Theorem ttglem 25756

Description: Lemma for ttgbas 25757 and ttgvsca 25760. (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.

Step	Hyp	Ref	Expression
1		ttgval.n	. . . . . 6 |- G = (toTG ` H )
2		eqid 2622	. . . . . 6 |- ( Base ` H ) = ( Base ` H )
3		eqid 2622	. . . . . 6 |- ( -g ` H ) = ( -g ` H )
4		eqid 2622	. . . . . 6 |- ( .s ` H ) = ( .s ` H )
5		eqid 2622	. . . . . 6 |- (Itv ` G ) = (Itv ` G )
6	1, 2, 3, 4, 5	ttgval 25755	. . . . 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 ) ) } ) ) )
7	6	simpld 475	. . . 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 ) ) } ) >. ) )
8	7	fveq2d 6195	. . 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
11	9, 10	ndxid 15883	. . . . 5 |- E = Slot ( E ` ndx )
12	10	nnrei 11029	. . . . . . 7 |- N e. RR
13		ttglem.4	. . . . . . 7 |- N < ; 1 6
14	12, 13	ltneii 10150	. . . . . 6 |- N =/= ; 1 6
15	9, 10	ndxarg 15882	. . . . . . 7 |- ( E ` ndx ) = N
16		itvndx 25339	. . . . . . 7 |- (Itv ` ndx ) = ; 1 6
17	15, 16	neeq12i 2860	. . . . . 6 |- ( ( E ` ndx ) =/= (Itv ` ndx ) <-> N =/= ; 1 6 )
18	14, 17	mpbir 221	. . . . 5 |- ( E ` ndx ) =/= (Itv ` ndx )
19	11, 18	setsnid 15915	. . . 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 11308	. . . . . . . . 9 |- 1 e. NN0
21		6nn0 11313	. . . . . . . . 9 |- 6 e. NN0
22		7nn 11190	. . . . . . . . 9 |- 7 e. NN
23		6lt7 11209	. . . . . . . . 9 |- 6 < 7
24	20, 21, 22, 23	declt 11530	. . . . . . . 8 |- ; 1 6 < ; 1 7
25		6nn 11189	. . . . . . . . . . 11 |- 6 e. NN
26	20, 25	decnncl 11518	. . . . . . . . . 10 |- ; 1 6 e. NN
27	26	nnrei 11029	. . . . . . . . 9 |- ; 1 6 e. RR
28	20, 22	decnncl 11518	. . . . . . . . . 10 |- ; 1 7 e. NN
29	28	nnrei 11029	. . . . . . . . 9 |- ; 1 7 e. RR
30	12, 27, 29	lttri 10163	. . . . . . . 8 |- ( ( N < ; 1 6 /\ ; 1 6 < ; 1 7 ) -> N < ; 1 7 )
31	13, 24, 30	mp2an 708	. . . . . . 7 |- N < ; 1 7
32	12, 31	ltneii 10150	. . . . . 6 |- N =/= ; 1 7
33		lngndx 25340	. . . . . . 7 |- (LineG ` ndx ) = ; 1 7
34	15, 33	neeq12i 2860	. . . . . 6 |- ( ( E ` ndx ) =/= (LineG ` ndx ) <-> N =/= ; 1 7 )
35	32, 34	mpbir 221	. . . . 5 |- ( E ` ndx ) =/= (LineG ` ndx )
36	11, 35	setsnid 15915	. . . 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 ) ) } ) >. ) )
37	19, 36	eqtri 2644	. . 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 ) ) } ) >. ) )
38	8, 37	syl6reqr 2675	. 2 |- ( H e. _V -> ( E ` H ) = ( E ` G ) )
39	9	str0 15911	. . 3 |- (/) = ( E ` (/) )
40		fvprc 6185	. . 3 |- ( -. H e. _V -> ( E ` H ) = (/) )
41		fvprc 6185	. . . . 5 |- ( -. H e. _V -> (toTG ` H ) = (/) )
42	1, 41	syl5eq 2668	. . . 4 |- ( -. H e. _V -> G = (/) )
43	42	fveq2d 6195	. . 3 |- ( -. H e. _V -> ( E ` G ) = ( E ` (/) ) )
44	39, 40, 43	3eqtr4a 2682	. 2 |- ( -. H e. _V -> ( E ` H ) = ( E ` G ) )
45	38, 44	pm2.61i 176	1 |- ( E ` H ) = ( E ` G )

Syntax hints:   -. wn 3   \/ w3o 1036   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   (/)c0 3915   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   < clt 10074   NNcn 11020   6c6 11074   7c7 11075  ;cdc 11493   [,]cicc 12178   ndxcnx 15854   sSet csts 15855  Slot cslot 15856   Basecbs 15857   .scvsca 15945   -gcsg 17424  Itvcitv 25335  LineGclng 25336  toTGcttg 25753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-dec 11494  df-ndx 15860  df-slot 15861  df-sets 15864  df-itv 25337  df-lng 25338  df-ttg 25754

This theorem is referenced by:  ttgbas  25757  ttgplusg  25758  ttgvsca  25760  ttgds  25761