Theorem dvds2lem 11765

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 11751	. . . . . . 7 |- ( ( I e. ZZ /\ J e. ZZ ) -> ( I || J <-> E. x e. ZZ ( x x. I ) = J ) )
4		divides 11751	. . . . . . 7 |- ( ( K e. ZZ /\ L e. ZZ ) -> ( K || L <-> E. y e. ZZ ( y x. K ) = L ) )
5	3, 4	bi2anan9 601	. . . . . 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 409	. . . . 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 143	. . . 4 |- ( ph -> ( ( I || J /\ K || L ) -> ( E. x e. ZZ ( x x. I ) = J /\ E. y e. ZZ ( y x. K ) = L ) ) )
8		reeanv 2639	. . . 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 161	. . 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 5860	. . . . . . 7 |- ( z = Z -> ( z x. M ) = ( Z x. M ) )
13	12	eqeq1d 2179	. . . . . 6 |- ( z = Z -> ( ( z x. M ) = N <-> ( Z x. M ) = N ) )
14	13	rspcev 2834	. . . . 5 |- ( ( Z e. ZZ /\ ( Z x. M ) = N ) -> E. z e. ZZ ( z x. M ) = N )
15	10, 11, 14	syl6an 1427	. . . 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 2595	. . 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 45	. 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 11751	. . 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 168	1 |- ( ph -> ( ( I || J /\ K || L ) -> M || N ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   E.wrex 2449   class class class wbr 3989  (class class class)co 5853   x. cmul 7779   ZZcz 9212   || cdvds 11749

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-iota 5160  df-fv 5206  df-ov 5856  df-dvds 11750

This theorem is referenced by:  dvds2ln  11786  dvds2add  11787  dvds2sub  11788  dvdstr  11790