Theorem dvds0lem 12546

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 6082	. . . . . . . . 9 |- ( x = K -> ( x x. M ) = ( K x. M ) )
2	1	eqeq1d 2247	. . . . . . . 8 |- ( x = K -> ( ( x x. M ) = N <-> ( K x. M ) = N ) )
3	2	rspcev 2929	. . . . . . 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 12534	. . . . . . 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 1225	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( ( K x. M ) = N -> M || N ) )
10	9	3comr 1242	. 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 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4125  (class class class)co 6075   x. cmul 8174   ZZcz 9623   || cdvds 12532

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-iota 5332  df-fv 5380  df-ov 6078  df-dvds 12533

This theorem is referenced by:  iddvds  12549  1dvds  12550  dvds0  12551  dvdsmul1  12558  dvdsmul2  12559  divalgmod  12672  oddpwdclemxy  12925  ex-dvds  16658