Theorem tgcgreqb 26387

Description: Congruence and equality. (Contributed by Thierry Arnoux, 27-Aug-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
tgcgreqb	⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))

Proof of Theorem tgcgreqb

Step	Hyp	Ref	Expression
1		tkgeom.p	. . 3 ⊢ 𝑃 = (Base‘𝐺)
2		tkgeom.d	. . 3 ⊢ − = (dist‘𝐺)
3		tkgeom.i	. . 3 ⊢ 𝐼 = (Itv‘𝐺)
4		tkgeom.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG)
5	4	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
6		tgcgrcomlr.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
7	6	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 ∈ 𝑃)
8		tgcgrcomlr.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
9	8	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 ∈ 𝑃)
10		tgcgrcomlr.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
11	10	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑃)
12		tgcgrcomlr.6	. . . . 5 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷))
13	12	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 − 𝐵) = (𝐶 − 𝐷))
14		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
15	14	oveq1d 7171	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 − 𝐵) = (𝐵 − 𝐵))
16	13, 15	eqtr3d 2795	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐶 − 𝐷) = (𝐵 − 𝐵))
17	1, 2, 3, 5, 7, 9, 11, 16	axtgcgrid 26369	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 = 𝐷)
18	4	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐺 ∈ TarskiG)
19		tgcgrcomlr.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
20	19	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐴 ∈ 𝑃)
21	10	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐵 ∈ 𝑃)
22	8	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐷 ∈ 𝑃)
23	12	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐴 − 𝐵) = (𝐶 − 𝐷))
24		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 = 𝐷)
25	24	oveq1d 7171	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐶 − 𝐷) = (𝐷 − 𝐷))
26	23, 25	eqtrd 2793	. . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐴 − 𝐵) = (𝐷 − 𝐷))
27	1, 2, 3, 18, 20, 21, 22, 26	axtgcgrid 26369	. 2 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐴 = 𝐵)
28	17, 27	impbida 800	1 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ‘cfv 6340  (class class class)co 7156  Basecbs 16554  distcds 16645  TarskiGcstrkg 26336  Itvcitv 26342

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-nul 5180

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-iota 6299  df-fv 6348  df-ov 7159  df-trkgc 26354  df-trkg 26359

This theorem is referenced by:  tgcgreq  26388  tgcgrneq  26389