Theorem ltgseg 25491

Description: The set E denotes the possible values of the congruence. (Contributed by Thierry Arnoux, 15-Dec-2019.)

Hypotheses

Ref	Expression
legval.p	|- P = ( Base ` G )
legval.d	|- .- = ( dist ` G )
legval.i	|- I = (Itv ` G )
legval.l	|- .<_ = (≤G ` G )
legval.g	|- ( ph -> G e. TarskiG )
legso.a	|- E = ( .- " ( P X. P ) )
legso.f	|- ( ph -> Fun .- )
ltgseg.p	|- ( ph -> A e. E )

Assertion

Ref	Expression
ltgseg	|- ( ph -> E. x e. P E. y e. P A = ( x .- y ) )

Distinct variable groups:   x, .- , y   x, A, y   x, P, y   ph, x, y

Allowed substitution hints:   E( x, y)   G( x, y)   I( x, y)   .<_ ( x, y)

Proof of Theorem ltgseg

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp-4r 807	. . . . 5 |- ( ( ( ( ( ( ph /\ a e. ( P X. P ) ) /\ ( .- ` a ) = A ) /\ x e. P ) /\ y e. P ) /\ a = <. x , y >. ) -> ( .- ` a ) = A )
2		simpr 477	. . . . . 6 |- ( ( ( ( ( ( ph /\ a e. ( P X. P ) ) /\ ( .- ` a ) = A ) /\ x e. P ) /\ y e. P ) /\ a = <. x , y >. ) -> a = <. x , y >. )
3	2	fveq2d 6195	. . . . 5 |- ( ( ( ( ( ( ph /\ a e. ( P X. P ) ) /\ ( .- ` a ) = A ) /\ x e. P ) /\ y e. P ) /\ a = <. x , y >. ) -> ( .- ` a ) = ( .- ` <. x , y >. ) )
4	1, 3	eqtr3d 2658	. . . 4 |- ( ( ( ( ( ( ph /\ a e. ( P X. P ) ) /\ ( .- ` a ) = A ) /\ x e. P ) /\ y e. P ) /\ a = <. x , y >. ) -> A = ( .- ` <. x , y >. ) )
5		df-ov 6653	. . . 4 |- ( x .- y ) = ( .- ` <. x , y >. )
6	4, 5	syl6eqr 2674	. . 3 |- ( ( ( ( ( ( ph /\ a e. ( P X. P ) ) /\ ( .- ` a ) = A ) /\ x e. P ) /\ y e. P ) /\ a = <. x , y >. ) -> A = ( x .- y ) )
7		simplr 792	. . . 4 |- ( ( ( ph /\ a e. ( P X. P ) ) /\ ( .- ` a ) = A ) -> a e. ( P X. P ) )
8		elxp2 5132	. . . 4 |- ( a e. ( P X. P ) <-> E. x e. P E. y e. P a = <. x , y >. )
9	7, 8	sylib 208	. . 3 |- ( ( ( ph /\ a e. ( P X. P ) ) /\ ( .- ` a ) = A ) -> E. x e. P E. y e. P a = <. x , y >. )
10	6, 9	reximddv2 3020	. 2 |- ( ( ( ph /\ a e. ( P X. P ) ) /\ ( .- ` a ) = A ) -> E. x e. P E. y e. P A = ( x .- y ) )
11		legso.f	. . 3 |- ( ph -> Fun .- )
12		ltgseg.p	. . . 4 |- ( ph -> A e. E )
13		legso.a	. . . 4 |- E = ( .- " ( P X. P ) )
14	12, 13	syl6eleq 2711	. . 3 |- ( ph -> A e. ( .- " ( P X. P ) ) )
15		fvelima 6248	. . 3 |- ( ( Fun .- /\ A e. ( .- " ( P X. P ) ) ) -> E. a e. ( P X. P ) ( .- ` a ) = A )
16	11, 14, 15	syl2anc 693	. 2 |- ( ph -> E. a e. ( P X. P ) ( .- ` a ) = A )
17	10, 16	r19.29a 3078	1 |- ( ph -> E. x e. P E. y e. P A = ( x .- y ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   <.cop 4183   X. cxp 5112   "cima 5117   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   Basecbs 15857   distcds 15950  TarskiGcstrkg 25329  Itvcitv 25335  ≤Gcleg 25477

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-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  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-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-opab 4713  df-id 5024  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-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653

This theorem is referenced by:  legso  25494