Theorem pcpremul 12991

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 3323	. . . . . 6 |- { n e. NN0 | ( P ^ n ) || ( M x. N ) } C_ NN0
2		nn0ssz 9595	. . . . . 6 |- NN0 C_ ZZ
3	1, 2	sstri 3247	. . . . 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 12828	. . . . . 6 |- ( P e. Prime -> P e. ( ZZ>= ` 2 ) )
6	5	3ad2ant1 1045	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. ( ZZ>= ` 2 ) )
7		zmulcl 9631	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M x. N ) e. ZZ )
8	7	ad2ant2r 509	. . . . . 6 |- ( ( ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M x. N ) e. ZZ )
9	8	3adant1 1042	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M x. N ) e. ZZ )
10		simp2l 1050	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> M e. ZZ )
11	10	zcnd 9701	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> M e. CC )
12		simp3l 1052	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> N e. ZZ )
13	12	zcnd 9701	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> N e. CC )
14		simp2r 1051	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> M =/= 0 )
15		0zd 9589	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> 0 e. ZZ )
16		zapne 9652	. . . . . . . . 9 |- ( ( M e. ZZ /\ 0 e. ZZ ) -> ( M # 0 <-> M =/= 0 ) )
17	10, 15, 16	syl2anc 411	. . . . . . . 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 1053	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> N =/= 0 )
20		zapne 9652	. . . . . . . . 9 |- ( ( N e. ZZ /\ 0 e. ZZ ) -> ( N # 0 <-> N =/= 0 ) )
21	12, 15, 20	syl2anc 411	. . . . . . . 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 8932	. . . . . 6 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M x. N ) # 0 )
24		zapne 9652	. . . . . . 7 |- ( ( ( M x. N ) e. ZZ /\ 0 e. ZZ ) -> ( ( M x. N ) # 0 <-> ( M x. N ) =/= 0 ) )
25	9, 15, 24	syl2anc 411	. . . . . 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 2232	. . . . . 6 |- { n e. NN0 | ( P ^ n ) || ( M x. N ) } = { n e. NN0 | ( P ^ n ) || ( M x. N ) }
28	27	pclemdc 12986	. . . . 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 1272	. . . 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 12985	. . . . 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 1272	. . . 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 6058	. . . . . . 7 |- ( x = ( S + T ) -> ( P ^ x ) = ( P ^ ( S + T ) ) )
33	32	breq1d 4119	. . . . . 6 |- ( x = ( S + T ) -> ( ( P ^ x ) || ( M x. N ) <-> ( P ^ ( S + T ) ) || ( M x. N ) ) )
34		eqid 2232	. . . . . . . . . 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 12987	. . . . . . . . 9 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( M e. ZZ /\ M =/= 0 ) ) -> ( S e. NN0 /\ ( P ^ S ) || M ) )
37	6, 10, 14, 36	syl12anc 1272	. . . . . . . 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 2232	. . . . . . . . . 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 12987	. . . . . . . . 9 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( T e. NN0 /\ ( P ^ T ) || N ) )
42	6, 12, 19, 41	syl12anc 1272	. . . . . . . 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 9557	. . . . . 6 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + T ) e. NN0 )
45		prmnn 12807	. . . . . . . . . 10 |- ( P e. Prime -> P e. NN )
46	45	3ad2ant1 1045	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. NN )
47	46, 44	nnexpcld 11057	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + T ) ) e. NN )
48	47	nnzd 9699	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + T ) ) e. ZZ )
49	46, 43	nnexpcld 11057	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ T ) e. NN )
50	49	nnzd 9699	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ T ) e. ZZ )
51	10, 50	zmulcld 9706	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M x. ( P ^ T ) ) e. ZZ )
52	46	nncnd 9251	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. CC )
53	52, 43, 38	expaddd 11037	. . . . . . . 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 11057	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) e. NN )
56	55	nnzd 9699	. . . . . . . . . 10 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) e. ZZ )
57		dvdsmulc 12505	. . . . . . . . . 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 1274	. . . . . . . . 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 4131	. . . . . . 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 12504	. . . . . . . . 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 1274	. . . . . . . 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 12516	. . . . . 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 2975	. . . . 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 6058	. . . . . . 7 |- ( x = n -> ( P ^ x ) = ( P ^ n ) )
68	67	breq1d 4119	. . . . . 6 |- ( x = n -> ( ( P ^ x ) || ( M x. N ) <-> ( P ^ n ) || ( M x. N ) ) )
69	68	cbvrabv 2812	. . . . 5 |- { x e. NN0 | ( P ^ x ) || ( M x. N ) } = { n e. NN0 | ( P ^ n ) || ( M x. N ) }
70	66, 69	eleqtrdi 2325	. . . 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 10596	. . 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 4151	. 2 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + T ) <_ U )
74	34, 35	pcprendvds2 12989	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( M e. ZZ /\ M =/= 0 ) ) -> -. P || ( M / ( P ^ S ) ) )
75	6, 10, 14, 74	syl12anc 1272	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. P || ( M / ( P ^ S ) ) )
76	39, 40	pcprendvds2 12989	. . . . . 6 |- ( ( P e. ( ZZ>= ` 2 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. P || ( N / ( P ^ T ) ) )
77	6, 12, 19, 76	syl12anc 1272	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. P || ( N / ( P ^ T ) ) )
78		ioran 760	. . . . 5 |- ( -. ( P || ( M / ( P ^ S ) ) \/ P || ( N / ( P ^ T ) ) ) <-> ( -. P || ( M / ( P ^ S ) ) /\ -. P || ( N / ( P ^ T ) ) ) )
79	75, 77, 78	sylanbrc 417	. . . 4 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( P || ( M / ( P ^ S ) ) \/ P || ( N / ( P ^ T ) ) ) )
80		simp1 1024	. . . . 5 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. Prime )
81	55	nnne0d 9282	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) =/= 0 )
82		dvdsval2 12476	. . . . . . 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 1274	. . . . . 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 9282	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ T ) =/= 0 )
86		dvdsval2 12476	. . . . . . 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 1274	. . . . . 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 12843	. . . . 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 1274	. . . 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 680	. . 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 12987	. . . . . . 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 1272	. . . . . 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 9640	. . . . 5 |- ( ( ( S + T ) e. NN0 /\ U e. NN0 ) -> ( ( S + T ) < U <-> ( ( S + T ) + 1 ) <_ U ) )
96	44, 94, 95	syl2anc 411	. . . 4 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( S + T ) < U <-> ( ( S + T ) + 1 ) <_ U ) )
97	46	nnzd 9699	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> P e. ZZ )
98		peano2nn0 9536	. . . . . . . 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 12547	. . . . . . . 8 |- ( ( P e. ZZ /\ ( ( S + T ) + 1 ) e. NN0 /\ U e. ( ZZ>= ` ( ( S + T ) + 1 ) ) ) -> ( P ^ ( ( S + T ) + 1 ) ) || ( P ^ U ) )
101	100	3expia 1232	. . . . . . 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 411	. . . . . 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 11057	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( ( S + T ) + 1 ) ) e. NN )
105	104	nnzd 9699	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( ( S + T ) + 1 ) ) e. ZZ )
106	46, 94	nnexpcld 11057	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ U ) e. NN )
107	106	nnzd 9699	. . . . . . . 8 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ U ) e. ZZ )
108		dvdstr 12514	. . . . . . . 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 1274	. . . . . . 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 428	. . . . . 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 9698	. . . . . 6 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( ( S + T ) + 1 ) e. ZZ )
113	94	nn0zd 9698	. . . . . 6 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> U e. ZZ )
114		eluz 9867	. . . . . 6 |- ( ( ( ( S + T ) + 1 ) e. ZZ /\ U e. ZZ ) -> ( U e. ( ZZ>= ` ( ( S + T ) + 1 ) ) <-> ( ( S + T ) + 1 ) <_ U ) )
115	112, 113, 114	syl2anc 411	. . . . 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 11036	. . . . . . 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 8294	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( M x. N ) e. CC )
118	47	nncnd 9251	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + T ) ) e. CC )
119	47	nnap0d 9283	. . . . . . . . 9 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + T ) ) # 0 )
120	117, 118, 119	divcanap2d 9066	. . . . . . . 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 6066	. . . . . . . . . 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 9251	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) e. CC )
123	49	nncnd 9251	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ T ) e. CC )
124	55	nnap0d 9283	. . . . . . . . . . 11 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ S ) # 0 )
125	49	nnap0d 9283	. . . . . . . . . . 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 9106	. . . . . . . . . 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 2268	. . . . . . . . 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 6066	. . . . . . . 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 2267	. . . . . . 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 4122	. . . . . 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 9706	. . . . . . 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 9282	. . . . . . 7 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( P ^ ( S + T ) ) =/= 0 )
133		dvdscmulr 12506	. . . . . . 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 1278	. . . . . 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 669	. 2 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> -. ( S + T ) < U )
139	44	nn0red 9554	. . 3 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + T ) e. RR )
140	94	nn0red 9554	. . 3 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> U e. RR )
141	139, 140	eqleltd 8390	. 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 953	1 |- ( ( P e. Prime /\ ( M e. ZZ /\ M =/= 0 ) /\ ( N e. ZZ /\ N =/= 0 ) ) -> ( S + T ) = U )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2203   =/= wne 2412   A.wral 2520   E.wrex 2521   {crab 2524   C_ wss 3211   class class class wbr 4109   ` cfv 5352  (class class class)co 6050   supcsup 7273   RRcr 8126   0cc0 8127   1c1 8128   + caddc 8130   x. cmul 8132   < clt 8308   <_ cle 8309   # cap 8855   / cdiv 8946   NNcn 9237   2c2 9288   NN0cn0 9496   ZZcz 9577   ZZ>=cuz 9853   ^cexp 10900   || cdvds 12473   Primecprime 12804

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-2o 6648  df-er 6767  df-en 6976  df-sup 7275  df-inf 7276  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-fl 10630  df-mod 10685  df-seqfrec 10810  df-exp 10901  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-dvds 12474  df-gcd 12650  df-prm 12805

This theorem is referenced by:  pceulem  12992  pcmul  12999