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Theorem dmncan1 26701
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 26682 . . . . . 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 2283 . . . . . . 7  |-  (  /g  `  G )  =  (  /g  `  G )
62, 3, 4, 5rngosubdi 26584 . . . . . 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 457 . . . . 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 451 . . . 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 2291 . . 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 21057 . . . . . . . . . . . 12  |-  ( R  e.  RingOps  ->  G  e.  GrpOp )
111, 10syl 15 . . . . . . . . . . 11  |-  ( R  e.  Dmn  ->  G  e.  GrpOp )
124, 5grpodivcl 20914 . . . . . . . . . . . 12  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B (  /g  `  G
) C )  e.  X )
13123expb 1152 . . . . . . . . . . 11  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( B
(  /g  `  G ) C )  e.  X
)
1411, 13sylan 457 . . . . . . . . . 10  |-  ( ( R  e.  Dmn  /\  ( B  e.  X  /\  C  e.  X
) )  ->  ( B (  /g  `  G
) C )  e.  X )
1514adantlr 695 . . . . . . . . 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 26700 . . . . . . . . . . 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 1169 . . . . . . . . . 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 421 . . . . . . . . 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 456 . . . . . . . 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 595 . . . . . . 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 1166 . . . . . 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 2530 . . . . . 6  |-  ( ( A  =  Z  \/  ( B (  /g  `  G
) C )  =  Z )  <->  ( A  =/=  Z  ->  ( B
(  /g  `  G ) C )  =  Z ) )
2422, 23syl6ib 217 . . . . 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 72 . . . 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 418 . . 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 226 . 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 451 . . . 4  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  G  e.  GrpOp )
292, 3, 4rngocl 21049 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  e.  X )
30293adant3r3 1162 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H B )  e.  X
)
311, 30sylan 457 . . . 4  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A H B )  e.  X )
322, 3, 4rngocl 21049 . . . . . 6  |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  C  e.  X )  ->  ( A H C )  e.  X )
33323adant3r2 1161 . . . . 5  |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A H C )  e.  X
)
341, 33sylan 457 . . . 4  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( A H C )  e.  X )
354, 16, 5grpoeqdivid 26571 . . . 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 1182 . . 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 451 . 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 26571 . . . . . 6  |-  ( ( G  e.  GrpOp  /\  B  e.  X  /\  C  e.  X )  ->  ( B  =  C  <->  ( B
(  /g  `  G ) C )  =  Z ) )
39383expb 1152 . . . . 5  |-  ( ( G  e.  GrpOp  /\  ( B  e.  X  /\  C  e.  X )
)  ->  ( B  =  C  <->  ( B (  /g  `  G ) C )  =  Z ) )
4011, 39sylan 457 . . . 4  |-  ( ( R  e.  Dmn  /\  ( B  e.  X  /\  C  e.  X
) )  ->  ( B  =  C  <->  ( B
(  /g  `  G ) C )  =  Z ) )
41403adantr1 1114 . . 3  |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X
) )  ->  ( B  =  C  <->  ( B
(  /g  `  G ) C )  =  Z ) )
4241adantr 451 . 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 259 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 set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   ran crn 4690   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121   GrpOpcgr 20853  GIdcgi 20854    /g cgs 20856   RingOpscrngo 21042   Dmncdmn 26672
This theorem is referenced by:  dmncan2  26702
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-1o 6479  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-grpo 20858  df-gid 20859  df-ginv 20860  df-gdiv 20861  df-ablo 20949  df-ass 20980  df-exid 20982  df-mgm 20986  df-sgr 20998  df-mndo 21005  df-rngo 21043  df-com2 21078  df-crngo 26621  df-idl 26635  df-pridl 26636  df-prrngo 26673  df-dmn 26674  df-igen 26685
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