Theorem dvds2lem 11353

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 11343	. . . . . . 7 |- ( ( I e. ZZ /\ J e. ZZ ) -> ( I || J <-> E. x e. ZZ ( x x. I ) = J ) )
4		divides 11343	. . . . . . 7 |- ( ( K e. ZZ /\ L e. ZZ ) -> ( K || L <-> E. y e. ZZ ( y x. K ) = L ) )
5	3, 4	bi2anan9 578	. . . . . 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 406	. . . . 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 2574	. . . 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 5735	. . . . . . 7 |- ( z = Z -> ( z x. M ) = ( Z x. M ) )
13	12	eqeq1d 2123	. . . . . 6 |- ( z = Z -> ( ( z x. M ) = N <-> ( Z x. M ) = N ) )
14	13	rspcev 2760	. . . . 5 |- ( ( Z e. ZZ /\ ( Z x. M ) = N ) -> E. z e. ZZ ( z x. M ) = N )
15	10, 11, 14	syl6an 1393	. . . 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 2531	. . 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 11343	. . 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 1314   e. wcel 1463   E.wrex 2391   class class class wbr 3895  (class class class)co 5728   x. cmul 7552   ZZcz 8958   || cdvds 11341

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-iota 5046  df-fv 5089  df-ov 5731  df-dvds 11342

This theorem is referenced by:  dvds2ln  11374  dvds2add  11375  dvds2sub  11376  dvdstr  11378