ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  strndxid Unicode version

Theorem strndxid 12540
Description: The value of a structure component extractor is the value of the corresponding slot of the structure. (Contributed by AV, 13-Mar-2020.)
Hypotheses
Ref Expression
strndxid.s  |-  ( ph  ->  S  e.  V )
strndxid.e  |-  E  = Slot 
N
strndxid.n  |-  N  e.  NN
Assertion
Ref Expression
strndxid  |-  ( ph  ->  ( S `  ( E `  ndx ) )  =  ( E `  S ) )

Proof of Theorem strndxid
StepHypRef Expression
1 strndxid.e . . . 4  |-  E  = Slot 
N
2 strndxid.n . . . 4  |-  N  e.  NN
31, 2ndxid 12536 . . 3  |-  E  = Slot  ( E `  ndx )
4 strndxid.s . . 3  |-  ( ph  ->  S  e.  V )
51, 2ndxarg 12535 . . . . 5  |-  ( E `
 ndx )  =  N
65, 2eqeltri 2262 . . . 4  |-  ( E `
 ndx )  e.  NN
76a1i 9 . . 3  |-  ( ph  ->  ( E `  ndx )  e.  NN )
83, 4, 7strnfvnd 12532 . 2  |-  ( ph  ->  ( E `  S
)  =  ( S `
 ( E `  ndx ) ) )
98eqcomd 2195 1  |-  ( ph  ->  ( S `  ( E `  ndx ) )  =  ( E `  S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 2160   ` cfv 5235   NNcn 8949   ndxcnx 12509  Slot cslot 12511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-cnex 7932  ax-resscn 7933  ax-1re 7935  ax-addrcl 7938
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-iota 5196  df-fun 5237  df-fv 5243  df-inn 8950  df-ndx 12515  df-slot 12516
This theorem is referenced by:  imasbas  12784  imasplusg  12785  imasmulr  12786
  Copyright terms: Public domain W3C validator