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Theorem dmnnzd 30634
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 2457 . . . . . 6  |-  (GId `  H )  =  (GId
`  H )
61, 2, 3, 4, 5isdmn3 30633 . . . . 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 1013 . . . 4  |-  ( R  e.  Dmn  ->  A. a  e.  X  A. b  e.  X  ( (
a H b )  =  Z  ->  (
a  =  Z  \/  b  =  Z )
) )
8 oveq1 6303 . . . . . . 7  |-  ( a  =  A  ->  (
a H b )  =  ( A H b ) )
98eqeq1d 2459 . . . . . 6  |-  ( a  =  A  ->  (
( a H b )  =  Z  <->  ( A H b )  =  Z ) )
10 eqeq1 2461 . . . . . . 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 6304 . . . . . . 7  |-  ( b  =  B  ->  ( A H b )  =  ( A H B ) )
1413eqeq1d 2459 . . . . . 6  |-  ( b  =  B  ->  (
( A H b )  =  Z  <->  ( A H B )  =  Z ) )
15 eqeq1 2461 . . . . . . 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 3219 . . . 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 1211 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 973    = wceq 1395    e. wcel 1819    =/= wne 2652   A.wral 2807   ran crn 5009   ` cfv 5594  (class class class)co 6296   1stc1st 6797   2ndc2nd 6798  GIdcgi 25315  CRingOpsccring 30554   Dmncdmn 30606
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-1o 7148  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-grpo 25319  df-gid 25320  df-ginv 25321  df-ablo 25410  df-ass 25441  df-exid 25443  df-mgmOLD 25447  df-sgrOLD 25459  df-mndo 25466  df-rngo 25504  df-com2 25539  df-crngo 30555  df-idl 30569  df-pridl 30570  df-prrngo 30607  df-dmn 30608  df-igen 30619
This theorem is referenced by:  dmncan1  30635
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