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Theorem ttglem 24848
Description: Lemma for ttgbas 24849 and ttgvsca 24852. (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 2428 . . . . . 6  |-  ( Base `  H )  =  (
Base `  H )
3 eqid 2428 . . . . . 6  |-  ( -g `  H )  =  (
-g `  H )
4 eqid 2428 . . . . . 6  |-  ( .s
`  H )  =  ( .s `  H
)
5 eqid 2428 . . . . . 6  |-  (Itv `  G )  =  (Itv
`  G )
61, 2, 3, 4, 5ttgval 24847 . . . . 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 460 . . . 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 5829 . . 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 15085 . . . . 5  |-  E  = Slot  ( E `  ndx )
1210nnrei 10569 . . . . . . 7  |-  N  e.  RR
13 ttglem.4 . . . . . . 7  |-  N  < ; 1 6
1412, 13ltneii 9698 . . . . . 6  |-  N  =/= ; 1 6
159, 10ndxarg 15084 . . . . . . 7  |-  ( E `
 ndx )  =  N
16 itvndx 24430 . . . . . . 7  |-  (Itv `  ndx )  = ; 1 6
1715, 16neeq12i 2667 . . . . . 6  |-  ( ( E `  ndx )  =/=  (Itv `  ndx )  <->  N  =/= ; 1 6 )
1814, 17mpbir 212 . . . . 5  |-  ( E `
 ndx )  =/=  (Itv `  ndx )
1911, 18setsnid 15108 . . . 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 10836 . . . . . . . . 9  |-  1  e.  NN0
21 6nn0 10841 . . . . . . . . 9  |-  6  e.  NN0
22 7nn 10723 . . . . . . . . 9  |-  7  e.  NN
23 6lt7 10742 . . . . . . . . 9  |-  6  <  7
2420, 21, 22, 23declt 11023 . . . . . . . 8  |- ; 1 6  < ; 1 7
25 6nn 10722 . . . . . . . . . . 11  |-  6  e.  NN
2620, 25decnncl 11015 . . . . . . . . . 10  |- ; 1 6  e.  NN
2726nnrei 10569 . . . . . . . . 9  |- ; 1 6  e.  RR
2820, 22decnncl 11015 . . . . . . . . . 10  |- ; 1 7  e.  NN
2928nnrei 10569 . . . . . . . . 9  |- ; 1 7  e.  RR
3012, 27, 29lttri 9711 . . . . . . . 8  |-  ( ( N  < ; 1 6  /\ ; 1 6  < ; 1 7 )  ->  N  < ; 1 7 )
3113, 24, 30mp2an 676 . . . . . . 7  |-  N  < ; 1 7
3212, 31ltneii 9698 . . . . . 6  |-  N  =/= ; 1 7
33 lngndx 24431 . . . . . . 7  |-  (LineG `  ndx )  = ; 1 7
3415, 33neeq12i 2667 . . . . . 6  |-  ( ( E `  ndx )  =/=  (LineG `  ndx )  <->  N  =/= ; 1 7 )
3532, 34mpbir 212 . . . . 5  |-  ( E `
 ndx )  =/=  (LineG `  ndx )
3611, 35setsnid 15108 . . . 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 2450 . . 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 2481 . 2  |-  ( H  e.  _V  ->  ( E `  H )  =  ( E `  G ) )
399str0 15104 . . 3  |-  (/)  =  ( E `  (/) )
40 fvprc 5819 . . 3  |-  ( -.  H  e.  _V  ->  ( E `  H )  =  (/) )
41 fvprc 5819 . . . . 5  |-  ( -.  H  e.  _V  ->  (toTG `  H )  =  (/) )
421, 41syl5eq 2474 . . . 4  |-  ( -.  H  e.  _V  ->  G  =  (/) )
4342fveq2d 5829 . . 3  |-  ( -.  H  e.  _V  ->  ( E `  G )  =  ( E `  (/) ) )
4439, 40, 433eqtr4a 2488 . 2  |-  ( -.  H  e.  _V  ->  ( E `  H )  =  ( E `  G ) )
4538, 44pm2.61i 167 1  |-  ( E `
 H )  =  ( E `  G
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ w3o 981    = wceq 1437    e. wcel 1872    =/= wne 2599   E.wrex 2715   {crab 2718   _Vcvv 3022   (/)c0 3704   <.cop 3947   class class class wbr 4366   ` cfv 5544  (class class class)co 6249    |-> cmpt2 6251   0cc0 9490   1c1 9491    < clt 9626   NNcn 10560   6c6 10614   7c7 10615  ;cdc 11002   [,]cicc 11589   ndxcnx 15061   sSet csts 15062  Slot cslot 15063   Basecbs 15064   .scvsca 15137   -gcsg 16614  Itvcitv 24426  LineGclng 24427  toTGcttg 24845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-rep 4479  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-iun 4244  df-br 4367  df-opab 4426  df-mpt 4427  df-tr 4462  df-eprel 4707  df-id 4711  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-pred 5342  df-ord 5388  df-on 5389  df-lim 5390  df-suc 5391  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-riota 6211  df-ov 6252  df-oprab 6253  df-mpt2 6254  df-om 6651  df-1st 6751  df-2nd 6752  df-wrecs 6983  df-recs 7045  df-rdg 7083  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9813  df-neg 9814  df-nn 10561  df-2 10619  df-3 10620  df-4 10621  df-5 10622  df-6 10623  df-7 10624  df-8 10625  df-9 10626  df-10 10627  df-n0 10821  df-dec 11003  df-ndx 15067  df-slot 15068  df-sets 15070  df-itv 24428  df-lng 24429  df-ttg 24846
This theorem is referenced by:  ttgbas  24849  ttgplusg  24850  ttgvsca  24852  ttgds  24853
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