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Theorem 2stropg 12075
Description: The other slot of a constructed two-slot structure. (Contributed by Mario Carneiro, 29-Aug-2015.) (Revised by Jim Kingdon, 28-Jan-2023.)
Hypotheses
Ref Expression
2str.g  |-  G  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. }
2str.e  |-  E  = Slot 
N
2str.l  |-  1  <  N
2str.n  |-  N  e.  NN
Assertion
Ref Expression
2stropg  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  .+  =  ( E `  G ) )

Proof of Theorem 2stropg
StepHypRef Expression
1 2str.e . . 3  |-  E  = Slot 
N
2 2str.n . . 3  |-  N  e.  NN
31, 2ndxslid 11998 . 2  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
4 2str.g . . 3  |-  G  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. }
5 basendxnn 12028 . . . . . 6  |-  ( Base `  ndx )  e.  NN
65a1i 9 . . . . 5  |-  ( ( B  e.  V  /\  .+  e.  W )  -> 
( Base `  ndx )  e.  NN )
7 simpl 108 . . . . 5  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  B  e.  V )
8 opexg 4150 . . . . 5  |-  ( ( ( Base `  ndx )  e.  NN  /\  B  e.  V )  ->  <. ( Base `  ndx ) ,  B >.  e.  _V )
96, 7, 8syl2anc 408 . . . 4  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  <. ( Base `  ndx ) ,  B >.  e. 
_V )
101, 2ndxarg 11996 . . . . . . 7  |-  ( E `
 ndx )  =  N
1110, 2eqeltri 2212 . . . . . 6  |-  ( E `
 ndx )  e.  NN
1211a1i 9 . . . . 5  |-  ( ( B  e.  V  /\  .+  e.  W )  -> 
( E `  ndx )  e.  NN )
13 simpr 109 . . . . 5  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  .+  e.  W )
14 opexg 4150 . . . . 5  |-  ( ( ( E `  ndx )  e.  NN  /\  .+  e.  W )  ->  <. ( E `  ndx ) , 
.+  >.  e.  _V )
1512, 13, 14syl2anc 408 . . . 4  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  <. ( E `  ndx ) ,  .+  >.  e.  _V )
16 prexg 4133 . . . 4  |-  ( (
<. ( Base `  ndx ) ,  B >.  e. 
_V  /\  <. ( E `
 ndx ) , 
.+  >.  e.  _V )  ->  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. }  e.  _V )
179, 15, 16syl2anc 408 . . 3  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. }  e.  _V )
184, 17eqeltrid 2226 . 2  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  G  e.  _V )
195nnrei 8741 . . . . . 6  |-  ( Base `  ndx )  e.  RR
20 2str.l . . . . . . 7  |-  1  <  N
21 basendx 12027 . . . . . . 7  |-  ( Base `  ndx )  =  1
2220, 21, 103brtr4i 3958 . . . . . 6  |-  ( Base `  ndx )  <  ( E `  ndx )
2319, 22ltneii 7872 . . . . 5  |-  ( Base `  ndx )  =/=  ( E `  ndx )
2423a1i 9 . . . 4  |-  ( ( B  e.  V  /\  .+  e.  W )  -> 
( Base `  ndx )  =/=  ( E `  ndx ) )
25 funprg 5173 . . . 4  |-  ( ( ( ( Base `  ndx )  e.  NN  /\  ( E `  ndx )  e.  NN )  /\  ( B  e.  V  /\  .+  e.  W )  /\  ( Base `  ndx )  =/=  ( E `  ndx ) )  ->  Fun  {
<. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. } )
266, 12, 7, 13, 24, 25syl221anc 1227 . . 3  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  Fun  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. } )
274funeqi 5144 . . 3  |-  ( Fun 
G  <->  Fun  { <. ( Base `  ndx ) ,  B >. ,  <. ( E `  ndx ) , 
.+  >. } )
2826, 27sylibr 133 . 2  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  Fun  G )
29 prid2g 3628 . . . 4  |-  ( <.
( E `  ndx ) ,  .+  >.  e.  _V  -> 
<. ( E `  ndx ) ,  .+  >.  e.  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. } )
3015, 29syl 14 . . 3  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  <. ( E `  ndx ) ,  .+  >.  e.  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. } )
3130, 4eleqtrrdi 2233 . 2  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  <. ( E `  ndx ) ,  .+  >.  e.  G
)
323, 18, 28, 31strslfvd 12014 1  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  .+  =  ( E `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1331    e. wcel 1480    =/= wne 2308   _Vcvv 2686   {cpr 3528   <.cop 3530   class class class wbr 3929   Fun wfun 5117   ` cfv 5123   1c1 7633    < clt 7812   NNcn 8732   ndxcnx 11970  Slot cslot 11972   Basecbs 11973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1re 7726  ax-addrcl 7729  ax-pre-ltirr 7744
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-iota 5088  df-fun 5125  df-fv 5131  df-pnf 7814  df-mnf 7815  df-ltxr 7817  df-inn 8733  df-ndx 11976  df-slot 11977  df-base 11979
This theorem is referenced by:  grpplusgg  12082  eltpsg  12221
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