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Theorem ttglem 24381
Description: Lemma for ttgbas 24382 and ttgvsca 24385. (Contributed by Thierry Arnoux, 15-Apr-2019.)
Hypotheses
Ref Expression
ttgval.n  |-  G  =  (toTG `  H )
ttglem.2  |-  E  = Slot 
N
ttglem.3  |-  N  e.  NN
ttglem.4  |-  N  < ; 1 6
Assertion
Ref Expression
ttglem  |-  ( E `
 H )  =  ( E `  G
)

Proof of Theorem ttglem
Dummy variables  k  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ttgval.n . . . . . 6  |-  G  =  (toTG `  H )
2 eqid 2454 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
3 eqid 2454 . . . . . 6  |-  ( -g `  H )  =  (
-g `  H )
4 eqid 2454 . . . . . 6  |-  ( .s
`  H )  =  ( .s `  H
)
5 eqid 2454 . . . . . 6  |-  (Itv `  G )  =  (Itv
`  G )
61, 2, 3, 4, 5ttgval 24380 . . . . 5  |-  ( H  e.  _V  ->  ( G  =  ( ( H sSet  <. (Itv `  ndx ) ,  ( x  e.  ( Base `  H
) ,  y  e.  ( Base `  H
)  |->  { z  e.  ( Base `  H
)  |  E. k  e.  ( 0 [,] 1
) ( z (
-g `  H )
x )  =  ( k ( .s `  H ) ( y ( -g `  H
) x ) ) } ) >. ) sSet  <.
(LineG `  ndx ) ,  ( x  e.  (
Base `  H ) ,  y  e.  ( Base `  H )  |->  { z  e.  ( Base `  H )  |  ( z  e.  ( x (Itv `  G )
y )  \/  x  e.  ( z (Itv `  G ) y )  \/  y  e.  ( x (Itv `  G
) z ) ) } ) >. )  /\  (Itv `  G )  =  ( x  e.  ( Base `  H
) ,  y  e.  ( Base `  H
)  |->  { z  e.  ( Base `  H
)  |  E. k  e.  ( 0 [,] 1
) ( z (
-g `  H )
x )  =  ( k ( .s `  H ) ( y ( -g `  H
) x ) ) } ) ) )
76simpld 457 . . . 4  |-  ( H  e.  _V  ->  G  =  ( ( H sSet  <. (Itv `  ndx ) ,  ( x  e.  (
Base `  H ) ,  y  e.  ( Base `  H )  |->  { z  e.  ( Base `  H )  |  E. k  e.  ( 0 [,] 1 ) ( z ( -g `  H
) x )  =  ( k ( .s
`  H ) ( y ( -g `  H
) x ) ) } ) >. ) sSet  <.
(LineG `  ndx ) ,  ( x  e.  (
Base `  H ) ,  y  e.  ( Base `  H )  |->  { z  e.  ( Base `  H )  |  ( z  e.  ( x (Itv `  G )
y )  \/  x  e.  ( z (Itv `  G ) y )  \/  y  e.  ( x (Itv `  G
) z ) ) } ) >. )
)
87fveq2d 5852 . . 3  |-  ( H  e.  _V  ->  ( E `  G )  =  ( E `  ( ( H sSet  <. (Itv
`  ndx ) ,  ( x  e.  ( Base `  H ) ,  y  e.  ( Base `  H
)  |->  { z  e.  ( Base `  H
)  |  E. k  e.  ( 0 [,] 1
) ( z (
-g `  H )
x )  =  ( k ( .s `  H ) ( y ( -g `  H
) x ) ) } ) >. ) sSet  <.
(LineG `  ndx ) ,  ( x  e.  (
Base `  H ) ,  y  e.  ( Base `  H )  |->  { z  e.  ( Base `  H )  |  ( z  e.  ( x (Itv `  G )
y )  \/  x  e.  ( z (Itv `  G ) y )  \/  y  e.  ( x (Itv `  G
) z ) ) } ) >. )
) )
9 ttglem.2 . . . . . 6  |-  E  = Slot 
N
10 ttglem.3 . . . . . 6  |-  N  e.  NN
119, 10ndxid 14737 . . . . 5  |-  E  = Slot  ( E `  ndx )
1210nnrei 10540 . . . . . . 7  |-  N  e.  RR
13 ttglem.4 . . . . . . 7  |-  N  < ; 1 6
1412, 13ltneii 9686 . . . . . 6  |-  N  =/= ; 1 6
159, 10ndxarg 14736 . . . . . . 7  |-  ( E `
 ndx )  =  N
16 itvndx 24034 . . . . . . 7  |-  (Itv `  ndx )  = ; 1 6
1715, 16neeq12i 2743 . . . . . 6  |-  ( ( E `  ndx )  =/=  (Itv `  ndx )  <->  N  =/= ; 1 6 )
1814, 17mpbir 209 . . . . 5  |-  ( E `
 ndx )  =/=  (Itv `  ndx )
1911, 18setsnid 14760 . . . 4  |-  ( E `
 H )  =  ( E `  ( H sSet  <. (Itv `  ndx ) ,  ( x  e.  ( Base `  H
) ,  y  e.  ( Base `  H
)  |->  { z  e.  ( Base `  H
)  |  E. k  e.  ( 0 [,] 1
) ( z (
-g `  H )
x )  =  ( k ( .s `  H ) ( y ( -g `  H
) x ) ) } ) >. )
)
20 1nn0 10807 . . . . . . . . 9  |-  1  e.  NN0
21 6nn0 10812 . . . . . . . . 9  |-  6  e.  NN0
22 7nn 10694 . . . . . . . . 9  |-  7  e.  NN
23 6lt7 10713 . . . . . . . . 9  |-  6  <  7
2420, 21, 22, 23declt 10997 . . . . . . . 8  |- ; 1 6  < ; 1 7
25 6nn 10693 . . . . . . . . . . 11  |-  6  e.  NN
2620, 25decnncl 10989 . . . . . . . . . 10  |- ; 1 6  e.  NN
2726nnrei 10540 . . . . . . . . 9  |- ; 1 6  e.  RR
2820, 22decnncl 10989 . . . . . . . . . 10  |- ; 1 7  e.  NN
2928nnrei 10540 . . . . . . . . 9  |- ; 1 7  e.  RR
3012, 27, 29lttri 9699 . . . . . . . 8  |-  ( ( N  < ; 1 6  /\ ; 1 6  < ; 1 7 )  ->  N  < ; 1 7 )
3113, 24, 30mp2an 670 . . . . . . 7  |-  N  < ; 1 7
3212, 31ltneii 9686 . . . . . 6  |-  N  =/= ; 1 7
33 lngndx 24035 . . . . . . 7  |-  (LineG `  ndx )  = ; 1 7
3415, 33neeq12i 2743 . . . . . 6  |-  ( ( E `  ndx )  =/=  (LineG `  ndx )  <->  N  =/= ; 1 7 )
3532, 34mpbir 209 . . . . 5  |-  ( E `
 ndx )  =/=  (LineG `  ndx )
3611, 35setsnid 14760 . . . 4  |-  ( E `
 ( H sSet  <. (Itv
`  ndx ) ,  ( x  e.  ( Base `  H ) ,  y  e.  ( Base `  H
)  |->  { z  e.  ( Base `  H
)  |  E. k  e.  ( 0 [,] 1
) ( z (
-g `  H )
x )  =  ( k ( .s `  H ) ( y ( -g `  H
) x ) ) } ) >. )
)  =  ( E `
 ( ( H sSet  <. (Itv `  ndx ) ,  ( x  e.  (
Base `  H ) ,  y  e.  ( Base `  H )  |->  { z  e.  ( Base `  H )  |  E. k  e.  ( 0 [,] 1 ) ( z ( -g `  H
) x )  =  ( k ( .s
`  H ) ( y ( -g `  H
) x ) ) } ) >. ) sSet  <.
(LineG `  ndx ) ,  ( x  e.  (
Base `  H ) ,  y  e.  ( Base `  H )  |->  { z  e.  ( Base `  H )  |  ( z  e.  ( x (Itv `  G )
y )  \/  x  e.  ( z (Itv `  G ) y )  \/  y  e.  ( x (Itv `  G
) z ) ) } ) >. )
)
3719, 36eqtri 2483 . . 3  |-  ( E `
 H )  =  ( E `  (
( H sSet  <. (Itv `  ndx ) ,  ( x  e.  ( Base `  H
) ,  y  e.  ( Base `  H
)  |->  { z  e.  ( Base `  H
)  |  E. k  e.  ( 0 [,] 1
) ( z (
-g `  H )
x )  =  ( k ( .s `  H ) ( y ( -g `  H
) x ) ) } ) >. ) sSet  <.
(LineG `  ndx ) ,  ( x  e.  (
Base `  H ) ,  y  e.  ( Base `  H )  |->  { z  e.  ( Base `  H )  |  ( z  e.  ( x (Itv `  G )
y )  \/  x  e.  ( z (Itv `  G ) y )  \/  y  e.  ( x (Itv `  G
) z ) ) } ) >. )
)
388, 37syl6reqr 2514 . 2  |-  ( H  e.  _V  ->  ( E `  H )  =  ( E `  G ) )
399str0 14756 . . 3  |-  (/)  =  ( E `  (/) )
40 fvprc 5842 . . 3  |-  ( -.  H  e.  _V  ->  ( E `  H )  =  (/) )
41 fvprc 5842 . . . . 5  |-  ( -.  H  e.  _V  ->  (toTG `  H )  =  (/) )
421, 41syl5eq 2507 . . . 4  |-  ( -.  H  e.  _V  ->  G  =  (/) )
4342fveq2d 5852 . . 3  |-  ( -.  H  e.  _V  ->  ( E `  G )  =  ( E `  (/) ) )
4439, 40, 433eqtr4a 2521 . 2  |-  ( -.  H  e.  _V  ->  ( E `  H )  =  ( E `  G ) )
4538, 44pm2.61i 164 1  |-  ( E `
 H )  =  ( E `  G
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ w3o 970    = wceq 1398    e. wcel 1823    =/= wne 2649   E.wrex 2805   {crab 2808   _Vcvv 3106   (/)c0 3783   <.cop 4022   class class class wbr 4439   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   0cc0 9481   1c1 9482    < clt 9617   NNcn 10531   6c6 10585   7c7 10586  ;cdc 10976   [,]cicc 11535   ndxcnx 14713   sSet csts 14714  Slot cslot 14715   Basecbs 14716   .scvsca 14788   -gcsg 16254  Itvcitv 24030  LineGclng 24031  toTGcttg 24378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-dec 10977  df-ndx 14719  df-slot 14720  df-sets 14722  df-itv 24032  df-lng 24033  df-ttg 24379
This theorem is referenced by:  ttgbas  24382  ttgplusg  24383  ttgvsca  24385  ttgds  24386
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