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Theorem dmnnzd 26686
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 2437 . . . . . 6  |-  (GId `  H )  =  (GId
`  H )
61, 2, 3, 4, 5isdmn3 26685 . . . . 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 975 . . . 4  |-  ( R  e.  Dmn  ->  A. a  e.  X  A. b  e.  X  ( (
a H b )  =  Z  ->  (
a  =  Z  \/  b  =  Z )
) )
8 oveq1 6089 . . . . . . 7  |-  ( a  =  A  ->  (
a H b )  =  ( A H b ) )
98eqeq1d 2445 . . . . . 6  |-  ( a  =  A  ->  (
( a H b )  =  Z  <->  ( A H b )  =  Z ) )
10 eqeq1 2443 . . . . . . 7  |-  ( a  =  A  ->  (
a  =  Z  <->  A  =  Z ) )
1110orbi1d 685 . . . . . 6  |-  ( a  =  A  ->  (
( a  =  Z  \/  b  =  Z )  <->  ( A  =  Z  \/  b  =  Z ) ) )
129, 11imbi12d 313 . . . . 5  |-  ( a  =  A  ->  (
( ( a H b )  =  Z  ->  ( a  =  Z  \/  b  =  Z ) )  <->  ( ( A H b )  =  Z  ->  ( A  =  Z  \/  b  =  Z ) ) ) )
13 oveq2 6090 . . . . . . 7  |-  ( b  =  B  ->  ( A H b )  =  ( A H B ) )
1413eqeq1d 2445 . . . . . 6  |-  ( b  =  B  ->  (
( A H b )  =  Z  <->  ( A H B )  =  Z ) )
15 eqeq1 2443 . . . . . . 7  |-  ( b  =  B  ->  (
b  =  Z  <->  B  =  Z ) )
1615orbi2d 684 . . . . . 6  |-  ( b  =  B  ->  (
( A  =  Z  \/  b  =  Z )  <->  ( A  =  Z  \/  B  =  Z ) ) )
1714, 16imbi12d 313 . . . . 5  |-  ( b  =  B  ->  (
( ( A H b )  =  Z  ->  ( A  =  Z  \/  b  =  Z ) )  <->  ( ( A H B )  =  Z  ->  ( A  =  Z  \/  B  =  Z ) ) ) )
1812, 17rspc2v 3059 . . . 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 29 . . 3  |-  ( R  e.  Dmn  ->  (
( A  e.  X  /\  B  e.  X
)  ->  ( ( A H B )  =  Z  ->  ( A  =  Z  \/  B  =  Z ) ) ) )
2019exp3a 427 . 2  |-  ( R  e.  Dmn  ->  ( A  e.  X  ->  ( B  e.  X  -> 
( ( A H B )  =  Z  ->  ( A  =  Z  \/  B  =  Z ) ) ) ) )
21203imp2 1169 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 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726    =/= wne 2600   A.wral 2706   ran crn 4880   ` cfv 5455  (class class class)co 6082   1stc1st 6348   2ndc2nd 6349  GIdcgi 21776  CRingOpsccring 26606   Dmncdmn 26658
This theorem is referenced by:  dmncan1  26687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-int 4052  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-1o 6725  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-fin 7114  df-grpo 21780  df-gid 21781  df-ginv 21782  df-ablo 21871  df-ass 21902  df-exid 21904  df-mgm 21908  df-sgr 21920  df-mndo 21927  df-rngo 21965  df-com2 22000  df-crngo 26607  df-idl 26621  df-pridl 26622  df-prrngo 26659  df-dmn 26660  df-igen 26671
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