Theorem dvds1lem 11777

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 5872	. . . . . 6 |- ( z = Z -> ( z x. M ) = ( Z x. M ) )
4	3	eqeq1d 2184	. . . . 5 |- ( z = Z -> ( ( z x. M ) = N <-> ( Z x. M ) = N ) )
5	4	rspcev 2839	. . . 4 |- ( ( Z e. ZZ /\ ( Z x. M ) = N ) -> E. z e. ZZ ( z x. M ) = N )
6	1, 2, 5	syl6an 1432	. . 3 |- ( ( ph /\ x e. ZZ ) -> ( ( x x. J ) = K -> E. z e. ZZ ( z x. M ) = N ) )
7	6	rexlimdva 2592	. 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 11764	. . 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 11764	. . 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 203	1 |- ( ph -> ( J || K -> M || N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2146   E.wrex 2454   class class class wbr 3998  (class class class)co 5865   x. cmul 7791   ZZcz 9226   || cdvds 11762

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-iota 5170  df-fv 5216  df-ov 5868  df-dvds 11763

This theorem is referenced by:  negdvdsb  11782  dvdsnegb  11783  muldvds1  11791  muldvds2  11792  dvdscmul  11793  dvdsmulc  11794  dvdscmulr  11795  dvdsmulcr  11796