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Theorem dmncan1 32042
Description: Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.)
Hypotheses
Ref Expression
dmncan.1  |-  G  =  ( 1st `  R
)
dmncan.2  |-  H  =  ( 2nd `  R
)
dmncan.3  |-  X  =  ran  G
dmncan.4  |-  Z  =  (GId `  G )
Assertion
Ref Expression
dmncan1  |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A  =/=  Z )  ->  (
( A H B )  =  ( A H C )  ->  B  =  C )
)

Proof of Theorem dmncan1
StepHypRef Expression
1 dmnrngo 32023 . . . . . 6  |-  ( R  e.  Dmn  ->  R  e.  RingOps )
2 dmncan.1 . . . . . . 7  |-  G  =  ( 1st `  R
)
3 dmncan.2 . . . . . . 7  |-  H  =  ( 2nd `  R
)
4 dmncan.3 . . . . . . 7  |-  X  =  ran  G
5 eqid 2420 . . . . . . 7  |-  (  /g  `  G )  =  (  /g  `  G )
62, 3, 4, 5rngosubdi 31925 . . . . . 6  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H ( B (  /g  `  G ) C ) )  =  ( ( A H B ) (  /g  `  G ) ( A H C ) ) )
71, 6sylan 473 . . . . 5  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A H ( B (  /g  `  G ) C ) )  =  ( ( A H B ) (  /g  `  G ) ( A H C ) ) )
87adantr 466 . . . 4  |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A  =/=  Z )  ->  ( A H ( B (  /g  `  G ) C ) )  =  ( ( A H B ) (  /g  `  G ) ( A H C ) ) )
98eqeq1d 2422 . . 3  |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A  =/=  Z )  ->  (
( A H ( B (  /g  `  G
) C ) )  =  Z  <->  ( ( A H B ) (  /g  `  G ) ( A H C ) )  =  Z ) )
102rngogrpo 25989 . . . . . . . . . . . 12  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
111, 10syl 17 . . . . . . . . . . 11  |-  ( R  e.  Dmn  ->  G  e.  GrpOp )
124, 5grpodivcl 25846 . . . . . . . . . . . 12  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B (  /g  `  G
) C )  e.  X )
13123expb 1206 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( B
(  /g  `  G ) C )  e.  X
)
1411, 13sylan 473 . . . . . . . . . 10  |-  ( ( R  e.  Dmn  /\  ( B  e.  X  /\  C  e.  X
) )  ->  ( B (  /g  `  G
) C )  e.  X )
1514adantlr 719 . . . . . . . . 9  |-  ( ( ( R  e.  Dmn  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  -> 
( B (  /g  `  G ) C )  e.  X )
16 dmncan.4 . . . . . . . . . . . 12  |-  Z  =  (GId `  G )
172, 3, 4, 16dmnnzd 32041 . . . . . . . . . . 11  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  ( B (  /g  `  G ) C )  e.  X  /\  ( A H ( B (  /g  `  G ) C ) )  =  Z ) )  -> 
( A  =  Z  \/  ( B (  /g  `  G ) C )  =  Z ) )
18173exp2 1223 . . . . . . . . . 10  |-  ( R  e.  Dmn  ->  ( A  e.  X  ->  ( ( B (  /g  `  G ) C )  e.  X  ->  (
( A H ( B (  /g  `  G
) C ) )  =  Z  ->  ( A  =  Z  \/  ( B (  /g  `  G
) C )  =  Z ) ) ) ) )
1918imp31 433 . . . . . . . . 9  |-  ( ( ( R  e.  Dmn  /\  A  e.  X )  /\  ( B (  /g  `  G ) C )  e.  X
)  ->  ( ( A H ( B (  /g  `  G ) C ) )  =  Z  ->  ( A  =  Z  \/  ( B (  /g  `  G
) C )  =  Z ) ) )
2015, 19syldan 472 . . . . . . . 8  |-  ( ( ( R  e.  Dmn  /\  A  e.  X )  /\  ( B  e.  X  /\  C  e.  X ) )  -> 
( ( A H ( B (  /g  `  G ) C ) )  =  Z  -> 
( A  =  Z  \/  ( B (  /g  `  G ) C )  =  Z ) ) )
2120exp43 615 . . . . . . 7  |-  ( R  e.  Dmn  ->  ( A  e.  X  ->  ( B  e.  X  -> 
( C  e.  X  ->  ( ( A H ( B (  /g  `  G ) C ) )  =  Z  -> 
( A  =  Z  \/  ( B (  /g  `  G ) C )  =  Z ) ) ) ) ) )
22213imp2 1220 . . . . . 6  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  (
( A H ( B (  /g  `  G
) C ) )  =  Z  ->  ( A  =  Z  \/  ( B (  /g  `  G
) C )  =  Z ) ) )
23 neor 2746 . . . . . 6  |-  ( ( A  =  Z  \/  ( B (  /g  `  G
) C )  =  Z )  <->  ( A  =/=  Z  ->  ( B
(  /g  `  G ) C )  =  Z ) )
2422, 23syl6ib 229 . . . . 5  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  (
( A H ( B (  /g  `  G
) C ) )  =  Z  ->  ( A  =/=  Z  ->  ( B (  /g  `  G
) C )  =  Z ) ) )
2524com23 81 . . . 4  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A  =/=  Z  ->  (
( A H ( B (  /g  `  G
) C ) )  =  Z  ->  ( B (  /g  `  G
) C )  =  Z ) ) )
2625imp 430 . . 3  |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A  =/=  Z )  ->  (
( A H ( B (  /g  `  G
) C ) )  =  Z  ->  ( B (  /g  `  G
) C )  =  Z ) )
279, 26sylbird 238 . 2  |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A  =/=  Z )  ->  (
( ( A H B ) (  /g  `  G ) ( A H C ) )  =  Z  ->  ( B (  /g  `  G
) C )  =  Z ) )
2811adantr 466 . . . 4  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  G  e.  GrpOp )
292, 3, 4rngocl 25981 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
30293adant3r3 1216 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H B )  e.  X
)
311, 30sylan 473 . . . 4  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A H B )  e.  X )
322, 3, 4rngocl 25981 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  e.  X )
33323adant3r2 1215 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H C )  e.  X
)
341, 33sylan 473 . . . 4  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A H C )  e.  X )
354, 16, 5grpoeqdivid 31912 . . . 4  |-  ( ( G  e.  GrpOp  /\  ( A H B )  e.  X  /\  ( A H C )  e.  X )  ->  (
( A H B )  =  ( A H C )  <->  ( ( A H B ) (  /g  `  G ) ( A H C ) )  =  Z ) )
3628, 31, 34, 35syl3anc 1264 . . 3  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  (
( A H B )  =  ( A H C )  <->  ( ( A H B ) (  /g  `  G ) ( A H C ) )  =  Z ) )
3736adantr 466 . 2  |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A  =/=  Z )  ->  (
( A H B )  =  ( A H C )  <->  ( ( A H B ) (  /g  `  G ) ( A H C ) )  =  Z ) )
384, 16, 5grpoeqdivid 31912 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B  =  C  <->  ( B
(  /g  `  G ) C )  =  Z ) )
39383expb 1206 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( B  =  C  <->  ( B (  /g  `  G ) C )  =  Z ) )
4011, 39sylan 473 . . . 4  |-  ( ( R  e.  Dmn  /\  ( B  e.  X  /\  C  e.  X
) )  ->  ( B  =  C  <->  ( B
(  /g  `  G ) C )  =  Z ) )
41403adantr1 1164 . . 3  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( B  =  C  <->  ( B
(  /g  `  G ) C )  =  Z ) )
4241adantr 466 . 2  |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A  =/=  Z )  ->  ( B  =  C  <->  ( B
(  /g  `  G ) C )  =  Z ) )
4327, 37, 423imtr4d 271 1  |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  /\  A  =/=  Z )  ->  (
( A H B )  =  ( A H C )  ->  B  =  C )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    =/= wne 2616   ran crn 4846   ` cfv 5592  (class class class)co 6296   1stc1st 6796   2ndc2nd 6797   GrpOpcgr 25785  GIdcgi 25786    /g cgs 25788   RingOpscrngo 25974   Dmncdmn 32013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  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-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-1o 7181  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-grpo 25790  df-gid 25791  df-ginv 25792  df-gdiv 25793  df-ablo 25881  df-ass 25912  df-exid 25914  df-mgmOLD 25918  df-sgrOLD 25930  df-mndo 25937  df-rngo 25975  df-com2 26010  df-crngo 31962  df-idl 31976  df-pridl 31977  df-prrngo 32014  df-dmn 32015  df-igen 32026
This theorem is referenced by:  dmncan2  32043
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