Theorem dvds0lem 12307

Description: A lemma to assist theorems of || with no antecedents. (Contributed by Paul Chapman, 21-Mar-2011.)

Assertion

Ref	Expression
dvds0lem	|- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K x. M ) = N ) -> M || N )

Proof of Theorem dvds0lem

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 6007	. . . . . . . . 9 |- ( x = K -> ( x x. M ) = ( K x. M ) )
2	1	eqeq1d 2238	. . . . . . . 8 |- ( x = K -> ( ( x x. M ) = N <-> ( K x. M ) = N ) )
3	2	rspcev 2907	. . . . . . 7 |- ( ( K e. ZZ /\ ( K x. M ) = N ) -> E. x e. ZZ ( x x. M ) = N )
4	3	adantl 277	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ ( K x. M ) = N ) ) -> E. x e. ZZ ( x x. M ) = N )
5		divides 12295	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> E. x e. ZZ ( x x. M ) = N ) )
6	5	adantr 276	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ ( K x. M ) = N ) ) -> ( M || N <-> E. x e. ZZ ( x x. M ) = N ) )
7	4, 6	mpbird 167	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ ( K x. M ) = N ) ) -> M || N )
8	7	expr 375	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ K e. ZZ ) -> ( ( K x. M ) = N -> M || N ) )
9	8	3impa 1218	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( ( K x. M ) = N -> M || N ) )
10	9	3comr 1235	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) = N -> M || N ) )
11	10	imp 124	1 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K x. M ) = N ) -> M || N )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   E.wrex 2509   class class class wbr 4082  (class class class)co 6000   x. cmul 8000   ZZcz 9442   || cdvds 12293

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-iota 5277  df-fv 5325  df-ov 6003  df-dvds 12294

This theorem is referenced by:  iddvds  12310  1dvds  12311  dvds0  12312  dvdsmul1  12319  dvdsmul2  12320  divalgmod  12433  oddpwdclemxy  12686  ex-dvds  16052