Theorem tgcgrcomlr 25792

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 25774	. 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 25774	. 2 ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐷 − 𝐶))
12	1, 8, 11	3eqtr3d 2869	1 ⊢ (𝜑 → (𝐵 − 𝐴) = (𝐷 − 𝐶))

Syntax hints:   → wi 4   = wceq 1658   ∈ wcel 2166  ‘cfv 6123  (class class class)co 6905  Basecbs 16222  distcds 16314  TarskiGcstrkg 25742  Itvcitv 25748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-nul 5013

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-iota 6086  df-fv 6131  df-ov 6908  df-trkgc 25760  df-trkg 25765

This theorem is referenced by:  tgcgrextend  25797  tgifscgr  25820  tgcgrsub  25821  iscgrglt  25826  trgcgrg  25827  tgcgrxfr  25830  cgr3swap12  25835  cgr3swap23  25836  tgbtwnxfr  25842  lnext  25879  tgbtwnconn1lem1  25884  tgbtwnconn1lem2  25885  tgbtwnconn1lem3  25886  tgbtwnconn1  25887  legov2  25898  legtri3  25902  legbtwn  25906  tgcgrsub2  25907  miriso  25982  mircgrextend  25994  mirtrcgr  25995  miduniq  25997  colmid  26000  symquadlem  26001  krippenlem  26002  midexlem  26004  ragcom  26010  ragflat  26016  ragcgr  26019  footex  26030  colperpexlem1  26039  mideulem2  26043  opphllem  26044  opphllem3  26058  lmiisolem  26105  hypcgrlem1  26108  trgcopy  26113  trgcopyeulem  26114  iscgra1  26119  cgracgr  26127  cgraswap  26129  cgrcgra  26130  cgracom  26131  cgratr  26132  dfcgra2  26138  sacgrOLD  26140  acopy  26142  acopyeu  26143  cgrg3col4  26152  tgsas1  26153  tgsas3  26156  tgasa1  26157