Theorem dvds1lem 11967

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 5929	. . . . . 6 |- ( z = Z -> ( z x. M ) = ( Z x. M ) )
4	3	eqeq1d 2205	. . . . 5 |- ( z = Z -> ( ( z x. M ) = N <-> ( Z x. M ) = N ) )
5	4	rspcev 2868	. . . 4 |- ( ( Z e. ZZ /\ ( Z x. M ) = N ) -> E. z e. ZZ ( z x. M ) = N )
6	1, 2, 5	syl6an 1445	. . 3 |- ( ( ph /\ x e. ZZ ) -> ( ( x x. J ) = K -> E. z e. ZZ ( z x. M ) = N ) )
7	6	rexlimdva 2614	. 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 11954	. . 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 11954	. . 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 1364   e. wcel 2167   E.wrex 2476   class class class wbr 4033  (class class class)co 5922   x. cmul 7884   ZZcz 9326   || cdvds 11952

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-iota 5219  df-fv 5266  df-ov 5925  df-dvds 11953

This theorem is referenced by:  negdvdsb  11972  dvdsnegb  11973  muldvds1  11981  muldvds2  11982  dvdscmul  11983  dvdsmulc  11984  dvdscmulr  11985  dvdsmulcr  11986