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Theorem oecan 7275
Description: Left cancellation law for ordinal exponentiation. (Contributed by NM, 6-Jan-2005.) (Revised by Mario Carneiro, 24-May-2015.)
Assertion
Ref Expression
oecan  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  (
( A  ^o  B
)  =  ( A  ^o  C )  <->  B  =  C ) )

Proof of Theorem oecan
StepHypRef Expression
1 oeordi 7273 . . . . . . 7  |-  ( ( C  e.  On  /\  A  e.  ( On  \  2o ) )  -> 
( B  e.  C  ->  ( A  ^o  B
)  e.  ( A  ^o  C ) ) )
21ancoms 451 . . . . . 6  |-  ( ( A  e.  ( On 
\  2o )  /\  C  e.  On )  ->  ( B  e.  C  ->  ( A  ^o  B
)  e.  ( A  ^o  C ) ) )
323adant2 1016 . . . . 5  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  ( B  e.  C  ->  ( A  ^o  B )  e.  ( A  ^o  C ) ) )
4 oeordi 7273 . . . . . . 7  |-  ( ( B  e.  On  /\  A  e.  ( On  \  2o ) )  -> 
( C  e.  B  ->  ( A  ^o  C
)  e.  ( A  ^o  B ) ) )
54ancoms 451 . . . . . 6  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On )  ->  ( C  e.  B  ->  ( A  ^o  C
)  e.  ( A  ^o  B ) ) )
653adant3 1017 . . . . 5  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  ( C  e.  B  ->  ( A  ^o  C )  e.  ( A  ^o  B ) ) )
73, 6orim12d 839 . . . 4  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  (
( B  e.  C  \/  C  e.  B
)  ->  ( ( A  ^o  B )  e.  ( A  ^o  C
)  \/  ( A  ^o  C )  e.  ( A  ^o  B
) ) ) )
87con3d 133 . . 3  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  ( -.  ( ( A  ^o  B )  e.  ( A  ^o  C )  \/  ( A  ^o  C )  e.  ( A  ^o  B ) )  ->  -.  ( B  e.  C  \/  C  e.  B )
) )
9 eldifi 3565 . . . . . 6  |-  ( A  e.  ( On  \  2o )  ->  A  e.  On )
1093ad2ant1 1018 . . . . 5  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  A  e.  On )
11 simp2 998 . . . . 5  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  B  e.  On )
12 oecl 7224 . . . . 5  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  e.  On )
1310, 11, 12syl2anc 659 . . . 4  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  ( A  ^o  B )  e.  On )
14 simp3 999 . . . . 5  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  C  e.  On )
15 oecl 7224 . . . . 5  |-  ( ( A  e.  On  /\  C  e.  On )  ->  ( A  ^o  C
)  e.  On )
1610, 14, 15syl2anc 659 . . . 4  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  ( A  ^o  C )  e.  On )
17 eloni 5420 . . . . 5  |-  ( ( A  ^o  B )  e.  On  ->  Ord  ( A  ^o  B ) )
18 eloni 5420 . . . . 5  |-  ( ( A  ^o  C )  e.  On  ->  Ord  ( A  ^o  C ) )
19 ordtri3 5446 . . . . 5  |-  ( ( Ord  ( A  ^o  B )  /\  Ord  ( A  ^o  C ) )  ->  ( ( A  ^o  B )  =  ( A  ^o  C
)  <->  -.  ( ( A  ^o  B )  e.  ( A  ^o  C
)  \/  ( A  ^o  C )  e.  ( A  ^o  B
) ) ) )
2017, 18, 19syl2an 475 . . . 4  |-  ( ( ( A  ^o  B
)  e.  On  /\  ( A  ^o  C )  e.  On )  -> 
( ( A  ^o  B )  =  ( A  ^o  C )  <->  -.  ( ( A  ^o  B )  e.  ( A  ^o  C )  \/  ( A  ^o  C )  e.  ( A  ^o  B ) ) ) )
2113, 16, 20syl2anc 659 . . 3  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  (
( A  ^o  B
)  =  ( A  ^o  C )  <->  -.  (
( A  ^o  B
)  e.  ( A  ^o  C )  \/  ( A  ^o  C
)  e.  ( A  ^o  B ) ) ) )
22 eloni 5420 . . . . 5  |-  ( B  e.  On  ->  Ord  B )
23 eloni 5420 . . . . 5  |-  ( C  e.  On  ->  Ord  C )
24 ordtri3 5446 . . . . 5  |-  ( ( Ord  B  /\  Ord  C )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B ) ) )
2522, 23, 24syl2an 475 . . . 4  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B
) ) )
26253adant1 1015 . . 3  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  ( B  =  C  <->  -.  ( B  e.  C  \/  C  e.  B )
) )
278, 21, 263imtr4d 268 . 2  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  (
( A  ^o  B
)  =  ( A  ^o  C )  ->  B  =  C )
)
28 oveq2 6286 . 2  |-  ( B  =  C  ->  ( A  ^o  B )  =  ( A  ^o  C
) )
2927, 28impbid1 203 1  |-  ( ( A  e.  ( On 
\  2o )  /\  B  e.  On  /\  C  e.  On )  ->  (
( A  ^o  B
)  =  ( A  ^o  C )  <->  B  =  C ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 366    /\ w3a 974    = wceq 1405    e. wcel 1842    \ cdif 3411   Ord word 5409   Oncon0 5410  (class class class)co 6278   2oc2o 7161    ^o coe 7166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-omul 7172  df-oexp 7173
This theorem is referenced by:  oeword  7276  infxpenc2lem1  8428
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