Theorem tgcgreqb 27987

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 481	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
6		tgcgrcomlr.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
7	6	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 ∈ 𝑃)
8		tgcgrcomlr.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
9	8	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 ∈ 𝑃)
10		tgcgrcomlr.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
11	10	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑃)
12		tgcgrcomlr.6	. . . . 5 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷))
13	12	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 − 𝐵) = (𝐶 − 𝐷))
14		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
15	14	oveq1d 7426	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 − 𝐵) = (𝐵 − 𝐵))
16	13, 15	eqtr3d 2774	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐶 − 𝐷) = (𝐵 − 𝐵))
17	1, 2, 3, 5, 7, 9, 11, 16	axtgcgrid 27969	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 = 𝐷)
18	4	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐺 ∈ TarskiG)
19		tgcgrcomlr.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
20	19	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐴 ∈ 𝑃)
21	10	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐵 ∈ 𝑃)
22	8	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐷 ∈ 𝑃)
23	12	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐴 − 𝐵) = (𝐶 − 𝐷))
24		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 = 𝐷)
25	24	oveq1d 7426	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐶 − 𝐷) = (𝐷 − 𝐷))
26	23, 25	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐴 − 𝐵) = (𝐷 − 𝐷))
27	1, 2, 3, 18, 20, 21, 22, 26	axtgcgrid 27969	. 2 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐴 = 𝐵)
28	17, 27	impbida 799	1 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ‘cfv 6543  (class class class)co 7411  Basecbs 17148  distcds 17210  TarskiGcstrkg 27933  Itvcitv 27939

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7414  df-trkgc 27954  df-trkg 27959

This theorem is referenced by:  tgcgreq  27988  tgcgrneq  27989