p pCnt A ) + ( p pCnt B ) ) ) = ( ( 1 - ( p pCnt A ) ) + ( 1 - ( p pCnt B ) ) ) )
51	50	eqeq1d 2473	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( ( 1 + 1 ) - ( ( p pCnt A ) + ( p pCnt B ) ) ) = 0 <-> ( ( 1 - ( p pCnt A ) ) + ( 1 - ( p pCnt B ) ) ) = 0 ) )
52	48, 51	bitr3d 263	. . . . . . . . . . . . 13 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( ( p pCnt A ) + ( p pCnt B ) ) = ( 1 + 1 ) <-> ( ( 1 - ( p pCnt A ) ) + ( 1 - ( p pCnt B ) ) ) = 0 ) )
53	52	adantr 472	. . . . . . . . . . . 12 |- ( ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) /\ ( ( p pCnt A ) <_ 1 /\ ( p pCnt B ) <_ 1 ) ) -> ( ( ( p pCnt A ) + ( p pCnt B ) ) = ( 1 + 1 ) <-> ( ( 1 - ( p pCnt A ) ) + ( 1 - ( p pCnt B ) ) ) = 0 ) )
54		subge0 10148	. . . . . . . . . . . . . . . 16 |- ( ( 1 e. RR /\ ( p pCnt A ) e. RR ) -> ( 0 <_ ( 1 - ( p pCnt A ) ) <-> ( p pCnt A ) <_ 1 ) )
55	40, 5, 54	sylancr 676	. . . . . . . . . . . . . . 15 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( 0 <_ ( 1 - ( p pCnt A ) ) <-> ( p pCnt A ) <_ 1 ) )
56		subge0 10148	. . . . . . . . . . . . . . . 16 |- ( ( 1 e. RR /\ ( p pCnt B ) e. RR ) -> ( 0 <_ ( 1 - ( p pCnt B ) ) <-> ( p pCnt B ) <_ 1 ) )
57	40, 8, 56	sylancr 676	. . . . . . . . . . . . . . 15 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( 0 <_ ( 1 - ( p pCnt B ) ) <-> ( p pCnt B ) <_ 1 ) )
58	55, 57	anbi12d 725	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( 0 <_ ( 1 - ( p pCnt A ) ) /\ 0 <_ ( 1 - ( p pCnt B ) ) ) <-> ( ( p pCnt A ) <_ 1 /\ ( p pCnt B ) <_ 1 ) ) )
59		resubcl 9958	. . . . . . . . . . . . . . . 16 |- ( ( 1 e. RR /\ ( p pCnt A ) e. RR ) -> ( 1 - ( p pCnt A ) ) e. RR )
60	40, 5, 59	sylancr 676	. . . . . . . . . . . . . . 15 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( 1 - ( p pCnt A ) ) e. RR )
61		resubcl 9958	. . . . . . . . . . . . . . . 16 |- ( ( 1 e. RR /\ ( p pCnt B ) e. RR ) -> ( 1 - ( p pCnt B ) ) e. RR )
62	40, 8, 61	sylancr 676	. . . . . . . . . . . . . . 15 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( 1 - ( p pCnt B ) ) e. RR )
63		add20 10147	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( 1 - ( p pCnt A ) ) e. RR /\ 0 <_ ( 1 - ( p pCnt A ) ) ) /\ ( ( 1 - ( p pCnt B ) ) e. RR /\ 0 <_ ( 1 - ( p pCnt B ) ) ) ) -> ( ( ( 1 - ( p pCnt A ) ) + ( 1 - ( p pCnt B ) ) ) = 0 <-> ( ( 1 - ( p pCnt A ) ) = 0 /\ ( 1 - ( p pCnt B ) ) = 0 ) ) )
64	63	an4s 842	. . . . . . . . . . . . . . . 16 |- ( ( ( ( 1 - ( p pCnt A ) ) e. RR /\ ( 1 - ( p pCnt B ) ) e. RR ) /\ ( 0 <_ ( 1 - ( p pCnt A ) ) /\ 0 <_ ( 1 - ( p pCnt B ) ) ) ) -> ( ( ( 1 - ( p pCnt A ) ) + ( 1 - ( p pCnt B ) ) ) = 0 <-> ( ( 1 - ( p pCnt A ) ) = 0 /\ ( 1 - ( p pCnt B ) ) = 0 ) ) )
65	64	ex 441	. . . . . . . . . . . . . . 15 |- ( ( ( 1 - ( p pCnt A ) ) e. RR /\ ( 1 - ( p pCnt B ) ) e. RR ) -> ( ( 0 <_ ( 1 - ( p pCnt A ) ) /\ 0 <_ ( 1 - ( p pCnt B ) ) ) -> ( ( ( 1 - ( p pCnt A ) ) + ( 1 - ( p pCnt B ) ) ) = 0 <-> ( ( 1 - ( p pCnt A ) ) = 0 /\ ( 1 - ( p pCnt B ) ) = 0 ) ) ) )
66	60, 62, 65	syl2anc 673	. . . . . . . . . . . . . 14 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( 0 <_ ( 1 - ( p pCnt A ) ) /\ 0 <_ ( 1 - ( p pCnt B ) ) ) -> ( ( ( 1 - ( p pCnt A ) ) + ( 1 - ( p pCnt B ) ) ) = 0 <-> ( ( 1 - ( p pCnt A ) ) = 0 /\ ( 1 - ( p pCnt B ) ) = 0 ) ) ) )
67	58, 66	sylbird 243	. . . . . . . . . . . . 13 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( ( p pCnt A ) <_ 1 /\ ( p pCnt B ) <_ 1 ) -> ( ( ( 1 - ( p pCnt A ) ) + ( 1 - ( p pCnt B ) ) ) = 0 <-> ( ( 1 - ( p pCnt A ) ) = 0 /\ ( 1 - ( p pCnt B ) ) = 0 ) ) ) )
68	67	imp 436	. . . . . . . . . . . 12 |- ( ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) /\ ( ( p pCnt A ) <_ 1 /\ ( p pCnt B ) <_ 1 ) ) -> ( ( ( 1 - ( p pCnt A ) ) + ( 1 - ( p pCnt B ) ) ) = 0 <-> ( ( 1 - ( p pCnt A ) ) = 0 /\ ( 1 - ( p pCnt B ) ) = 0 ) ) )
69	53, 68	bitrd 261	. . . . . . . . . . 11 |- ( ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) /\ ( ( p pCnt A ) <_ 1 /\ ( p pCnt B ) <_ 1 ) ) -> ( ( ( p pCnt A ) + ( p pCnt B ) ) = ( 1 + 1 ) <-> ( ( 1 - ( p pCnt A ) ) = 0 /\ ( 1 - ( p pCnt B ) ) = 0 ) ) )
70	39, 69	syl5bb 265	. . . . . . . . . 10 |- ( ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) /\ ( ( p pCnt A ) <_ 1 /\ ( p pCnt B ) <_ 1 ) ) -> ( ( 1 + 1 ) = ( ( p pCnt A ) + ( p pCnt B ) ) <-> ( ( 1 - ( p pCnt A ) ) = 0 /\ ( 1 - ( p pCnt B ) ) = 0 ) ) )
71	70	necon3abid 2679	. . . . . . . . 9 |- ( ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) /\ ( ( p pCnt A ) <_ 1 /\ ( p pCnt B ) <_ 1 ) ) -> ( ( 1 + 1 ) =/= ( ( p pCnt A ) + ( p pCnt B ) ) <-> -. ( ( 1 - ( p pCnt A ) ) = 0 /\ ( 1 - ( p pCnt B ) ) = 0 ) ) )
72	38, 71	mpbird 240	. . . . . . . 8 |- ( ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) /\ ( ( p pCnt A ) <_ 1 /\ ( p pCnt B ) <_ 1 ) ) -> ( 1 + 1 ) =/= ( ( p pCnt A ) + ( p pCnt B ) ) )
73	72	ex 441	. . . . . . 7 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( ( p pCnt A ) <_ 1 /\ ( p pCnt B ) <_ 1 ) -> ( 1 + 1 ) =/= ( ( p pCnt A ) + ( p pCnt B ) ) ) )
74	11, 73	jcad 542	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( ( p pCnt A ) <_ 1 /\ ( p pCnt B ) <_ 1 ) -> ( ( ( p pCnt A ) + ( p pCnt B ) ) <_ ( 1 + 1 ) /\ ( 1 + 1 ) =/= ( ( p pCnt A ) + ( p pCnt B ) ) ) ) )
75		nnz 10983	. . . . . . . . . . 11 |- ( A e. NN -> A e. ZZ )
76		nnne0 10664	. . . . . . . . . . 11 |- ( A e. NN -> A =/= 0 )
77	75, 76	jca 541	. . . . . . . . . 10 |- ( A e. NN -> ( A e. ZZ /\ A =/= 0 ) )
78	3, 77	syl 17	. . . . . . . . 9 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( A e. ZZ /\ A =/= 0 ) )
79		nnz 10983	. . . . . . . . . . 11 |- ( B e. NN -> B e. ZZ )
80		nnne0 10664	. . . . . . . . . . 11 |- ( B e. NN -> B =/= 0 )
81	79, 80	jca 541	. . . . . . . . . 10 |- ( B e. NN -> ( B e. ZZ /\ B =/= 0 ) )
82	6, 81	syl 17	. . . . . . . . 9 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( B e. ZZ /\ B =/= 0 ) )
83		pcmul 14880	. . . . . . . . 9 |- ( ( p e. Prime /\ ( A e. ZZ /\ A =/= 0 ) /\ ( B e. ZZ /\ B =/= 0 ) ) -> ( p pCnt ( A x. B ) ) = ( ( p pCnt A ) + ( p pCnt B ) ) )
84	2, 78, 82, 83	syl3anc 1292	. . . . . . . 8 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( p pCnt ( A x. B ) ) = ( ( p pCnt A ) + ( p pCnt B ) ) )
85	84	breq1d 4405	. . . . . . 7 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( p pCnt ( A x. B ) ) <_ 1 <-> ( ( p pCnt A ) + ( p pCnt B ) ) <_ 1 ) )
86		1nn0 10909	. . . . . . . 8 |- 1 e. NN0
87		nn0leltp1 11019	. . . . . . . 8 |- ( ( ( ( p pCnt A ) + ( p pCnt B ) ) e. NN0 /\ 1 e. NN0 ) -> ( ( ( p pCnt A ) + ( p pCnt B ) ) <_ 1 <-> ( ( p pCnt A ) + ( p pCnt B ) ) < ( 1 + 1 ) ) )
88	43, 86, 87	sylancl 675	. . . . . . 7 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( ( p pCnt A ) + ( p pCnt B ) ) <_ 1 <-> ( ( p pCnt A ) + ( p pCnt B ) ) < ( 1 + 1 ) ) )
89		ltlen 9753	. . . . . . . 8 |- ( ( ( ( p pCnt A ) + ( p pCnt B ) ) e. RR /\ ( 1 + 1 ) e. RR ) -> ( ( ( p pCnt A ) + ( p pCnt B ) ) < ( 1 + 1 ) <-> ( ( ( p pCnt A ) + ( p pCnt B ) ) <_ ( 1 + 1 ) /\ ( 1 + 1 ) =/= ( ( p pCnt A ) + ( p pCnt B ) ) ) ) )
90	44, 41, 89	sylancl 675	. . . . . . 7 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( ( p pCnt A ) + ( p pCnt B ) ) < ( 1 + 1 ) <-> ( ( ( p pCnt A ) + ( p pCnt B ) ) <_ ( 1 + 1 ) /\ ( 1 + 1 ) =/= ( ( p pCnt A ) + ( p pCnt B ) ) ) ) )
91	85, 88, 90	3bitrd 287	. . . . . 6 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( p pCnt ( A x. B ) ) <_ 1 <-> ( ( ( p pCnt A ) + ( p pCnt B ) ) <_ ( 1 + 1 ) /\ ( 1 + 1 ) =/= ( ( p pCnt A ) + ( p pCnt B ) ) ) ) )
92	74, 91	sylibrd 242	. . . . 5 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ p e. Prime ) -> ( ( ( p pCnt A ) <_ 1 /\ ( p pCnt B ) <_ 1 ) -> ( p pCnt ( A x. B ) ) <_ 1 ) )
93	92	ralimdva 2805	. . . 4 |- ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) -> ( A. p e. Prime ( ( p pCnt A ) <_ 1 /\ ( p pCnt B ) <_ 1 ) -> A. p e. Prime ( p pCnt ( A x. B ) ) <_ 1 ) )
94	1, 93	syl5bir 226	. . 3 |- ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) -> ( ( A. p e. Prime ( p pCnt A ) <_ 1 /\ A. p e. Prime ( p pCnt B ) <_ 1 ) -> A. p e. Prime ( p pCnt ( A x. B ) ) <_ 1 ) )
95		issqf 24142	. . . . 5 |- ( A e. NN -> ( ( mmu ` A ) =/= 0 <-> A. p e. Prime ( p pCnt A ) <_ 1 ) )
96		issqf 24142	. . . . 5 |- ( B e. NN -> ( ( mmu ` B ) =/= 0 <-> A. p e. Prime ( p pCnt B ) <_ 1 ) )
97	95, 96	bi2anan9 890	. . . 4 |- ( ( A e. NN /\ B e. NN ) -> ( ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) <-> ( A. p e. Prime ( p pCnt A ) <_ 1 /\ A. p e. Prime ( p pCnt B ) <_ 1 ) ) )
98	97	3adant3 1050	. . 3 |- ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) -> ( ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) <-> ( A. p e. Prime ( p pCnt A ) <_ 1 /\ A. p e. Prime ( p pCnt B ) <_ 1 ) ) )
99		nnmulcl 10654	. . . . 5 |- ( ( A e. NN /\ B e. NN ) -> ( A x. B ) e. NN )
100	99	3adant3 1050	. . . 4 |- ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) -> ( A x. B ) e. NN )
101		issqf 24142	. . . 4 |- ( ( A x. B ) e. NN -> ( ( mmu ` ( A x. B ) ) =/= 0 <-> A. p e. Prime ( p pCnt ( A x. B ) ) <_ 1 ) )
102	100, 101	syl 17	. . 3 |- ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) -> ( ( mmu ` ( A x. B ) ) =/= 0 <-> A. p e. Prime ( p pCnt ( A x. B ) ) <_ 1 ) )
103	94, 98, 102	3imtr4d 276	. 2 |- ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) -> ( ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) -> ( mmu ` ( A x. B ) ) =/= 0 ) )
104	103	imp 436	1 |- ( ( ( A e. NN /\ B e. NN /\ ( A gcd B ) = 1 ) /\ ( ( mmu ` A ) =/= 0 /\ ( mmu ` B ) =/= 0 ) ) -> ( mmu ` ( A x. B ) ) =/= 0 )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376   /\ w3a 1007   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   ifcif 3872   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558   + caddc 9560   x. cmul 9562   < clt 9693   <_ cle 9694   - cmin 9880   NNcn 10631   NN0cn0 10893   ZZcz 10961   gcd cgcd 14547   Primecprime 14701   pCnt cpc 14865   mmucmu 24100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-sup 7974  df-inf 7975  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-n0 10894  df-z 10962  df-uz 11183  df-q 11288  df-rp 11326  df-fz 11811  df-fl 12061  df-mod 12130  df-seq 12252  df-exp 12311  df-hash 12554  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-dvds 14383  df-gcd 14548  df-prm 14702  df-pc 14866  df-mu 24106

This theorem is referenced by:  mumul  24187