Theorem dvds0lem 11492

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 5774	. . . . . . . . 9 |- ( x = K -> ( x x. M ) = ( K x. M ) )
2	1	eqeq1d 2146	. . . . . . . 8 |- ( x = K -> ( ( x x. M ) = N <-> ( K x. M ) = N ) )
3	2	rspcev 2784	. . . . . . 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 11484	. . . . . . 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 372	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ K e. ZZ ) -> ( ( K x. M ) = N -> M || N ) )
9	8	3impa 1176	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( ( K x. M ) = N -> M || N ) )
10	9	3comr 1189	. 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 962   = wceq 1331   e. wcel 1480   E.wrex 2415   class class class wbr 3924  (class class class)co 5767   x. cmul 7618   ZZcz 9047   || cdvds 11482

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-iota 5083  df-fv 5126  df-ov 5770  df-dvds 11483

This theorem is referenced by:  iddvds  11495  1dvds  11496  dvds0  11497  dvdsmul1  11504  dvdsmul2  11505  divalgmod  11613  oddpwdclemxy  11836  ex-dvds  12931