Theorem istrkgl 25357

Description: Building lines from the segment property. (Contributed by Thierry Arnoux, 14-Mar-2019.)

Hypotheses

Ref	Expression
istrkg.p	|- P = ( Base ` G )
istrkg.d	|- .- = ( dist ` G )
istrkg.i	|- I = (Itv ` G )

Assertion

Ref	Expression
istrkgl	|- ( G e. { f | [. ( Base ` f ) / p ]. [. (Itv ` f ) / i ]. (LineG ` f ) = ( x e. p , y e. ( p \ { x } ) |-> { z e. p | ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) } ) } <-> ( G e. _V /\ (LineG ` G ) = ( x e. P , y e. ( P \ { x } ) |-> { z e. P | ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) } ) ) )

Distinct variable groups:   f, i, p, G   x, f, y, z, I, i, p   P, f, i, p, x, y, z   .- , f, i, p, x, y, z

Allowed substitution hints:   G( x, y, z)

Proof of Theorem istrkgl

Step	Hyp	Ref	Expression
1		istrkg.p	. . . 4 |- P = ( Base ` G )
2		istrkg.i	. . . 4 |- I = (Itv ` G )
3		simpl 473	. . . . . . 7 |- ( ( p = P /\ i = I ) -> p = P )
4	3	eqcomd 2628	. . . . . 6 |- ( ( p = P /\ i = I ) -> P = p )
5	4	adantr 481	. . . . . . 7 |- ( ( ( p = P /\ i = I ) /\ x e. P ) -> P = p )
6	5	difeq1d 3727	. . . . . 6 |- ( ( ( p = P /\ i = I ) /\ x e. P ) -> ( P \ { x } ) = ( p \ { x } ) )
7		simpr 477	. . . . . . . . . . . 12 |- ( ( p = P /\ i = I ) -> i = I )
8	7	eqcomd 2628	. . . . . . . . . . 11 |- ( ( p = P /\ i = I ) -> I = i )
9	8	oveqd 6667	. . . . . . . . . 10 |- ( ( p = P /\ i = I ) -> ( x I y ) = ( x i y ) )
10	9	eleq2d 2687	. . . . . . . . 9 |- ( ( p = P /\ i = I ) -> ( z e. ( x I y ) <-> z e. ( x i y ) ) )
11	8	oveqd 6667	. . . . . . . . . 10 |- ( ( p = P /\ i = I ) -> ( z I y ) = ( z i y ) )
12	11	eleq2d 2687	. . . . . . . . 9 |- ( ( p = P /\ i = I ) -> ( x e. ( z I y ) <-> x e. ( z i y ) ) )
13	8	oveqd 6667	. . . . . . . . . 10 |- ( ( p = P /\ i = I ) -> ( x I z ) = ( x i z ) )
14	13	eleq2d 2687	. . . . . . . . 9 |- ( ( p = P /\ i = I ) -> ( y e. ( x I z ) <-> y e. ( x i z ) ) )
15	10, 12, 14	3orbi123d 1398	. . . . . . . 8 |- ( ( p = P /\ i = I ) -> ( ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) <-> ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) ) )
16	4, 15	rabeqbidv 3195	. . . . . . 7 |- ( ( p = P /\ i = I ) -> { z e. P | ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) } = { z e. p | ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) } )
17	16	adantr 481	. . . . . 6 |- ( ( ( p = P /\ i = I ) /\ ( x e. P /\ y e. ( P \ { x } ) ) ) -> { z e. P | ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) } = { z e. p | ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) } )
18	4, 6, 17	mpt2eq123dva 6716	. . . . 5 |- ( ( p = P /\ i = I ) -> ( x e. P , y e. ( P \ { x } ) |-> { z e. P | ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) } ) = ( x e. p , y e. ( p \ { x } ) |-> { z e. p | ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) } ) )
19	18	eqeq2d 2632	. . . 4 |- ( ( p = P /\ i = I ) -> ( (LineG ` f ) = ( x e. P , y e. ( P \ { x } ) |-> { z e. P | ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) } ) <-> (LineG ` f ) = ( x e. p , y e. ( p \ { x } ) |-> { z e. p | ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) } ) ) )
20	1, 2, 19	sbcie2s 15916	. . 3 |- ( f = G -> ( [. ( Base ` f ) / p ]. [. (Itv ` f ) / i ]. (LineG ` f ) = ( x e. p , y e. ( p \ { x } ) |-> { z e. p | ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) } ) <-> (LineG ` f ) = ( x e. P , y e. ( P \ { x } ) |-> { z e. P | ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) } ) ) )
21		fveq2 6191	. . . 4 |- ( f = G -> (LineG ` f ) = (LineG ` G ) )
22	21	eqeq1d 2624	. . 3 |- ( f = G -> ( (LineG ` f ) = ( x e. P , y e. ( P \ { x } ) |-> { z e. P | ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) } ) <-> (LineG ` G ) = ( x e. P , y e. ( P \ { x } ) |-> { z e. P | ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) } ) ) )
23	20, 22	bitrd 268	. 2 |- ( f = G -> ( [. ( Base ` f ) / p ]. [. (Itv ` f ) / i ]. (LineG ` f ) = ( x e. p , y e. ( p \ { x } ) |-> { z e. p | ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) } ) <-> (LineG ` G ) = ( x e. P , y e. ( P \ { x } ) |-> { z e. P | ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) } ) ) )
24		eqid 2622	. 2 |- { f | [. ( Base ` f ) / p ]. [. (Itv ` f ) / i ]. (LineG ` f ) = ( x e. p , y e. ( p \ { x } ) |-> { z e. p | ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) } ) } = { f | [. ( Base ` f ) / p ]. [. (Itv ` f ) / i ]. (LineG ` f ) = ( x e. p , y e. ( p \ { x } ) |-> { z e. p | ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) } ) }
25	23, 24	elab4g 3355	1 |- ( G e. { f | [. ( Base ` f ) / p ]. [. (Itv ` f ) / i ]. (LineG ` f ) = ( x e. p , y e. ( p \ { x } ) |-> { z e. p | ( z e. ( x i y ) \/ x e. ( z i y ) \/ y e. ( x i z ) ) } ) } <-> ( G e. _V /\ (LineG ` G ) = ( x e. P , y e. ( P \ { x } ) |-> { z e. P | ( z e. ( x I y ) \/ x e. ( z I y ) \/ y e. ( x I z ) ) } ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   _Vcvv 3200   [.wsbc 3435   \ cdif 3571   {csn 4177   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Basecbs 15857   distcds 15950  Itvcitv 25335  LineGclng 25336

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655

This theorem is referenced by:  tglng  25441  f1otrg  25751  eengtrkg  25865