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Theorem dmnnzd 28880
Description: A domain has no zero-divisors (besides zero). (Contributed by Jeff Madsen, 19-Jun-2010.)
Hypotheses
Ref Expression
dmnnzd.1  |-  G  =  ( 1st `  R
)
dmnnzd.2  |-  H  =  ( 2nd `  R
)
dmnnzd.3  |-  X  =  ran  G
dmnnzd.4  |-  Z  =  (GId `  G )
Assertion
Ref Expression
dmnnzd  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  ( A H B )  =  Z ) )  ->  ( A  =  Z  \/  B  =  Z ) )

Proof of Theorem dmnnzd
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmnnzd.1 . . . . . 6  |-  G  =  ( 1st `  R
)
2 dmnnzd.2 . . . . . 6  |-  H  =  ( 2nd `  R
)
3 dmnnzd.3 . . . . . 6  |-  X  =  ran  G
4 dmnnzd.4 . . . . . 6  |-  Z  =  (GId `  G )
5 eqid 2443 . . . . . 6  |-  (GId `  H )  =  (GId
`  H )
61, 2, 3, 4, 5isdmn3 28879 . . . . 5  |-  ( R  e.  Dmn  <->  ( R  e. CRingOps 
/\  (GId `  H
)  =/=  Z  /\  A. a  e.  X  A. b  e.  X  (
( a H b )  =  Z  -> 
( a  =  Z  \/  b  =  Z ) ) ) )
76simp3bi 1005 . . . 4  |-  ( R  e.  Dmn  ->  A. a  e.  X  A. b  e.  X  ( (
a H b )  =  Z  ->  (
a  =  Z  \/  b  =  Z )
) )
8 oveq1 6103 . . . . . . 7  |-  ( a  =  A  ->  (
a H b )  =  ( A H b ) )
98eqeq1d 2451 . . . . . 6  |-  ( a  =  A  ->  (
( a H b )  =  Z  <->  ( A H b )  =  Z ) )
10 eqeq1 2449 . . . . . . 7  |-  ( a  =  A  ->  (
a  =  Z  <->  A  =  Z ) )
1110orbi1d 702 . . . . . 6  |-  ( a  =  A  ->  (
( a  =  Z  \/  b  =  Z )  <->  ( A  =  Z  \/  b  =  Z ) ) )
129, 11imbi12d 320 . . . . 5  |-  ( a  =  A  ->  (
( ( a H b )  =  Z  ->  ( a  =  Z  \/  b  =  Z ) )  <->  ( ( A H b )  =  Z  ->  ( A  =  Z  \/  b  =  Z ) ) ) )
13 oveq2 6104 . . . . . . 7  |-  ( b  =  B  ->  ( A H b )  =  ( A H B ) )
1413eqeq1d 2451 . . . . . 6  |-  ( b  =  B  ->  (
( A H b )  =  Z  <->  ( A H B )  =  Z ) )
15 eqeq1 2449 . . . . . . 7  |-  ( b  =  B  ->  (
b  =  Z  <->  B  =  Z ) )
1615orbi2d 701 . . . . . 6  |-  ( b  =  B  ->  (
( A  =  Z  \/  b  =  Z )  <->  ( A  =  Z  \/  B  =  Z ) ) )
1714, 16imbi12d 320 . . . . 5  |-  ( b  =  B  ->  (
( ( A H b )  =  Z  ->  ( A  =  Z  \/  b  =  Z ) )  <->  ( ( A H B )  =  Z  ->  ( A  =  Z  \/  B  =  Z ) ) ) )
1812, 17rspc2v 3084 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A. a  e.  X  A. b  e.  X  ( ( a H b )  =  Z  ->  ( a  =  Z  \/  b  =  Z ) )  -> 
( ( A H B )  =  Z  ->  ( A  =  Z  \/  B  =  Z ) ) ) )
197, 18syl5com 30 . . 3  |-  ( R  e.  Dmn  ->  (
( A  e.  X  /\  B  e.  X
)  ->  ( ( A H B )  =  Z  ->  ( A  =  Z  \/  B  =  Z ) ) ) )
2019expd 436 . 2  |-  ( R  e.  Dmn  ->  ( A  e.  X  ->  ( B  e.  X  -> 
( ( A H B )  =  Z  ->  ( A  =  Z  \/  B  =  Z ) ) ) ) )
21203imp2 1202 1  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  ( A H B )  =  Z ) )  ->  ( A  =  Z  \/  B  =  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2611   A.wral 2720   ran crn 4846   ` cfv 5423  (class class class)co 6096   1stc1st 6580   2ndc2nd 6581  GIdcgi 23679  CRingOpsccring 28800   Dmncdmn 28852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-1o 6925  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-fin 7319  df-grpo 23683  df-gid 23684  df-ginv 23685  df-ablo 23774  df-ass 23805  df-exid 23807  df-mgm 23811  df-sgr 23823  df-mndo 23830  df-rngo 23868  df-com2 23903  df-crngo 28801  df-idl 28815  df-pridl 28816  df-prrngo 28853  df-dmn 28854  df-igen 28865
This theorem is referenced by:  dmncan1  28881
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