Step |
Hyp |
Ref |
Expression |
1 |
|
nnnn0 |
|- ( A e. NN -> A e. NN0 ) |
2 |
|
nnnn0 |
|- ( B e. NN -> B e. NN0 ) |
3 |
|
pc11 |
|- ( ( A e. NN0 /\ B e. NN0 ) -> ( A = B <-> A. p e. Prime ( p pCnt A ) = ( p pCnt B ) ) ) |
4 |
1 2 3
|
syl2an |
|- ( ( A e. NN /\ B e. NN ) -> ( A = B <-> A. p e. Prime ( p pCnt A ) = ( p pCnt B ) ) ) |
5 |
4
|
ad2ant2r |
|- ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) -> ( A = B <-> A. p e. Prime ( p pCnt A ) = ( p pCnt B ) ) ) |
6 |
|
eleq1 |
|- ( ( p pCnt A ) = ( p pCnt B ) -> ( ( p pCnt A ) e. NN <-> ( p pCnt B ) e. NN ) ) |
7 |
|
dfbi3 |
|- ( ( ( p pCnt A ) e. NN <-> ( p pCnt B ) e. NN ) <-> ( ( ( p pCnt A ) e. NN /\ ( p pCnt B ) e. NN ) \/ ( -. ( p pCnt A ) e. NN /\ -. ( p pCnt B ) e. NN ) ) ) |
8 |
|
sqfpc |
|- ( ( A e. NN /\ ( mmu ` A ) =/= 0 /\ p e. Prime ) -> ( p pCnt A ) <_ 1 ) |
9 |
8
|
ad4ant124 |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( p pCnt A ) <_ 1 ) |
10 |
|
nnle1eq1 |
|- ( ( p pCnt A ) e. NN -> ( ( p pCnt A ) <_ 1 <-> ( p pCnt A ) = 1 ) ) |
11 |
9 10
|
syl5ibcom |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( ( p pCnt A ) e. NN -> ( p pCnt A ) = 1 ) ) |
12 |
|
simprl |
|- ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) -> B e. NN ) |
13 |
12
|
adantr |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> B e. NN ) |
14 |
|
simplrr |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( mmu ` B ) =/= 0 ) |
15 |
|
simpr |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> p e. Prime ) |
16 |
|
sqfpc |
|- ( ( B e. NN /\ ( mmu ` B ) =/= 0 /\ p e. Prime ) -> ( p pCnt B ) <_ 1 ) |
17 |
13 14 15 16
|
syl3anc |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( p pCnt B ) <_ 1 ) |
18 |
|
nnle1eq1 |
|- ( ( p pCnt B ) e. NN -> ( ( p pCnt B ) <_ 1 <-> ( p pCnt B ) = 1 ) ) |
19 |
17 18
|
syl5ibcom |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( ( p pCnt B ) e. NN -> ( p pCnt B ) = 1 ) ) |
20 |
11 19
|
anim12d |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( ( ( p pCnt A ) e. NN /\ ( p pCnt B ) e. NN ) -> ( ( p pCnt A ) = 1 /\ ( p pCnt B ) = 1 ) ) ) |
21 |
|
eqtr3 |
|- ( ( ( p pCnt A ) = 1 /\ ( p pCnt B ) = 1 ) -> ( p pCnt A ) = ( p pCnt B ) ) |
22 |
20 21
|
syl6 |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( ( ( p pCnt A ) e. NN /\ ( p pCnt B ) e. NN ) -> ( p pCnt A ) = ( p pCnt B ) ) ) |
23 |
|
id |
|- ( p e. Prime -> p e. Prime ) |
24 |
|
simpll |
|- ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) -> A e. NN ) |
25 |
|
pccl |
|- ( ( p e. Prime /\ A e. NN ) -> ( p pCnt A ) e. NN0 ) |
26 |
23 24 25
|
syl2anr |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( p pCnt A ) e. NN0 ) |
27 |
|
elnn0 |
|- ( ( p pCnt A ) e. NN0 <-> ( ( p pCnt A ) e. NN \/ ( p pCnt A ) = 0 ) ) |
28 |
26 27
|
sylib |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( ( p pCnt A ) e. NN \/ ( p pCnt A ) = 0 ) ) |
29 |
28
|
ord |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( -. ( p pCnt A ) e. NN -> ( p pCnt A ) = 0 ) ) |
30 |
|
pccl |
|- ( ( p e. Prime /\ B e. NN ) -> ( p pCnt B ) e. NN0 ) |
31 |
23 12 30
|
syl2anr |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( p pCnt B ) e. NN0 ) |
32 |
|
elnn0 |
|- ( ( p pCnt B ) e. NN0 <-> ( ( p pCnt B ) e. NN \/ ( p pCnt B ) = 0 ) ) |
33 |
31 32
|
sylib |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( ( p pCnt B ) e. NN \/ ( p pCnt B ) = 0 ) ) |
34 |
33
|
ord |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( -. ( p pCnt B ) e. NN -> ( p pCnt B ) = 0 ) ) |
35 |
29 34
|
anim12d |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( ( -. ( p pCnt A ) e. NN /\ -. ( p pCnt B ) e. NN ) -> ( ( p pCnt A ) = 0 /\ ( p pCnt B ) = 0 ) ) ) |
36 |
|
eqtr3 |
|- ( ( ( p pCnt A ) = 0 /\ ( p pCnt B ) = 0 ) -> ( p pCnt A ) = ( p pCnt B ) ) |
37 |
35 36
|
syl6 |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( ( -. ( p pCnt A ) e. NN /\ -. ( p pCnt B ) e. NN ) -> ( p pCnt A ) = ( p pCnt B ) ) ) |
38 |
22 37
|
jaod |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( ( ( ( p pCnt A ) e. NN /\ ( p pCnt B ) e. NN ) \/ ( -. ( p pCnt A ) e. NN /\ -. ( p pCnt B ) e. NN ) ) -> ( p pCnt A ) = ( p pCnt B ) ) ) |
39 |
7 38
|
syl5bi |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( ( ( p pCnt A ) e. NN <-> ( p pCnt B ) e. NN ) -> ( p pCnt A ) = ( p pCnt B ) ) ) |
40 |
6 39
|
impbid2 |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( ( p pCnt A ) = ( p pCnt B ) <-> ( ( p pCnt A ) e. NN <-> ( p pCnt B ) e. NN ) ) ) |
41 |
|
pcelnn |
|- ( ( p e. Prime /\ A e. NN ) -> ( ( p pCnt A ) e. NN <-> p || A ) ) |
42 |
23 24 41
|
syl2anr |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( ( p pCnt A ) e. NN <-> p || A ) ) |
43 |
|
pcelnn |
|- ( ( p e. Prime /\ B e. NN ) -> ( ( p pCnt B ) e. NN <-> p || B ) ) |
44 |
23 12 43
|
syl2anr |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( ( p pCnt B ) e. NN <-> p || B ) ) |
45 |
42 44
|
bibi12d |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( ( ( p pCnt A ) e. NN <-> ( p pCnt B ) e. NN ) <-> ( p || A <-> p || B ) ) ) |
46 |
40 45
|
bitrd |
|- ( ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) /\ p e. Prime ) -> ( ( p pCnt A ) = ( p pCnt B ) <-> ( p || A <-> p || B ) ) ) |
47 |
46
|
ralbidva |
|- ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) -> ( A. p e. Prime ( p pCnt A ) = ( p pCnt B ) <-> A. p e. Prime ( p || A <-> p || B ) ) ) |
48 |
5 47
|
bitrd |
|- ( ( ( A e. NN /\ ( mmu ` A ) =/= 0 ) /\ ( B e. NN /\ ( mmu ` B ) =/= 0 ) ) -> ( A = B <-> A. p e. Prime ( p || A <-> p || B ) ) ) |