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Theorem ttglem 23269
Description: Lemma for ttgbas 23270 and ttgvsca 23273. (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 2452 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
3 eqid 2452 . . . . . 6  |-  ( -g `  H )  =  (
-g `  H )
4 eqid 2452 . . . . . 6  |-  ( .s
`  H )  =  ( .s `  H
)
5 eqid 2452 . . . . . 6  |-  (Itv `  G )  =  (Itv
`  G )
61, 2, 3, 4, 5ttgval 23268 . . . . 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 459 . . . 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 5798 . . 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 14308 . . . . 5  |-  E  = Slot  ( E `  ndx )
1210nnrei 10437 . . . . . . 7  |-  N  e.  RR
13 ttglem.4 . . . . . . 7  |-  N  < ; 1 6
1412, 13ltneii 9593 . . . . . 6  |-  N  =/= ; 1 6
159, 10ndxarg 14307 . . . . . . 7  |-  ( E `
 ndx )  =  N
16 itvndx 23028 . . . . . . 7  |-  (Itv `  ndx )  = ; 1 6
1715, 16neeq12i 2738 . . . . . 6  |-  ( ( E `  ndx )  =/=  (Itv `  ndx )  <->  N  =/= ; 1 6 )
1814, 17mpbir 209 . . . . 5  |-  ( E `
 ndx )  =/=  (Itv `  ndx )
1911, 18setsnid 14329 . . . 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 10701 . . . . . . . . 9  |-  1  e.  NN0
21 6nn0 10706 . . . . . . . . 9  |-  6  e.  NN0
22 7nn 10590 . . . . . . . . 9  |-  7  e.  NN
23 6lt7 10609 . . . . . . . . 9  |-  6  <  7
2420, 21, 22, 23declt 10882 . . . . . . . 8  |- ; 1 6  < ; 1 7
25 6nn 10589 . . . . . . . . . . 11  |-  6  e.  NN
2620, 25decnncl 10874 . . . . . . . . . 10  |- ; 1 6  e.  NN
2726nnrei 10437 . . . . . . . . 9  |- ; 1 6  e.  RR
2820, 22decnncl 10874 . . . . . . . . . 10  |- ; 1 7  e.  NN
2928nnrei 10437 . . . . . . . . 9  |- ; 1 7  e.  RR
3012, 27, 29lttri 9606 . . . . . . . 8  |-  ( ( N  < ; 1 6  /\ ; 1 6  < ; 1 7 )  ->  N  < ; 1 7 )
3113, 24, 30mp2an 672 . . . . . . 7  |-  N  < ; 1 7
3212, 31ltneii 9593 . . . . . 6  |-  N  =/= ; 1 7
33 lngndx 23029 . . . . . . 7  |-  (LineG `  ndx )  = ; 1 7
3415, 33neeq12i 2738 . . . . . 6  |-  ( ( E `  ndx )  =/=  (LineG `  ndx )  <->  N  =/= ; 1 7 )
3532, 34mpbir 209 . . . . 5  |-  ( E `
 ndx )  =/=  (LineG `  ndx )
3611, 35setsnid 14329 . . . 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 2481 . . 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 2512 . 2  |-  ( H  e.  _V  ->  ( E `  H )  =  ( E `  G ) )
399str0 14325 . . 3  |-  (/)  =  ( E `  (/) )
40 fvprc 5788 . . 3  |-  ( -.  H  e.  _V  ->  ( E `  H )  =  (/) )
41 fvprc 5788 . . . . 5  |-  ( -.  H  e.  _V  ->  (toTG `  H )  =  (/) )
421, 41syl5eq 2505 . . . 4  |-  ( -.  H  e.  _V  ->  G  =  (/) )
4342fveq2d 5798 . . 3  |-  ( -.  H  e.  _V  ->  ( E `  G )  =  ( E `  (/) ) )
4439, 40, 433eqtr4a 2519 . 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 964    = wceq 1370    e. wcel 1758    =/= wne 2645   E.wrex 2797   {crab 2800   _Vcvv 3072   (/)c0 3740   <.cop 3986   class class class wbr 4395   ` cfv 5521  (class class class)co 6195    |-> cmpt2 6197   0cc0 9388   1c1 9389    < clt 9524   NNcn 10428   6c6 10481   7c7 10482  ;cdc 10861   [,]cicc 11409   ndxcnx 14284   sSet csts 14285  Slot cslot 14286   Basecbs 14287   .scvsca 14356   -gcsg 15527  Itvcitv 23024  LineGclng 23025  toTGcttg 23266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-2 10486  df-3 10487  df-4 10488  df-5 10489  df-6 10490  df-7 10491  df-8 10492  df-9 10493  df-10 10494  df-n0 10686  df-dec 10862  df-ndx 14290  df-slot 14291  df-sets 14293  df-itv 23026  df-lng 23027  df-ttg 23267
This theorem is referenced by:  ttgbas  23270  ttgplusg  23271  ttgvsca  23273  ttgds  23274
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