Theorem tgcgrcomlr 28567

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 28549	. 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 28549	. 2 ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐷 − 𝐶))
12	1, 8, 11	3eqtr3d 2782	1 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐷 − 𝐶))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ‘cfv 6486  (class class class)co 7357  Basecbs 17171  distcds 17221  TarskiGcstrkg 28514  Itvcitv 28520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-nul 5229

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4263  df-if 4456  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-br 5074  df-iota 6442  df-fv 6494  df-ov 7360  df-trkgc 28535  df-trkg 28540

This theorem is referenced by:  tgcgrextend  28572  tgifscgr  28595  tgcgrsub  28596  iscgrglt  28601  trgcgrg  28602  tgcgrxfr  28605  cgr3swap12  28610  cgr3swap23  28611  tgbtwnxfr  28617  lnext  28654  tgbtwnconn1lem1  28659  tgbtwnconn1lem2  28660  tgbtwnconn1lem3  28661  tgbtwnconn1  28662  legov2  28673  legtri3  28677  legbtwn  28681  tgcgrsub2  28682  miriso  28757  mircgrextend  28769  mirtrcgr  28770  miduniq  28772  colmid  28775  symquadlem  28776  krippenlem  28777  midexlem  28779  ragcom  28785  ragflat  28791  ragcgr  28794  footexALT  28805  footexlem1  28806  footexlem2  28807  colperpexlem1  28817  mideulem2  28821  opphllem  28822  opphllem3  28836  lmiisolem  28883  hypcgrlem1  28886  trgcopy  28891  trgcopyeulem  28892  iscgra1  28897  cgracgr  28905  cgraswap  28907  cgrcgra  28908  cgracom  28909  cgratr  28910  flatcgra  28911  dfcgra2  28917  acopy  28920  acopyeu  28921  cgrg3col4  28940  tgsas1  28941  tgsas3  28944  tgasa1  28945