Theorem pcpremul 12247

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 3232	. . . . . 6 |- { n e. NN0 | ( P ^ n ) || ( M x. N ) } C_ NN0
2		nn0ssz 9230	. . . . . 6 |- NN0 C_ ZZ
3	1, 2	sstri 3156	. . . . 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 12085	. . . . . 6 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
6	5	3ad2ant1 1013	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. ( ZZ>= ` 2 ) )
7		zmulcl 9265	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
8	7	ad2ant2r 506	. . . . . 6 |- ( ( ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M x. N ) e. ZZ )
9	8	3adant1 1010	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M x. N ) e. ZZ )
10		simp2l 1018	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> M e. ZZ )
11	10	zcnd 9335	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> M e. CC )
12		simp3l 1020	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> N e. ZZ )
13	12	zcnd 9335	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> N e. CC )
14		simp2r 1019	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> M =/= 0 )
15		0zd 9224	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> 0 e. ZZ )
16		zapne 9286	. . . . . . . . 9 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> ( M # 0 <-> M =/= 0 ) )
17	10, 15, 16	syl2anc 409	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M # 0 <-> M =/= 0 ) )
18	14, 17	mpbird 166	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> M # 0 )
19		simp3r 1021	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> N =/= 0 )
20		zapne 9286	. . . . . . . . 9 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N # 0 <-> N =/= 0 ) )
21	12, 15, 20	syl2anc 409	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( N # 0 <-> N =/= 0 ) )
22	19, 21	mpbird 166	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> N # 0 )
23	11, 13, 18, 22	mulap0d 8576	. . . . . 6 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M x. N ) # 0 )
24		zapne 9286	. . . . . . 7 |- ( ( ( M x. N ) e. ZZ /\ 0 e. ZZ ) -> ( ( M x. N ) # 0 <-> ( M x. N ) =/= 0 ) )
25	9, 15, 24	syl2anc 409	. . . . . 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 146	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M x. N ) =/= 0 )
27		eqid 2170	. . . . . 6 |- { n e. NN0 | ( P ^ n ) || ( M x. N ) } = { n e. NN0 | ( P ^ n ) || ( M x. N ) }
28	27	pclemdc 12242	. . . . 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 1231	. . . 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 12241	. . . . 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 1231	. . . 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 5861	. . . . . . 7 |- ( x = ( S + T ) -> ( P ^ x ) = ( P ^ ( S + T ) ) )
33	32	breq1d 3999	. . . . . 6 |- ( x = ( S + T ) -> ( ( P ^ x ) || ( M x. N ) <-> ( P ^ ( S + T ) ) || ( M x. N ) ) )
34		eqid 2170	. . . . . . . . . 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 12243	. . . . . . . . 9 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( M e. ZZ /\ M =/= 0 ) ) -> ( S e. NN0 /\ ( P ^ S ) || M ) )
37	6, 10, 14, 36	syl12anc 1231	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S e. NN0 /\ ( P ^ S ) || M ) )
38	37	simpld 111	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> S e. NN0 )
39		eqid 2170	. . . . . . . . . 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 12243	. . . . . . . . 9 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( T e. NN0 /\ ( P ^ T ) || N ) )
42	6, 12, 19, 41	syl12anc 1231	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( T e. NN0 /\ ( P ^ T ) || N ) )
43	42	simpld 111	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> T e. NN0 )
44	38, 43	nn0addcld 9192	. . . . . 6 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + T ) e. NN0 )
45		prmnn 12064	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
46	45	3ad2ant1 1013	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. NN )
47	46, 44	nnexpcld 10631	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + T ) ) e. NN )
48	47	nnzd 9333	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + T ) ) e. ZZ )
49	46, 43	nnexpcld 10631	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ T ) e. NN )
50	49	nnzd 9333	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ T ) e. ZZ )
51	10, 50	zmulcld 9340	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M x. ( P ^ T ) ) e. ZZ )
52	46	nncnd 8892	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. CC )
53	52, 43, 38	expaddd 10611	. . . . . . . 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 113	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) || M )
55	46, 38	nnexpcld 10631	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) e. NN )
56	55	nnzd 9333	. . . . . . . . . 10 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) e. ZZ )
57		dvdsmulc 11781	. . . . . . . . . 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 1233	. . . . . . . . 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 4011	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + T ) ) || ( M x. ( P ^ T ) ) )
61	42	simprd 113	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ T ) || N )
62		dvdscmul 11780	. . . . . . . . 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 1233	. . . . . . . 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 11792	. . . . . 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 2888	. . . . 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 5861	. . . . . . 7 |- ( x = n -> ( P ^ x ) = ( P ^ n ) )
68	67	breq1d 3999	. . . . . 6 |- ( x = n -> ( ( P ^ x ) || ( M x. N ) <-> ( P ^ n ) || ( M x. N ) ) )
69	68	cbvrabv 2729	. . . . 5 |- { x e. NN0 | ( P ^ x ) || ( M x. N ) } = { n e. NN0 | ( P ^ n ) || ( M x. N ) }
70	66, 69	eleqtrdi 2263	. . . 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 11907	. . 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 4031	. 2 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + T ) <_ U )
74	34, 35	pcprendvds2 12245	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( M e. ZZ /\ M =/= 0 ) ) -> -. P || ( M / ( P ^ S ) ) )
75	6, 10, 14, 74	syl12anc 1231	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. P || ( M / ( P ^ S ) ) )
76	39, 40	pcprendvds2 12245	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. P || ( N / ( P ^ T ) ) )
77	6, 12, 19, 76	syl12anc 1231	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. P || ( N / ( P ^ T ) ) )
78		ioran 747	. . . . 5 |- ( -. ( P || ( M / ( P ^ S ) ) \/ P || ( N / ( P ^ T ) ) ) <-> ( -. P || ( M / ( P ^ S ) ) /\ -. P || ( N / ( P ^ T ) ) ) )
79	75, 77, 78	sylanbrc 415	. . . 4 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( P || ( M / ( P ^ S ) ) \/ P || ( N / ( P ^ T ) ) ) )
80		simp1 992	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. Prime )
81	55	nnne0d 8923	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) =/= 0 )
82		dvdsval2 11752	. . . . . . 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 1233	. . . . . 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 146	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M / ( P ^ S ) ) e. ZZ )
85	49	nnne0d 8923	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ T ) =/= 0 )
86		dvdsval2 11752	. . . . . . 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 1233	. . . . . 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 146	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( N / ( P ^ T ) ) e. ZZ )
89		euclemma 12100	. . . . 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 1233	. . . 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 668	. . 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 12243	. . . . . . 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 1231	. . . . . 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 111	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> U e. NN0 )
95		nn0ltp1le 9274	. . . . 5 |- ( ( ( S + T ) e. NN0 /\ U e. NN0 ) -> ( ( S + T ) < U <-> ( ( S + T ) + 1 ) <_ U ) )
96	44, 94, 95	syl2anc 409	. . . 4 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( S + T ) < U <-> ( ( S + T ) + 1 ) <_ U ) )
97	46	nnzd 9333	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. ZZ )
98		peano2nn0 9175	. . . . . . . 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 11821	. . . . . . . 8 |- ( ( P e. ZZ /\ ( ( S + T ) + 1 ) e. NN0 /\ U e. ( ZZ>= ` ( ( S + T ) + 1 ) ) ) -> ( P ^ ( ( S + T ) + 1 ) ) || ( P ^ U ) )
101	100	3expia 1200	. . . . . . 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 409	. . . . . 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 113	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ U ) || ( M x. N ) )
104	46, 99	nnexpcld 10631	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( ( S + T ) + 1 ) ) e. NN )
105	104	nnzd 9333	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( ( S + T ) + 1 ) ) e. ZZ )
106	46, 94	nnexpcld 10631	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ U ) e. NN )
107	106	nnzd 9333	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ U ) e. ZZ )
108		dvdstr 11790	. . . . . . . 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 1233	. . . . . . 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 426	. . . . . 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 9332	. . . . . 6 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( S + T ) + 1 ) e. ZZ )
113	94	nn0zd 9332	. . . . . 6 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> U e. ZZ )
114		eluz 9500	. . . . . 6 |- ( ( ( ( S + T ) + 1 ) e. ZZ /\ U e. ZZ ) -> ( U e. ( ZZ>= ` ( ( S + T ) + 1 ) ) <-> ( ( S + T ) + 1 ) <_ U ) )
115	112, 113, 114	syl2anc 409	. . . . 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 10610	. . . . . . 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 7940	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M x. N ) e. CC )
118	47	nncnd 8892	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + T ) ) e. CC )
119	47	nnap0d 8924	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + T ) ) # 0 )
120	117, 118, 119	divcanap2d 8709	. . . . . . . 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 5869	. . . . . . . . . 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 8892	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) e. CC )
123	49	nncnd 8892	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ T ) e. CC )
124	55	nnap0d 8924	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) # 0 )
125	49	nnap0d 8924	. . . . . . . . . . 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 8749	. . . . . . . . . 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 2206	. . . . . . . . 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 5869	. . . . . . . 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 2205	. . . . . . 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 4002	. . . . . 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 9340	. . . . . . 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 8923	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + T ) ) =/= 0 )
133		dvdscmulr 11782	. . . . . . 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 1237	. . . . . 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 187	. . . . 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 201	. . . 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 149	. . 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 658	. 2 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( S + T ) < U )
139	44	nn0red 9189	. . 3 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + T ) e. RR )
140	94	nn0red 9189	. . 3 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> U e. RR )
141	139, 140	eqleltd 8036	. 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 939	1 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + T ) = U )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 703  DECID wdc 829   /\ w3a 973   = wceq 1348   e. wcel 2141   =/= wne 2340   A.wral 2448   E.wrex 2449   {crab 2452   C_ wss 3121   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   supcsup 6959   RRcr 7773   0cc0 7774   1c1 7775   + caddc 7777   x. cmul 7779   < clt 7954   <_ cle 7955   # cap 8500   / cdiv 8589   NNcn 8878   2c2 8929   NN0cn0 9135   ZZcz 9212   ZZ>=cuz 9487   ^cexp 10475   || cdvds 11749   Primecprime 12061

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

This theorem is referenced by:  pceulem  12248  pcmul  12255