Theorem tgcgrcomlr 28548

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 28530	. 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 28530	. 2 ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐷 − 𝐶))
12	1, 8, 11	3eqtr3d 2780	1 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐷 − 𝐶))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ‘cfv 6499  (class class class)co 7367  Basecbs 17179  distcds 17229  TarskiGcstrkg 28495  Itvcitv 28501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6455  df-fv 6507  df-ov 7370  df-trkgc 28516  df-trkg 28521

This theorem is referenced by:  tgcgrextend  28553  tgifscgr  28576  tgcgrsub  28577  iscgrglt  28582  trgcgrg  28583  tgcgrxfr  28586  cgr3swap12  28591  cgr3swap23  28592  tgbtwnxfr  28598  lnext  28635  tgbtwnconn1lem1  28640  tgbtwnconn1lem2  28641  tgbtwnconn1lem3  28642  tgbtwnconn1  28643  legov2  28654  legtri3  28658  legbtwn  28662  tgcgrsub2  28663  miriso  28738  mircgrextend  28750  mirtrcgr  28751  miduniq  28753  colmid  28756  symquadlem  28757  krippenlem  28758  midexlem  28760  ragcom  28766  ragflat  28772  ragcgr  28775  footexALT  28786  footexlem1  28787  footexlem2  28788  colperpexlem1  28798  mideulem2  28802  opphllem  28803  opphllem3  28817  lmiisolem  28864  hypcgrlem1  28867  trgcopy  28872  trgcopyeulem  28873  iscgra1  28878  cgracgr  28886  cgraswap  28888  cgrcgra  28889  cgracom  28890  cgratr  28891  flatcgra  28892  dfcgra2  28898  acopy  28901  acopyeu  28902  cgrg3col4  28921  tgsas1  28922  tgsas3  28925  tgasa1  28926