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Theorem dmnnzd 26803
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 2296 . . . . . 6  |-  (GId `  H )  =  (GId
`  H )
61, 2, 3, 4, 5isdmn3 26802 . . . . 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 972 . . . 4  |-  ( R  e.  Dmn  ->  A. a  e.  X  A. b  e.  X  ( (
a H b )  =  Z  ->  (
a  =  Z  \/  b  =  Z )
) )
8 oveq1 5881 . . . . . . 7  |-  ( a  =  A  ->  (
a H b )  =  ( A H b ) )
98eqeq1d 2304 . . . . . 6  |-  ( a  =  A  ->  (
( a H b )  =  Z  <->  ( A H b )  =  Z ) )
10 eqeq1 2302 . . . . . . 7  |-  ( a  =  A  ->  (
a  =  Z  <->  A  =  Z ) )
1110orbi1d 683 . . . . . 6  |-  ( a  =  A  ->  (
( a  =  Z  \/  b  =  Z )  <->  ( A  =  Z  \/  b  =  Z ) ) )
129, 11imbi12d 311 . . . . 5  |-  ( a  =  A  ->  (
( ( a H b )  =  Z  ->  ( a  =  Z  \/  b  =  Z ) )  <->  ( ( A H b )  =  Z  ->  ( A  =  Z  \/  b  =  Z ) ) ) )
13 oveq2 5882 . . . . . . 7  |-  ( b  =  B  ->  ( A H b )  =  ( A H B ) )
1413eqeq1d 2304 . . . . . 6  |-  ( b  =  B  ->  (
( A H b )  =  Z  <->  ( A H B )  =  Z ) )
15 eqeq1 2302 . . . . . . 7  |-  ( b  =  B  ->  (
b  =  Z  <->  B  =  Z ) )
1615orbi2d 682 . . . . . 6  |-  ( b  =  B  ->  (
( A  =  Z  \/  b  =  Z )  <->  ( A  =  Z  \/  B  =  Z ) ) )
1714, 16imbi12d 311 . . . . 5  |-  ( b  =  B  ->  (
( ( A H b )  =  Z  ->  ( A  =  Z  \/  b  =  Z ) )  <->  ( ( A H B )  =  Z  ->  ( A  =  Z  \/  B  =  Z ) ) ) )
1812, 17rspc2v 2903 . . . 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 26 . . 3  |-  ( R  e.  Dmn  ->  (
( A  e.  X  /\  B  e.  X
)  ->  ( ( A H B )  =  Z  ->  ( A  =  Z  \/  B  =  Z ) ) ) )
2019exp3a 425 . 2  |-  ( R  e.  Dmn  ->  ( A  e.  X  ->  ( B  e.  X  -> 
( ( A H B )  =  Z  ->  ( A  =  Z  \/  B  =  Z ) ) ) ) )
21203imp2 1166 1  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  ( A H B )  =  Z ) )  ->  ( A  =  Z  \/  B  =  Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556   ran crn 4706   ` cfv 5271  (class class class)co 5874   1stc1st 6136   2ndc2nd 6137  GIdcgi 20870  CRingOpsccring 26723   Dmncdmn 26775
This theorem is referenced by:  dmncan1  26804
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-1o 6495  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-grpo 20874  df-gid 20875  df-ginv 20876  df-ablo 20965  df-ass 20996  df-exid 20998  df-mgm 21002  df-sgr 21014  df-mndo 21021  df-rngo 21059  df-com2 21094  df-crngo 26724  df-idl 26738  df-pridl 26739  df-prrngo 26776  df-dmn 26777  df-igen 26788
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