Theorem dvds0lem 14992

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 6657	. . . . . . . . 9 |- ( x = K -> ( x x. M ) = ( K x. M ) )
2	1	eqeq1d 2624	. . . . . . . 8 |- ( x = K -> ( ( x x. M ) = N <-> ( K x. M ) = N ) )
3	2	rspcev 3309	. . . . . . 7 |- ( ( K e. ZZ /\ ( K x. M ) = N ) -> E. x e. ZZ ( x x. M ) = N )
4	3	adantl 482	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ ( K x. M ) = N ) ) -> E. x e. ZZ ( x x. M ) = N )
5		divides 14985	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> E. x e. ZZ ( x x. M ) = N ) )
6	5	adantr 481	. . . . . 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 247	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( K e. ZZ /\ ( K x. M ) = N ) ) -> M || N )
8	7	expr 643	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ K e. ZZ ) -> ( ( K x. M ) = N -> M || N ) )
9	8	3impa 1259	. . 3 |- ( ( M e. ZZ /\ N e. ZZ /\ K e. ZZ ) -> ( ( K x. M ) = N -> M || N ) )
10	9	3comr 1273	. 2 |- ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( ( K x. M ) = N -> M || N ) )
11	10	imp 445	1 |- ( ( ( K e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ ( K x. M ) = N ) -> M || N )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653  (class class class)co 6650   x. cmul 9941   ZZcz 11377   || cdvds 14983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-iota 5851  df-fv 5896  df-ov 6653  df-dvds 14984

This theorem is referenced by:  iddvds  14995  1dvds  14996  dvds0  14997  dvdsmul1  15003  dvdsmul2  15004  divalgmod  15129  divalgmodOLD  15130  isprm5  15419  ex-dvds  27313  oddpwdc  30416  inductionexd  38453