Theorem dvds2lem 13667

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

Proof of Theorem dvds2lem

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 13664	. . . . . . 7 |- ((I e. ZZ /\ J e. ZZ) -> (I||J <-> E.x e. ZZ (x x. I) = J))
4		divides 13664	. . . . . . 7 |- ((K e. ZZ /\ L e. ZZ) -> (K||L <-> E.y e. ZZ (y x. K) = L))
5	3, 4	bi2anan9 694	. . . . . 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	syl11anc 524	. . . . 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 170	. . . 4 |- (ph -> ((I||J /\ K||L) -> (E.x e. ZZ (x x. I) = J /\ E.y e. ZZ (y x. K) = L)))
8		reeanv 2249	. . . 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 230	. . 3 |- (ph -> ((I||J /\ K||L) -> E.x e. ZZ E.y e. ZZ ((x x. I) = J /\ (y x. K) = L)))
10		dvds2lem.5	. . . . . 6 |- ((ph /\ (x e. ZZ /\ y e. ZZ)) -> (((x x. I) = J /\ (y x. K) = L) -> (Z x. M) = N))
11		opreq1 4889	. . . . . . . . . 10 |- (z = Z -> (z x. M) = (Z x. M))
12	11	eqeq1d 1892	. . . . . . . . 9 |- (z = Z -> ((z x. M) = N <-> (Z x. M) = N))
13	12	rcla4ev 2381	. . . . . . . 8 |- ((Z e. ZZ /\ (Z x. M) = N) -> E.z e. ZZ (z x. M) = N)
14		dvds2lem.4	. . . . . . . 8 |- ((ph /\ (x e. ZZ /\ y e. ZZ)) -> Z e. ZZ)
15	13, 14	sylan 497	. . . . . . 7 |- (((ph /\ (x e. ZZ /\ y e. ZZ)) /\ (Z x. M) = N) -> E.z e. ZZ (z x. M) = N)
16	15	ex 402	. . . . . 6 |- ((ph /\ (x e. ZZ /\ y e. ZZ)) -> ((Z x. M) = N -> E.z e. ZZ (z x. M) = N))
17	10, 16	syld 30	. . . . 5 |- ((ph /\ (x e. ZZ /\ y e. ZZ)) -> (((x x. I) = J /\ (y x. K) = L) -> E.z e. ZZ (z x. M) = N))
18	17	ex 402	. . . 4 |- (ph -> ((x e. ZZ /\ y e. ZZ) -> (((x x. I) = J /\ (y x. K) = L) -> E.z e. ZZ (z x. M) = N)))
19	18	r19.23advv 2218	. . 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))
20	9, 19	syld 30	. 2 |- (ph -> ((I||J /\ K||L) -> E.z e. ZZ (z x. M) = N))
21		dvds2lem.3	. . 3 |- (ph -> (M e. ZZ /\ N e. ZZ))
22		divides 13664	. . 3 |- ((M e. ZZ /\ N e. ZZ) -> (M||N <-> E.z e. ZZ (z x. M) = N))
23	21, 22	syl 12	. 2 |- (ph -> (M||N <-> E.z e. ZZ (z x. M) = N))
24	20, 23	sylibrd 221	1 |- (ph -> ((I||J /\ K||L) -> M||N))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106   class class class wbr 3338  (class class class)co 4884   x. cmul 6391  ZZcz 6451  ||cdivides 13662

This theorem is referenced by:  dvds2ln 13684  dvds2add 13685  dvds2sub 13686  dvdstr 13687

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-opr 4886  df-divides 13663

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