Theorem pcpremul 13072

Description: Multiplicative property of the prime count pre-function. Note that the primality of P is essential for this property; ( 4 pCnt 2 ) = 0 but ( 4 pCnt ( 2 x. 2 ) ) = 1 =/= 2 x. ( 4 pCnt 2 ) = 0. Since this is needed to show uniqueness for the real prime count function (over QQ), we don't bother to define it off the primes. (Contributed by Mario Carneiro, 23-Feb-2014.)

Hypotheses

Ref	Expression
pcpremul.1	|- S = sup ( { n e. NN0 | ( P ^ n ) || M } , RR , < )
pcpremul.2	|- T = sup ( { n e. NN0 | ( P ^ n ) || N } , RR , < )
pcpremul.3	|- U = sup ( { n e. NN0 | ( P ^ n ) || ( M x. N ) } , RR , < )

Assertion

Ref	Expression
pcpremul	|- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + T ) = U )

Distinct variable groups:   n, M   n, N   P, n

Allowed substitution hints:   S( n)   T( n)   U( n)

Proof of Theorem pcpremul

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3333	. . . . . 6 |- { n e. NN0 | ( P ^ n ) || ( M x. N ) } C_ NN0
2		nn0ssz 9662	. . . . . 6 |- NN0 C_ ZZ
3	1, 2	sstri 3257	. . . . 5 |- { n e. NN0 | ( P ^ n ) || ( M x. N ) } C_ ZZ
4	3	a1i 9	. . . 4 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> { n e. NN0 | ( P ^ n ) || ( M x. N ) } C_ ZZ )
5		prmuz2 12909	. . . . . 6 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
6	5	3ad2ant1 1049	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. ( ZZ>= ` 2 ) )
7		zmulcl 9698	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
8	7	ad2ant2r 513	. . . . . 6 |- ( ( ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M x. N ) e. ZZ )
9	8	3adant1 1046	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M x. N ) e. ZZ )
10		simp2l 1054	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> M e. ZZ )
11	10	zcnd 9769	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> M e. CC )
12		simp3l 1056	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> N e. ZZ )
13	12	zcnd 9769	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> N e. CC )
14		simp2r 1055	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> M =/= 0 )
15		0zd 9656	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> 0 e. ZZ )
16		zapne 9719	. . . . . . . . 9 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> ( M # 0 <-> M =/= 0 ) )
17	10, 15, 16	syl2anc 415	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M # 0 <-> M =/= 0 ) )
18	14, 17	mpbird 167	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> M # 0 )
19		simp3r 1057	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> N =/= 0 )
20		zapne 9719	. . . . . . . . 9 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N # 0 <-> N =/= 0 ) )
21	12, 15, 20	syl2anc 415	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( N # 0 <-> N =/= 0 ) )
22	19, 21	mpbird 167	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> N # 0 )
23	11, 13, 18, 22	mulap0d 8986	. . . . . 6 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M x. N ) # 0 )
24		zapne 9719	. . . . . . 7 |- ( ( ( M x. N ) e. ZZ /\ 0 e. ZZ ) -> ( ( M x. N ) # 0 <-> ( M x. N ) =/= 0 ) )
25	9, 15, 24	syl2anc 415	. . . . . 6 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( M x. N ) # 0 <-> ( M x. N ) =/= 0 ) )
26	23, 25	mpbid 147	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M x. N ) =/= 0 )
27		eqid 2238	. . . . . 6 |- { n e. NN0 | ( P ^ n ) || ( M x. N ) } = { n e. NN0 | ( P ^ n ) || ( M x. N ) }
28	27	pclemdc 13067	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( ( M x. N ) e. ZZ /\ ( M x. N ) =/= 0 ) ) -> A. x e. ZZ DECID x e. { n e. NN0 | ( P ^ n ) || ( M x. N ) } )
29	6, 9, 26, 28	syl12anc 1276	. . . 4 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> A. x e. ZZ DECID x e. { n e. NN0 | ( P ^ n ) || ( M x. N ) } )
30	27	pclemub 13066	. . . . 5 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( ( M x. N ) e. ZZ /\ ( M x. N ) =/= 0 ) ) -> E. x e. ZZ A. y e. { n e. NN0 | ( P ^ n ) || ( M x. N ) } y <_ x )
31	6, 9, 26, 30	syl12anc 1276	. . . 4 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> E. x e. ZZ A. y e. { n e. NN0 | ( P ^ n ) || ( M x. N ) } y <_ x )
32		oveq2 6093	. . . . . . 7 |- ( x = ( S + T ) -> ( P ^ x ) = ( P ^ ( S + T ) ) )
33	32	breq1d 4140	. . . . . 6 |- ( x = ( S + T ) -> ( ( P ^ x ) || ( M x. N ) <-> ( P ^ ( S + T ) ) || ( M x. N ) ) )
34		eqid 2238	. . . . . . . . . 10 |- { n e. NN0 | ( P ^ n ) || M } = { n e. NN0 | ( P ^ n ) || M }
35		pcpremul.1	. . . . . . . . . 10 |- S = sup ( { n e. NN0 | ( P ^ n ) || M } , RR , < )
36	34, 35	pcprecl 13068	. . . . . . . . 9 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( M e. ZZ /\ M =/= 0 ) ) -> ( S e. NN0 /\ ( P ^ S ) || M ) )
37	6, 10, 14, 36	syl12anc 1276	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S e. NN0 /\ ( P ^ S ) || M ) )
38	37	simpld 112	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. NN0 )
39		eqid 2238	. . . . . . . . . 10 |- { n e. NN0 | ( P ^ n ) || N } = { n e. NN0 | ( P ^ n ) || N }
40		pcpremul.2	. . . . . . . . . 10 |- T = sup ( { n e. NN0 | ( P ^ n ) || N } , RR , < )
41	39, 40	pcprecl 13068	. . . . . . . . 9 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( T e. NN0 /\ ( P ^ T ) || N ) )
42	6, 12, 19, 41	syl12anc 1276	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( T e. NN0 /\ ( P ^ T ) || N ) )
43	42	simpld 112	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> T e. NN0 )
44	38, 43	nn0addcld 9624	. . . . . 6 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + T ) e. NN0 )
45		prmnn 12888	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
46	45	3ad2ant1 1049	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. NN )
47	46, 44	nnexpcld 11133	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + T ) ) e. NN )
48	47	nnzd 9767	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + T ) ) e. ZZ )
49	46, 43	nnexpcld 11133	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ T ) e. NN )
50	49	nnzd 9767	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ T ) e. ZZ )
51	10, 50	zmulcld 9774	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M x. ( P ^ T ) ) e. ZZ )
52	46	nncnd 9318	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. CC )
53	52, 43, 38	expaddd 11113	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + T ) ) = ( ( P ^ S ) x. ( P ^ T ) ) )
54	37	simprd 114	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) || M )
55	46, 38	nnexpcld 11133	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) e. NN )
56	55	nnzd 9767	. . . . . . . . . 10 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) e. ZZ )
57		dvdsmulc 12586	. . . . . . . . . 10 |- ( ( ( P ^ S ) e. ZZ /\ M e. ZZ /\ ( P ^ T ) e. ZZ ) -> ( ( P ^ S ) || M -> ( ( P ^ S ) x. ( P ^ T ) ) || ( M x. ( P ^ T ) ) ) )
58	56, 10, 50, 57	syl3anc 1278	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ S ) || M -> ( ( P ^ S ) x. ( P ^ T ) ) || ( M x. ( P ^ T ) ) ) )
59	54, 58	mpd 13	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ S ) x. ( P ^ T ) ) || ( M x. ( P ^ T ) ) )
60	53, 59	eqbrtrd 4152	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + T ) ) || ( M x. ( P ^ T ) ) )
61	42	simprd 114	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ T ) || N )
62		dvdscmul 12585	. . . . . . . . 9 |- ( ( ( P ^ T ) e. ZZ /\ N e. ZZ /\ M e. ZZ ) -> ( ( P ^ T ) || N -> ( M x. ( P ^ T ) ) || ( M x. N ) ) )
63	50, 12, 10, 62	syl3anc 1278	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ T ) || N -> ( M x. ( P ^ T ) ) || ( M x. N ) ) )
64	61, 63	mpd 13	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M x. ( P ^ T ) ) || ( M x. N ) )
65	48, 51, 9, 60, 64	dvdstrd 12597	. . . . . 6 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + T ) ) || ( M x. N ) )
66	33, 44, 65	elrabd 2984	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + T ) e. { x e. NN0 | ( P ^ x ) || ( M x. N ) } )
67		oveq2 6093	. . . . . . 7 |- ( x = n -> ( P ^ x ) = ( P ^ n ) )
68	67	breq1d 4140	. . . . . 6 |- ( x = n -> ( ( P ^ x ) || ( M x. N ) <-> ( P ^ n ) || ( M x. N ) ) )
69	68	cbvrabv 2820	. . . . 5 |- { x e. NN0 | ( P ^ x ) || ( M x. N ) } = { n e. NN0 | ( P ^ n ) || ( M x. N ) }
70	66, 69	eleqtrdi 2331	. . . 4 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + T ) e. { n e. NN0 | ( P ^ n ) || ( M x. N ) } )
71	4, 29, 31, 70	suprzubdc 10671	. . 3 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + T ) <_ sup ( { n e. NN0 | ( P ^ n ) || ( M x. N ) } , RR , < ) )
72		pcpremul.3	. . 3 |- U = sup ( { n e. NN0 | ( P ^ n ) || ( M x. N ) } , RR , < )
73	71, 72	breqtrrdi 4172	. 2 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + T ) <_ U )
74	34, 35	pcprendvds2 13070	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( M e. ZZ /\ M =/= 0 ) ) -> -. P || ( M / ( P ^ S ) ) )
75	6, 10, 14, 74	syl12anc 1276	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. P || ( M / ( P ^ S ) ) )
76	39, 40	pcprendvds2 13070	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. P || ( N / ( P ^ T ) ) )
77	6, 12, 19, 76	syl12anc 1276	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. P || ( N / ( P ^ T ) ) )
78		ioran 764	. . . . 5 |- ( -. ( P || ( M / ( P ^ S ) ) \/ P || ( N / ( P ^ T ) ) ) <-> ( -. P || ( M / ( P ^ S ) ) /\ -. P || ( N / ( P ^ T ) ) ) )
79	75, 77, 78	sylanbrc 421	. . . 4 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( P || ( M / ( P ^ S ) ) \/ P || ( N / ( P ^ T ) ) ) )
80		simp1 1028	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. Prime )
81	55	nnne0d 9349	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) =/= 0 )
82		dvdsval2 12557	. . . . . . 7 |- ( ( ( P ^ S ) e. ZZ /\ ( P ^ S ) =/= 0 /\ M e. ZZ ) -> ( ( P ^ S ) || M <-> ( M / ( P ^ S ) ) e. ZZ ) )
83	56, 81, 10, 82	syl3anc 1278	. . . . . 6 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ S ) || M <-> ( M / ( P ^ S ) ) e. ZZ ) )
84	54, 83	mpbid 147	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M / ( P ^ S ) ) e. ZZ )
85	49	nnne0d 9349	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ T ) =/= 0 )
86		dvdsval2 12557	. . . . . . 7 |- ( ( ( P ^ T ) e. ZZ /\ ( P ^ T ) =/= 0 /\ N e. ZZ ) -> ( ( P ^ T ) || N <-> ( N / ( P ^ T ) ) e. ZZ ) )
87	50, 85, 12, 86	syl3anc 1278	. . . . . 6 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ T ) || N <-> ( N / ( P ^ T ) ) e. ZZ ) )
88	61, 87	mpbid 147	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( N / ( P ^ T ) ) e. ZZ )
89		euclemma 12924	. . . . 5 |- ( ( P e. Prime /\ ( M / ( P ^ S ) ) e. ZZ /\ ( N / ( P ^ T ) ) e. ZZ ) -> ( P || ( ( M / ( P ^ S ) ) x. ( N / ( P ^ T ) ) ) <-> ( P || ( M / ( P ^ S ) ) \/ P || ( N / ( P ^ T ) ) ) ) )
90	80, 84, 88, 89	syl3anc 1278	. . . 4 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P || ( ( M / ( P ^ S ) ) x. ( N / ( P ^ T ) ) ) <-> ( P || ( M / ( P ^ S ) ) \/ P || ( N / ( P ^ T ) ) ) ) )
91	79, 90	mtbird 684	. . 3 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. P || ( ( M / ( P ^ S ) ) x. ( N / ( P ^ T ) ) ) )
92	27, 72	pcprecl 13068	. . . . . . 7 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( ( M x. N ) e. ZZ /\ ( M x. N ) =/= 0 ) ) -> ( U e. NN0 /\ ( P ^ U ) || ( M x. N ) ) )
93	6, 9, 26, 92	syl12anc 1276	. . . . . 6 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( U e. NN0 /\ ( P ^ U ) || ( M x. N ) ) )
94	93	simpld 112	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> U e. NN0 )
95		nn0ltp1le 9707	. . . . 5 |- ( ( ( S + T ) e. NN0 /\ U e. NN0 ) -> ( ( S + T ) < U <-> ( ( S + T ) + 1 ) <_ U ) )
96	44, 94, 95	syl2anc 415	. . . 4 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( S + T ) < U <-> ( ( S + T ) + 1 ) <_ U ) )
97	46	nnzd 9767	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. ZZ )
98		peano2nn0 9603	. . . . . . . 8 |- ( ( S + T ) e. NN0 -> ( ( S + T ) + 1 ) e. NN0 )
99	44, 98	syl 14	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( S + T ) + 1 ) e. NN0 )
100		dvdsexp 12628	. . . . . . . 8 |- ( ( P e. ZZ /\ ( ( S + T ) + 1 ) e. NN0 /\ U e. ( ZZ>= ` ( ( S + T ) + 1 ) ) ) -> ( P ^ ( ( S + T ) + 1 ) ) || ( P ^ U ) )
101	100	3expia 1236	. . . . . . 7 |- ( ( P e. ZZ /\ ( ( S + T ) + 1 ) e. NN0 ) -> ( U e. ( ZZ>= ` ( ( S + T ) + 1 ) ) -> ( P ^ ( ( S + T ) + 1 ) ) || ( P ^ U ) ) )
102	97, 99, 101	syl2anc 415	. . . . . 6 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( U e. ( ZZ>= ` ( ( S + T ) + 1 ) ) -> ( P ^ ( ( S + T ) + 1 ) ) || ( P ^ U ) ) )
103	93	simprd 114	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ U ) || ( M x. N ) )
104	46, 99	nnexpcld 11133	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( ( S + T ) + 1 ) ) e. NN )
105	104	nnzd 9767	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( ( S + T ) + 1 ) ) e. ZZ )
106	46, 94	nnexpcld 11133	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ U ) e. NN )
107	106	nnzd 9767	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ U ) e. ZZ )
108		dvdstr 12595	. . . . . . . 8 |- ( ( ( P ^ ( ( S + T ) + 1 ) ) e. ZZ /\ ( P ^ U ) e. ZZ /\ ( M x. N ) e. ZZ ) -> ( ( ( P ^ ( ( S + T ) + 1 ) ) || ( P ^ U ) /\ ( P ^ U ) || ( M x. N ) ) -> ( P ^ ( ( S + T ) + 1 ) ) || ( M x. N ) ) )
109	105, 107, 9, 108	syl3anc 1278	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( ( P ^ ( ( S + T ) + 1 ) ) || ( P ^ U ) /\ ( P ^ U ) || ( M x. N ) ) -> ( P ^ ( ( S + T ) + 1 ) ) || ( M x. N ) ) )
110	103, 109	mpan2d 432	. . . . . 6 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ ( ( S + T ) + 1 ) ) || ( P ^ U ) -> ( P ^ ( ( S + T ) + 1 ) ) || ( M x. N ) ) )
111	102, 110	syld 45	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( U e. ( ZZ>= ` ( ( S + T ) + 1 ) ) -> ( P ^ ( ( S + T ) + 1 ) ) || ( M x. N ) ) )
112	99	nn0zd 9766	. . . . . 6 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( S + T ) + 1 ) e. ZZ )
113	94	nn0zd 9766	. . . . . 6 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> U e. ZZ )
114		eluz 9935	. . . . . 6 |- ( ( ( ( S + T ) + 1 ) e. ZZ /\ U e. ZZ ) -> ( U e. ( ZZ>= ` ( ( S + T ) + 1 ) ) <-> ( ( S + T ) + 1 ) <_ U ) )
115	112, 113, 114	syl2anc 415	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( U e. ( ZZ>= ` ( ( S + T ) + 1 ) ) <-> ( ( S + T ) + 1 ) <_ U ) )
116	52, 44	expp1d 11112	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( ( S + T ) + 1 ) ) = ( ( P ^ ( S + T ) ) x. P ) )
117	11, 13	mulcld 8346	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M x. N ) e. CC )
118	47	nncnd 9318	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + T ) ) e. CC )
119	47	nnap0d 9350	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + T ) ) # 0 )
120	117, 118, 119	divcanap2d 9122	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ ( S + T ) ) x. ( ( M x. N ) / ( P ^ ( S + T ) ) ) ) = ( M x. N ) )
121	53	oveq2d 6101	. . . . . . . . . 10 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( M x. N ) / ( P ^ ( S + T ) ) ) = ( ( M x. N ) / ( ( P ^ S ) x. ( P ^ T ) ) ) )
122	55	nncnd 9318	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) e. CC )
123	49	nncnd 9318	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ T ) e. CC )
124	55	nnap0d 9350	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) # 0 )
125	49	nnap0d 9350	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ T ) # 0 )
126	11, 122, 13, 123, 124, 125	divmuldivapd 9162	. . . . . . . . . 10 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( M / ( P ^ S ) ) x. ( N / ( P ^ T ) ) ) = ( ( M x. N ) / ( ( P ^ S ) x. ( P ^ T ) ) ) )
127	121, 126	eqtr4d 2274	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( M x. N ) / ( P ^ ( S + T ) ) ) = ( ( M / ( P ^ S ) ) x. ( N / ( P ^ T ) ) ) )
128	127	oveq2d 6101	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ ( S + T ) ) x. ( ( M x. N ) / ( P ^ ( S + T ) ) ) ) = ( ( P ^ ( S + T ) ) x. ( ( M / ( P ^ S ) ) x. ( N / ( P ^ T ) ) ) ) )
129	120, 128	eqtr3d 2273	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M x. N ) = ( ( P ^ ( S + T ) ) x. ( ( M / ( P ^ S ) ) x. ( N / ( P ^ T ) ) ) ) )
130	116, 129	breq12d 4143	. . . . . 6 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ ( ( S + T ) + 1 ) ) || ( M x. N ) <-> ( ( P ^ ( S + T ) ) x. P ) || ( ( P ^ ( S + T ) ) x. ( ( M / ( P ^ S ) ) x. ( N / ( P ^ T ) ) ) ) ) )
131	84, 88	zmulcld 9774	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( M / ( P ^ S ) ) x. ( N / ( P ^ T ) ) ) e. ZZ )
132	47	nnne0d 9349	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + T ) ) =/= 0 )
133		dvdscmulr 12587	. . . . . . 7 |- ( ( P e. ZZ /\ ( ( M / ( P ^ S ) ) x. ( N / ( P ^ T ) ) ) e. ZZ /\ ( ( P ^ ( S + T ) ) e. ZZ /\ ( P ^ ( S + T ) ) =/= 0 ) ) -> ( ( ( P ^ ( S + T ) ) x. P ) || ( ( P ^ ( S + T ) ) x. ( ( M / ( P ^ S ) ) x. ( N / ( P ^ T ) ) ) ) <-> P || ( ( M / ( P ^ S ) ) x. ( N / ( P ^ T ) ) ) ) )
134	97, 131, 48, 132, 133	syl112anc 1282	. . . . . 6 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( ( P ^ ( S + T ) ) x. P ) || ( ( P ^ ( S + T ) ) x. ( ( M / ( P ^ S ) ) x. ( N / ( P ^ T ) ) ) ) <-> P || ( ( M / ( P ^ S ) ) x. ( N / ( P ^ T ) ) ) ) )
135	130, 134	bitrd 188	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( P ^ ( ( S + T ) + 1 ) ) || ( M x. N ) <-> P || ( ( M / ( P ^ S ) ) x. ( N / ( P ^ T ) ) ) ) )
136	111, 115, 135	3imtr3d 202	. . . 4 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( ( S + T ) + 1 ) <_ U -> P || ( ( M / ( P ^ S ) ) x. ( N / ( P ^ T ) ) ) ) )
137	96, 136	sylbid 150	. . 3 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( S + T ) < U -> P || ( ( M / ( P ^ S ) ) x. ( N / ( P ^ T ) ) ) ) )
138	91, 137	mtod 673	. 2 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( S + T ) < U )
139	44	nn0red 9621	. . 3 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + T ) e. RR )
140	94	nn0red 9621	. . 3 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> U e. RR )
141	139, 140	eqleltd 8443	. 2 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( S + T ) = U <-> ( ( S + T ) <_ U /\ -. ( S + T ) < U ) ) )
142	73, 138, 141	mpbir2and 957	1 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + T ) = U )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   =/= wne 2420   A.wral 2528   E.wrex 2529   {crab 2532   C_ wss 3220   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   supcsup 7322   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   < clt 8360   <_ cle 8361   # cap 8909   / cdiv 9002   NNcn 9304   2c2 9355   NN0cn0 9563   ZZcz 9644   ZZ>=cuz 9921   ^cexp 10975   || cdvds 12554   Primecprime 12885

This proof depends on 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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-2o 6688  df-er 6807  df-en 7023  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-dvds 12555  df-gcd 12731  df-prm 12886

This theorem is used by:  pceulem  13073  pcmul  13080