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Theorem 2strbasg 12512
Description: The base set 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
2strbasg  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  B  =  ( Base `  G ) )

Proof of Theorem 2strbasg
StepHypRef Expression
1 baseslid 12465 . 2  |-  ( Base 
= Slot  ( Base `  ndx )  /\  ( Base `  ndx )  e.  NN )
2 2str.g . . 3  |-  G  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. }
3 basendxnn 12464 . . . . . 6  |-  ( Base `  ndx )  e.  NN
43a1i 9 . . . . 5  |-  ( ( B  e.  V  /\  .+  e.  W )  -> 
( Base `  ndx )  e.  NN )
5 simpl 108 . . . . 5  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  B  e.  V )
6 opexg 4211 . . . . 5  |-  ( ( ( Base `  ndx )  e.  NN  /\  B  e.  V )  ->  <. ( Base `  ndx ) ,  B >.  e.  _V )
74, 5, 6syl2anc 409 . . . 4  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  <. ( Base `  ndx ) ,  B >.  e. 
_V )
8 2str.e . . . . . . . 8  |-  E  = Slot 
N
9 2str.n . . . . . . . 8  |-  N  e.  NN
108, 9ndxarg 12432 . . . . . . 7  |-  ( E `
 ndx )  =  N
1110, 9eqeltri 2243 . . . . . 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 4211 . . . . 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 4194 . . . 4  |-  ( (
<. ( Base `  ndx ) ,  B >.  e. 
_V  /\  <. ( E `
 ndx ) , 
.+  >.  e.  _V )  ->  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. }  e.  _V )
177, 15, 16syl2anc 409 . . 3  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. }  e.  _V )
182, 17eqeltrid 2257 . 2  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  G  e.  _V )
193nnrei 8880 . . . . . 6  |-  ( Base `  ndx )  e.  RR
20 2str.l . . . . . . 7  |-  1  <  N
21 basendx 12463 . . . . . . 7  |-  ( Base `  ndx )  =  1
2220, 21, 103brtr4i 4017 . . . . . 6  |-  ( Base `  ndx )  <  ( E `  ndx )
2319, 22ltneii 8009 . . . . 5  |-  ( Base `  ndx )  =/=  ( E `  ndx )
2423a1i 9 . . . 4  |-  ( ( B  e.  V  /\  .+  e.  W )  -> 
( Base `  ndx )  =/=  ( E `  ndx ) )
25 funprg 5246 . . . 4  |-  ( ( ( ( Base `  ndx )  e.  NN  /\  ( E `  ndx )  e.  NN )  /\  ( B  e.  V  /\  .+  e.  W )  /\  ( Base `  ndx )  =/=  ( E `  ndx ) )  ->  Fun  {
<. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. } )
264, 12, 5, 13, 24, 25syl221anc 1244 . . 3  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  Fun  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. } )
272funeqi 5217 . . 3  |-  ( Fun 
G  <->  Fun  { <. ( Base `  ndx ) ,  B >. ,  <. ( E `  ndx ) , 
.+  >. } )
2826, 27sylibr 133 . 2  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  Fun  G )
29 prid1g 3685 . . . 4  |-  ( <.
( Base `  ndx ) ,  B >.  e.  _V  -> 
<. ( Base `  ndx ) ,  B >.  e. 
{ <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. } )
307, 29syl 14 . . 3  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  <. ( Base `  ndx ) ,  B >.  e. 
{ <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. } )
3130, 2eleqtrrdi 2264 . 2  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  <. ( Base `  ndx ) ,  B >.  e.  G )
321, 18, 28, 31strslfvd 12450 1  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  B  =  ( Base `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1348    e. wcel 2141    =/= wne 2340   _Vcvv 2730   {cpr 3582   <.cop 3584   class class class wbr 3987   Fun wfun 5190   ` cfv 5196   1c1 7768    < clt 7947   NNcn 8871   ndxcnx 12406  Slot cslot 12408   Basecbs 12409
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7858  ax-resscn 7859  ax-1re 7861  ax-addrcl 7864  ax-pre-ltirr 7879
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-iota 5158  df-fun 5198  df-fv 5204  df-pnf 7949  df-mnf 7950  df-ltxr 7952  df-inn 8872  df-ndx 12412  df-slot 12413  df-base 12415
This theorem is referenced by:  grpbaseg  12519  eltpsg  12797
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