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Theorem 2stropg 12459
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 12382 . 2  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
4 2str.g . . 3  |-  G  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. }
5 basendxnn 12412 . . . . . 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 4201 . . . . 5  |-  ( ( ( Base `  ndx )  e.  NN  /\  B  e.  V )  ->  <. ( Base `  ndx ) ,  B >.  e.  _V )
96, 7, 8syl2anc 409 . . . 4  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  <. ( Base `  ndx ) ,  B >.  e. 
_V )
101, 2ndxarg 12380 . . . . . . 7  |-  ( E `
 ndx )  =  N
1110, 2eqeltri 2237 . . . . . 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 4201 . . . . 5  |-  ( ( ( E `  ndx )  e.  NN  /\  .+  e.  W )  ->  <. ( E `  ndx ) , 
.+  >.  e.  _V )
1512, 13, 14syl2anc 409 . . . 4  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  <. ( E `  ndx ) ,  .+  >.  e.  _V )
16 prexg 4184 . . . 4  |-  ( (
<. ( Base `  ndx ) ,  B >.  e. 
_V  /\  <. ( E `
 ndx ) , 
.+  >.  e.  _V )  ->  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. }  e.  _V )
179, 15, 16syl2anc 409 . . 3  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. }  e.  _V )
184, 17eqeltrid 2251 . 2  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  G  e.  _V )
195nnrei 8858 . . . . . 6  |-  ( Base `  ndx )  e.  RR
20 2str.l . . . . . . 7  |-  1  <  N
21 basendx 12411 . . . . . . 7  |-  ( Base `  ndx )  =  1
2220, 21, 103brtr4i 4007 . . . . . 6  |-  ( Base `  ndx )  <  ( E `  ndx )
2319, 22ltneii 7987 . . . . 5  |-  ( Base `  ndx )  =/=  ( E `  ndx )
2423a1i 9 . . . 4  |-  ( ( B  e.  V  /\  .+  e.  W )  -> 
( Base `  ndx )  =/=  ( E `  ndx ) )
25 funprg 5233 . . . 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 1238 . . 3  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  Fun  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. } )
274funeqi 5204 . . 3  |-  ( Fun 
G  <->  Fun  { <. ( Base `  ndx ) ,  B >. ,  <. ( E `  ndx ) , 
.+  >. } )
2826, 27sylibr 133 . 2  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  Fun  G )
29 prid2g 3676 . . . 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 2258 . 2  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  <. ( E `  ndx ) ,  .+  >.  e.  G
)
323, 18, 28, 31strslfvd 12398 1  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  .+  =  ( E `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1342    e. wcel 2135    =/= wne 2334   _Vcvv 2722   {cpr 3572   <.cop 3574   class class class wbr 3977   Fun wfun 5177   ` cfv 5183   1c1 7746    < clt 7925   NNcn 8849   ndxcnx 12354  Slot cslot 12356   Basecbs 12357
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182  ax-un 4406  ax-setind 4509  ax-cnex 7836  ax-resscn 7837  ax-1re 7839  ax-addrcl 7842  ax-pre-ltirr 7857
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2724  df-sbc 2948  df-dif 3114  df-un 3116  df-in 3118  df-ss 3125  df-nul 3406  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-uni 3785  df-int 3820  df-br 3978  df-opab 4039  df-mpt 4040  df-id 4266  df-xp 4605  df-rel 4606  df-cnv 4607  df-co 4608  df-dm 4609  df-rn 4610  df-res 4611  df-iota 5148  df-fun 5185  df-fv 5191  df-pnf 7927  df-mnf 7928  df-ltxr 7930  df-inn 8850  df-ndx 12360  df-slot 12361  df-base 12363
This theorem is referenced by:  grpplusgg  12466  eltpsg  12605
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