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Theorem 2stropg 12598
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 12505 . 2  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
4 2str.g . . 3  |-  G  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. }
5 basendxnn 12536 . . . . . 6  |-  ( Base `  ndx )  e.  NN
65a1i 9 . . . . 5  |-  ( ( B  e.  V  /\  .+  e.  W )  -> 
( Base `  ndx )  e.  NN )
7 simpl 109 . . . . 5  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  B  e.  V )
8 opexg 4243 . . . . 5  |-  ( ( ( Base `  ndx )  e.  NN  /\  B  e.  V )  ->  <. ( Base `  ndx ) ,  B >.  e.  _V )
96, 7, 8syl2anc 411 . . . 4  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  <. ( Base `  ndx ) ,  B >.  e. 
_V )
101, 2ndxarg 12503 . . . . . . 7  |-  ( E `
 ndx )  =  N
1110, 2eqeltri 2262 . . . . . 6  |-  ( E `
 ndx )  e.  NN
1211a1i 9 . . . . 5  |-  ( ( B  e.  V  /\  .+  e.  W )  -> 
( E `  ndx )  e.  NN )
13 simpr 110 . . . . 5  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  .+  e.  W )
14 opexg 4243 . . . . 5  |-  ( ( ( E `  ndx )  e.  NN  /\  .+  e.  W )  ->  <. ( E `  ndx ) , 
.+  >.  e.  _V )
1512, 13, 14syl2anc 411 . . . 4  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  <. ( E `  ndx ) ,  .+  >.  e.  _V )
16 prexg 4226 . . . 4  |-  ( (
<. ( Base `  ndx ) ,  B >.  e. 
_V  /\  <. ( E `
 ndx ) , 
.+  >.  e.  _V )  ->  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. }  e.  _V )
179, 15, 16syl2anc 411 . . 3  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. }  e.  _V )
184, 17eqeltrid 2276 . 2  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  G  e.  _V )
195nnrei 8946 . . . . . 6  |-  ( Base `  ndx )  e.  RR
20 2str.l . . . . . . 7  |-  1  <  N
21 basendx 12535 . . . . . . 7  |-  ( Base `  ndx )  =  1
2220, 21, 103brtr4i 4048 . . . . . 6  |-  ( Base `  ndx )  <  ( E `  ndx )
2319, 22ltneii 8072 . . . . 5  |-  ( Base `  ndx )  =/=  ( E `  ndx )
2423a1i 9 . . . 4  |-  ( ( B  e.  V  /\  .+  e.  W )  -> 
( Base `  ndx )  =/=  ( E `  ndx ) )
25 funprg 5281 . . . 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 1260 . . 3  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  Fun  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. } )
274funeqi 5252 . . 3  |-  ( Fun 
G  <->  Fun  { <. ( Base `  ndx ) ,  B >. ,  <. ( E `  ndx ) , 
.+  >. } )
2826, 27sylibr 134 . 2  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  Fun  G )
29 prid2g 3712 . . . 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 2283 . 2  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  <. ( E `  ndx ) ,  .+  >.  e.  G
)
323, 18, 28, 31strslfvd 12522 1  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  .+  =  ( E `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160    =/= wne 2360   _Vcvv 2752   {cpr 3608   <.cop 3610   class class class wbr 4018   Fun wfun 5225   ` cfv 5231   1c1 7830    < clt 8010   NNcn 8937   ndxcnx 12477  Slot cslot 12479   Basecbs 12480
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-in1 615  ax-in2 616  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 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7920  ax-resscn 7921  ax-1re 7923  ax-addrcl 7926  ax-pre-ltirr 7941
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  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-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  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 4308  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-iota 5193  df-fun 5233  df-fv 5239  df-pnf 8012  df-mnf 8013  df-ltxr 8015  df-inn 8938  df-ndx 12483  df-slot 12484  df-base 12486
This theorem is referenced by:  grpplusgg  12605  eltpsg  13937
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