Theorem dvds2lem 10207

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

Hypotheses

Ref	Expression
dvds2lem.1	|- ( ph -> ( I e. ZZ /\ J e. ZZ ) )
dvds2lem.2	|- ( ph -> ( K e. ZZ /\ L e. ZZ ) )
dvds2lem.3	|- ( ph -> ( M e. ZZ /\ N e. ZZ ) )
dvds2lem.4	|- ( ( ph /\ ( x e. ZZ /\ y e. ZZ ) ) -> Z e. ZZ )
dvds2lem.5	|- ( ( ph /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( ( x x. I ) = J /\ ( y x. K ) = L ) -> ( Z x. M ) = N ) )

Assertion

Ref	Expression
dvds2lem	|- ( ph -> ( ( I || J /\ K || L ) -> M || N ) )

Distinct variable groups:   x, I, y   x, J, y   x, K, y   x, L, y   x, M, y   x, N, y   ph, x, y

Allowed substitution hints:   Z( x, y)

Proof of Theorem dvds2lem

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvds2lem.1	. . . . . 6 |- ( ph -> ( I e. ZZ /\ J e. ZZ ) )
2		dvds2lem.2	. . . . . 6 |- ( ph -> ( K e. ZZ /\ L e. ZZ ) )
3		divides 10197	. . . . . . 7 |- ( ( I e. ZZ /\ J e. ZZ ) -> ( I || J <-> E. x e. ZZ ( x x. I ) = J ) )
4		divides 10197	. . . . . . 7 |- ( ( K e. ZZ /\ L e. ZZ ) -> ( K || L <-> E. y e. ZZ ( y x. K ) = L ) )
5	3, 4	bi2anan9 570	. . . . . 6 |- ( ( ( I e. ZZ /\ J e. ZZ ) /\ ( K e. ZZ /\ L e. ZZ ) ) -> ( ( I || J /\ K || L ) <-> ( E. x e. ZZ ( x x. I ) = J /\ E. y e. ZZ ( y x. K ) = L ) ) )
6	1, 2, 5	syl2anc 403	. . . . 5 |- ( ph -> ( ( I || J /\ K || L ) <-> ( E. x e. ZZ ( x x. I ) = J /\ E. y e. ZZ ( y x. K ) = L ) ) )
7	6	biimpd 142	. . . 4 |- ( ph -> ( ( I || J /\ K || L ) -> ( E. x e. ZZ ( x x. I ) = J /\ E. y e. ZZ ( y x. K ) = L ) ) )
8		reeanv 2523	. . . 4 |- ( E. x e. ZZ E. y e. ZZ ( ( x x. I ) = J /\ ( y x. K ) = L ) <-> ( E. x e. ZZ ( x x. I ) = J /\ E. y e. ZZ ( y x. K ) = L ) )
9	7, 8	syl6ibr 160	. . 3 |- ( ph -> ( ( I || J /\ K || L ) -> E. x e. ZZ E. y e. ZZ ( ( x x. I ) = J /\ ( y x. K ) = L ) ) )
10		dvds2lem.4	. . . . 5 |- ( ( ph /\ ( x e. ZZ /\ y e. ZZ ) ) -> Z e. ZZ )
11		dvds2lem.5	. . . . 5 |- ( ( ph /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( ( x x. I ) = J /\ ( y x. K ) = L ) -> ( Z x. M ) = N ) )
12		oveq1 5539	. . . . . . 7 |- ( z = Z -> ( z x. M ) = ( Z x. M ) )
13	12	eqeq1d 2089	. . . . . 6 |- ( z = Z -> ( ( z x. M ) = N <-> ( Z x. M ) = N ) )
14	13	rspcev 2701	. . . . 5 |- ( ( Z e. ZZ /\ ( Z x. M ) = N ) -> E. z e. ZZ ( z x. M ) = N )
15	10, 11, 14	syl6an 1363	. . . 4 |- ( ( ph /\ ( x e. ZZ /\ y e. ZZ ) ) -> ( ( ( x x. I ) = J /\ ( y x. K ) = L ) -> E. z e. ZZ ( z x. M ) = N ) )
16	15	rexlimdvva 2484	. . 3 |- ( ph -> ( E. x e. ZZ E. y e. ZZ ( ( x x. I ) = J /\ ( y x. K ) = L ) -> E. z e. ZZ ( z x. M ) = N ) )
17	9, 16	syld 44	. 2 |- ( ph -> ( ( I || J /\ K || L ) -> E. z e. ZZ ( z x. M ) = N ) )
18		dvds2lem.3	. . 3 |- ( ph -> ( M e. ZZ /\ N e. ZZ ) )
19		divides 10197	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M || N <-> E. z e. ZZ ( z x. M ) = N ) )
20	18, 19	syl 14	. 2 |- ( ph -> ( M || N <-> E. z e. ZZ ( z x. M ) = N ) )
21	17, 20	sylibrd 167	1 |- ( ph -> ( ( I || J /\ K || L ) -> M || N ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   E.wrex 2349   class class class wbr 3785  (class class class)co 5532   x. cmul 6986   ZZcz 8351   || cdvds 10195

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-iota 4887  df-fv 4930  df-ov 5535  df-dvds 10196

This theorem is referenced by:  dvds2ln  10228  dvds2add  10229  dvds2sub  10230  dvdstr  10232