Theorem dvds2lem 12548

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 12534	. . . . . . 7 |- ( ( I e. ZZ /\ J e. ZZ ) -> ( I || J <-> E. x e. ZZ ( x x. I ) = J ) )
4		divides 12534	. . . . . . 7 |- ( ( K e. ZZ /\ L e. ZZ ) -> ( K || L <-> E. y e. ZZ ( y x. K ) = L ) )
5	3, 4	bi2anan9 614	. . . . . 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 415	. . . . 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 144	. . . 4 |- ( ph -> ( ( I || J /\ K || L ) -> ( E. x e. ZZ ( x x. I ) = J /\ E. y e. ZZ ( y x. K ) = L ) ) )
8		reeanv 2721	. . . 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	imbitrrdi 162	. . 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 6082	. . . . . . 7 |- ( z = Z -> ( z x. M ) = ( Z x. M ) )
13	12	eqeq1d 2247	. . . . . 6 |- ( z = Z -> ( ( z x. M ) = N <-> ( Z x. M ) = N ) )
14	13	rspcev 2929	. . . . 5 |- ( ( Z e. ZZ /\ ( Z x. M ) = N ) -> E. z e. ZZ ( z x. M ) = N )
15	10, 11, 14	syl6an 1483	. . . 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 2676	. . 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 12534	. . 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 169	1 |- ( ph -> ( ( I || J /\ K || L ) -> M || N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4125  (class class class)co 6075   x. cmul 8174   ZZcz 9623   || cdvds 12532

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-iota 5332  df-fv 5380  df-ov 6078  df-dvds 12533

This theorem is referenced by:  dvds2ln  12569  dvds2add  12570  dvds2sub  12571  dvdstr  12573