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Theorem ndmaovcom 28065
Description: Any operation is commutative outside its domain, analogous to ndmovcom 6007. (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypothesis
Ref Expression
ndmaov.1  |-  dom  F  =  ( S  X.  S )
Assertion
Ref Expression
ndmaovcom  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  -> (( A F B))  = (( B F A))  )

Proof of Theorem ndmaovcom
StepHypRef Expression
1 opelxp 4719 . . . 4  |-  ( <. A ,  B >.  e.  ( S  X.  S
)  <->  ( A  e.  S  /\  B  e.  S ) )
2 ndmaov.1 . . . . . 6  |-  dom  F  =  ( S  X.  S )
32eqcomi 2287 . . . . 5  |-  ( S  X.  S )  =  dom  F
43eleq2i 2347 . . . 4  |-  ( <. A ,  B >.  e.  ( S  X.  S
)  <->  <. A ,  B >.  e.  dom  F )
51, 4bitr3i 242 . . 3  |-  ( ( A  e.  S  /\  B  e.  S )  <->  <. A ,  B >.  e. 
dom  F )
6 ndmaov 28043 . . 3  |-  ( -. 
<. A ,  B >.  e. 
dom  F  -> (( A F B))  =  _V )
75, 6sylnbi 297 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  -> (( A F B))  =  _V )
8 ancom 437 . . . 4  |-  ( ( A  e.  S  /\  B  e.  S )  <->  ( B  e.  S  /\  A  e.  S )
)
9 opelxp 4719 . . . 4  |-  ( <. B ,  A >.  e.  ( S  X.  S
)  <->  ( B  e.  S  /\  A  e.  S ) )
103eleq2i 2347 . . . 4  |-  ( <. B ,  A >.  e.  ( S  X.  S
)  <->  <. B ,  A >.  e.  dom  F )
118, 9, 103bitr2i 264 . . 3  |-  ( ( A  e.  S  /\  B  e.  S )  <->  <. B ,  A >.  e. 
dom  F )
12 ndmaov 28043 . . 3  |-  ( -. 
<. B ,  A >.  e. 
dom  F  -> (( B F A))  =  _V )
1311, 12sylnbi 297 . 2  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  -> (( B F A))  =  _V )
147, 13eqtr4d 2318 1  |-  ( -.  ( A  e.  S  /\  B  e.  S
)  -> (( A F B))  = (( B F A))  )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643    X. cxp 4687   dom cdm 4689   ((caov 27973
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-dfat 27974  df-afv 27975  df-aov 27976
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