Theorem gcdcllem2 15222

Description: Lemma for gcdn0cl 15224, gcddvds 15225 and dvdslegcd 15226. (Contributed by Paul Chapman, 21-Mar-2011.)

Hypotheses

Ref	Expression
gcdcllem2.1	|- S = { z e. ZZ | A. n e. { M , N } z || n }
gcdcllem2.2	|- R = { z e. ZZ | ( z || M /\ z || N ) }

Assertion

Ref	Expression
gcdcllem2	|- ( ( M e. ZZ /\ N e. ZZ ) -> R = S )

Distinct variable groups:   z, n, M   n, N, z

Allowed substitution hints:   R( z, n)   S( z, n)

Proof of Theorem gcdcllem2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4656	. . . . . 6 |- ( z = x -> ( z || n <-> x || n ) )
2	1	ralbidv 2986	. . . . 5 |- ( z = x -> ( A. n e. { M , N } z || n <-> A. n e. { M , N } x || n ) )
3		gcdcllem2.1	. . . . 5 |- S = { z e. ZZ | A. n e. { M , N } z || n }
4	2, 3	elrab2 3366	. . . 4 |- ( x e. S <-> ( x e. ZZ /\ A. n e. { M , N } x || n ) )
5		breq2 4657	. . . . . 6 |- ( n = M -> ( x || n <-> x || M ) )
6		breq2 4657	. . . . . 6 |- ( n = N -> ( x || n <-> x || N ) )
7	5, 6	ralprg 4234	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( A. n e. { M , N } x || n <-> ( x || M /\ x || N ) ) )
8	7	anbi2d 740	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( x e. ZZ /\ A. n e. { M , N } x || n ) <-> ( x e. ZZ /\ ( x || M /\ x || N ) ) ) )
9	4, 8	syl5bb 272	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( x e. S <-> ( x e. ZZ /\ ( x || M /\ x || N ) ) ) )
10		breq1 4656	. . . . 5 |- ( z = x -> ( z || M <-> x || M ) )
11		breq1 4656	. . . . 5 |- ( z = x -> ( z || N <-> x || N ) )
12	10, 11	anbi12d 747	. . . 4 |- ( z = x -> ( ( z || M /\ z || N ) <-> ( x || M /\ x || N ) ) )
13		gcdcllem2.2	. . . 4 |- R = { z e. ZZ | ( z || M /\ z || N ) }
14	12, 13	elrab2 3366	. . 3 |- ( x e. R <-> ( x e. ZZ /\ ( x || M /\ x || N ) ) )
15	9, 14	syl6rbbr 279	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( x e. R <-> x e. S ) )
16	15	eqrdv 2620	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> R = S )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   {cpr 4179   class class class wbr 4653   ZZcz 11377   || cdvds 14983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654

This theorem is referenced by:  gcdcllem3  15223