Theorem dvds1lem 12362

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 6024	. . . . . 6 |- ( z = Z -> ( z x. M ) = ( Z x. M ) )
4	3	eqeq1d 2240	. . . . 5 |- ( z = Z -> ( ( z x. M ) = N <-> ( Z x. M ) = N ) )
5	4	rspcev 2910	. . . 4 |- ( ( Z e. ZZ /\ ( Z x. M ) = N ) -> E. z e. ZZ ( z x. M ) = N )
6	1, 2, 5	syl6an 1478	. . 3 |- ( ( ph /\ x e. ZZ ) -> ( ( x x. J ) = K -> E. z e. ZZ ( z x. M ) = N ) )
7	6	rexlimdva 2650	. 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 12349	. . 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 12349	. . 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 1397   e. wcel 2202   E.wrex 2511   class class class wbr 4088  (class class class)co 6017   x. cmul 8036   ZZcz 9478   || cdvds 12347

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-iota 5286  df-fv 5334  df-ov 6020  df-dvds 12348

This theorem is referenced by:  negdvdsb  12367  dvdsnegb  12368  muldvds1  12376  muldvds2  12377  dvdscmul  12378  dvdsmulc  12379  dvdscmulr  12380  dvdsmulcr  12381