Theorem tgcgrcomlr 28760

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 28742	. 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 28742	. 2 ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐷 − 𝐶))
12	1, 8, 11	3eqtr3d 2805	1 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐷 − 𝐶))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  ‘cfv 6536  (class class class)co 7412  Basecbs 17275  distcds 17325  TarskiGcstrkg 28707  Itvcitv 28713

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-nul 5268

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-ov 7415  df-trkgc 28728  df-trkg 28733

This theorem is used by:  tgcgrextend  28765  tgifscgr  28788  tgcgrsub  28789  iscgrglt  28794  trgcgrg  28795  tgcgrxfr  28798  cgr3swap12  28803  cgr3swap23  28804  tgbtwnxfr  28810  lnext  28847  tgbtwnconn1lem1  28852  tgbtwnconn1lem2  28853  tgbtwnconn1lem3  28854  tgbtwnconn1  28855  legov2  28866  legtri3  28870  legbtwn  28874  tgcgrsub2  28875  miriso  28958  mircgrextend  28970  mirtrcgr  28971  miduniq  28973  colmid  28976  symquadlem  28977  krippenlem  28978  midexlem  28980  ragcom  28989  ragflat  28995  ragcgr  28998  footexALT  29009  footexlem1  29010  footexlem2  29011  colperpexlem1  29022  mideulem2  29026  opphllem  29027  opphllem3  29041  lmiisolem  29116  symquadmid  29119  hypcgrlem1  29120  trgcopy  29126  trgcopyeulem  29127  iscgra1  29132  cgracgr  29140  cgraswap  29142  cgrcgra  29143  cgracom  29144  cgratr  29145  flatcgra  29146  dfcgra2  29152  acopy  29155  acopyeu  29156  ragcgra  29157  ragsupplcgra  29159  cgrg3col4  29181  tgsas1  29182  tgsas3  29185  tgasa1  29186  symquadprlng  29223  tgaltai  29228