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Theorem 2stropg 11843
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 11766 . 2  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
4 2str.g . . 3  |-  G  =  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. }
5 basendxnn 11796 . . . . . 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 4088 . . . . 5  |-  ( ( ( Base `  ndx )  e.  NN  /\  B  e.  V )  ->  <. ( Base `  ndx ) ,  B >.  e.  _V )
96, 7, 8syl2anc 406 . . . 4  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  <. ( Base `  ndx ) ,  B >.  e. 
_V )
101, 2ndxarg 11764 . . . . . . 7  |-  ( E `
 ndx )  =  N
1110, 2eqeltri 2172 . . . . . 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 4088 . . . . 5  |-  ( ( ( E `  ndx )  e.  NN  /\  .+  e.  W )  ->  <. ( E `  ndx ) , 
.+  >.  e.  _V )
1512, 13, 14syl2anc 406 . . . 4  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  <. ( E `  ndx ) ,  .+  >.  e.  _V )
16 prexg 4071 . . . 4  |-  ( (
<. ( Base `  ndx ) ,  B >.  e. 
_V  /\  <. ( E `
 ndx ) , 
.+  >.  e.  _V )  ->  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. }  e.  _V )
179, 15, 16syl2anc 406 . . 3  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. }  e.  _V )
184, 17syl5eqel 2186 . 2  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  G  e.  _V )
195nnrei 8587 . . . . . 6  |-  ( Base `  ndx )  e.  RR
20 2str.l . . . . . . 7  |-  1  <  N
21 basendx 11795 . . . . . . 7  |-  ( Base `  ndx )  =  1
2220, 21, 103brtr4i 3903 . . . . . 6  |-  ( Base `  ndx )  <  ( E `  ndx )
2319, 22ltneii 7731 . . . . 5  |-  ( Base `  ndx )  =/=  ( E `  ndx )
2423a1i 9 . . . 4  |-  ( ( B  e.  V  /\  .+  e.  W )  -> 
( Base `  ndx )  =/=  ( E `  ndx ) )
25 funprg 5109 . . . 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 1195 . . 3  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  Fun  { <. ( Base `  ndx ) ,  B >. , 
<. ( E `  ndx ) ,  .+  >. } )
274funeqi 5080 . . 3  |-  ( Fun 
G  <->  Fun  { <. ( Base `  ndx ) ,  B >. ,  <. ( E `  ndx ) , 
.+  >. } )
2826, 27sylibr 133 . 2  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  Fun  G )
29 prid2g 3575 . . . 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, 4syl6eleqr 2193 . 2  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  <. ( E `  ndx ) ,  .+  >.  e.  G
)
323, 18, 28, 31strslfvd 11782 1  |-  ( ( B  e.  V  /\  .+  e.  W )  ->  .+  =  ( E `  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1299    e. wcel 1448    =/= wne 2267   _Vcvv 2641   {cpr 3475   <.cop 3477   class class class wbr 3875   Fun wfun 5053   ` cfv 5059   1c1 7501    < clt 7672   NNcn 8578   ndxcnx 11738  Slot cslot 11740   Basecbs 11741
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1re 7589  ax-addrcl 7592  ax-pre-ltirr 7607
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-iota 5024  df-fun 5061  df-fv 5067  df-pnf 7674  df-mnf 7675  df-ltxr 7677  df-inn 8579  df-ndx 11744  df-slot 11745  df-base 11747
This theorem is referenced by:  grpplusgg  11850  eltpsg  11989
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