Theorem dvds1lem 11504

Description: A lemma to assist theorems of || with one antecedent. (Contributed by Paul Chapman, 21-Mar-2011.)

Hypotheses

Ref	Expression
dvds1lem.1	|- ( ph -> ( J e. ZZ /\ K e. ZZ ) )
dvds1lem.2	|- ( ph -> ( M e. ZZ /\ N e. ZZ ) )
dvds1lem.3	|- ( ( ph /\ x e. ZZ ) -> Z e. ZZ )
dvds1lem.4	|- ( ( ph /\ x e. ZZ ) -> ( ( x x. J ) = K -> ( Z x. M ) = N ) )

Assertion

Ref	Expression
dvds1lem	|- ( ph -> ( J || K -> M || N ) )

Distinct variable groups:   x, J   x, K   x, M   x, N   ph, x

Allowed substitution hint:   Z( x)

Proof of Theorem dvds1lem

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvds1lem.3	. . . 4 |- ( ( ph /\ x e. ZZ ) -> Z e. ZZ )
2		dvds1lem.4	. . . 4 |- ( ( ph /\ x e. ZZ ) -> ( ( x x. J ) = K -> ( Z x. M ) = N ) )
3		oveq1 5781	. . . . . 6 |- ( z = Z -> ( z x. M ) = ( Z x. M ) )
4	3	eqeq1d 2148	. . . . 5 |- ( z = Z -> ( ( z x. M ) = N <-> ( Z x. M ) = N ) )
5	4	rspcev 2789	. . . 4 |- ( ( Z e. ZZ /\ ( Z x. M ) = N ) -> E. z e. ZZ ( z x. M ) = N )
6	1, 2, 5	syl6an 1410	. . 3 |- ( ( ph /\ x e. ZZ ) -> ( ( x x. J ) = K -> E. z e. ZZ ( z x. M ) = N ) )
7	6	rexlimdva 2549	. 2 |- ( ph -> ( E. x e. ZZ ( x x. J ) = K -> E. z e. ZZ ( z x. M ) = N ) )
8		dvds1lem.1	. . 3 |- ( ph -> ( J e. ZZ /\ K e. ZZ ) )
9		divides 11495	. . 3 |- ( ( J e. ZZ /\ K e. ZZ ) -> ( J || K <-> E. x e. ZZ ( x x. J ) = K ) )
10	8, 9	syl 14	. 2 |- ( ph -> ( J || K <-> E. x e. ZZ ( x x. J ) = K ) )
11		dvds1lem.2	. . 3 |- ( ph -> ( M e. ZZ /\ N e. ZZ ) )
12		divides 11495	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> E. z e. ZZ ( z x. M ) = N ) )
13	11, 12	syl 14	. 2 |- ( ph -> ( M || N <-> E. z e. ZZ ( z x. M ) = N ) )
14	7, 10, 13	3imtr4d 202	1 |- ( ph -> ( J || K -> M || N ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   E.wrex 2417   class class class wbr 3929  (class class class)co 5774   x. cmul 7625   ZZcz 9054   || cdvds 11493

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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-iota 5088  df-fv 5131  df-ov 5777  df-dvds 11494

This theorem is referenced by:  negdvdsb  11509  dvdsnegb  11510  muldvds1  11518  muldvds2  11519  dvdscmul  11520  dvdsmulc  11521  dvdscmulr  11522  dvdsmulcr  11523