Theorem tgcgrcomlr 28429

Description: Congruence commutes on both sides. (Contributed by Thierry Arnoux, 23-Mar-2019.)

Hypotheses

Ref	Expression
tkgeom.p	⊢ 𝑃 = (Base‘𝐺)
tkgeom.d	⊢ − = (dist‘𝐺)
tkgeom.i	⊢ 𝐼 = (Itv‘𝐺)
tkgeom.g	⊢ (𝜑 → 𝐺 ∈ TarskiG)
tgcgrcomlr.a	⊢ (𝜑 → 𝐴 ∈ 𝑃)
tgcgrcomlr.b	⊢ (𝜑 → 𝐵 ∈ 𝑃)
tgcgrcomlr.c	⊢ (𝜑 → 𝐶 ∈ 𝑃)
tgcgrcomlr.d	⊢ (𝜑 → 𝐷 ∈ 𝑃)
tgcgrcomlr.6	⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷))

Assertion

Ref	Expression
tgcgrcomlr	⊢ (𝜑 → (𝐵 − 𝐴) = (𝐷 − 𝐶))

Proof of Theorem tgcgrcomlr

Step	Hyp	Ref	Expression
1		tgcgrcomlr.6	. 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷))
2		tkgeom.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
3		tkgeom.d	. . 3 ⊢ − = (dist‘𝐺)
4		tkgeom.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
5		tkgeom.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
6		tgcgrcomlr.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
7		tgcgrcomlr.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
8	2, 3, 4, 5, 6, 7	axtgcgrrflx 28411	. 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐵 − 𝐴))
9		tgcgrcomlr.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
10		tgcgrcomlr.d	. . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
11	2, 3, 4, 5, 9, 10	axtgcgrrflx 28411	. 2 ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐷 − 𝐶))
12	1, 8, 11	3eqtr3d 2772	1 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐷 − 𝐶))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ‘cfv 6482  (class class class)co 7349  Basecbs 17120  distcds 17170  TarskiGcstrkg 28376  Itvcitv 28382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5245

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rab 3395  df-v 3438  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-iota 6438  df-fv 6490  df-ov 7352  df-trkgc 28397  df-trkg 28402

This theorem is referenced by:  tgcgrextend  28434  tgifscgr  28457  tgcgrsub  28458  iscgrglt  28463  trgcgrg  28464  tgcgrxfr  28467  cgr3swap12  28472  cgr3swap23  28473  tgbtwnxfr  28479  lnext  28516  tgbtwnconn1lem1  28521  tgbtwnconn1lem2  28522  tgbtwnconn1lem3  28523  tgbtwnconn1  28524  legov2  28535  legtri3  28539  legbtwn  28543  tgcgrsub2  28544  miriso  28619  mircgrextend  28631  mirtrcgr  28632  miduniq  28634  colmid  28637  symquadlem  28638  krippenlem  28639  midexlem  28641  ragcom  28647  ragflat  28653  ragcgr  28656  footexALT  28667  footexlem1  28668  footexlem2  28669  colperpexlem1  28679  mideulem2  28683  opphllem  28684  opphllem3  28698  lmiisolem  28745  hypcgrlem1  28748  trgcopy  28753  trgcopyeulem  28754  iscgra1  28759  cgracgr  28767  cgraswap  28769  cgrcgra  28770  cgracom  28771  cgratr  28772  flatcgra  28773  dfcgra2  28779  acopy  28782  acopyeu  28783  cgrg3col4  28802  tgsas1  28803  tgsas3  28806  tgasa1  28807