Theorem dvds0lem 11539

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 5789	. . . . . . . . 9 |- ( x = K -> ( x x. M ) = ( K x. M ) )
2	1	eqeq1d 2149	. . . . . . . 8 |- ( x = K -> ( ( x x. M ) = N <-> ( K x. M ) = N ) )
3	2	rspcev 2793	. . . . . . 7 |- ( ( K e. ZZ /\ ( K x. M ) = N ) -> E. x e. ZZ ( x x. M ) = N )
4	3	adantl 275	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ ( K x. M ) = N ) ) -> E. x e. ZZ ( x x. M ) = N )
5		divides 11531	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> E. x e. ZZ ( x x. M ) = N ) )
6	5	adantr 274	. . . . . 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 166	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ ( K x. M ) = N ) ) -> M || N )
8	7	expr 373	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ K e. ZZ ) -> ( ( K x. M ) = N -> M || N ) )
9	8	3impa 1177	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( ( K x. M ) = N -> M || N ) )
10	9	3comr 1190	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) = N -> M || N ) )
11	10	imp 123	1 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K x. M ) = N ) -> M || N )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 963   = wceq 1332   e. wcel 1481   E.wrex 2418   class class class wbr 3937  (class class class)co 5782   x. cmul 7649   ZZcz 9078   || cdvds 11529

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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-iota 5096  df-fv 5139  df-ov 5785  df-dvds 11530

This theorem is referenced by:  iddvds  11542  1dvds  11543  dvds0  11544  dvdsmul1  11551  dvdsmul2  11552  divalgmod  11660  oddpwdclemxy  11883  ex-dvds  13113