Theorem tgcgreqb 25176

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 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐺 ∈ TarskiG)
6		tgcgrcomlr.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃)
7	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 ∈ 𝑃)
8		tgcgrcomlr.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃)
9	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐷 ∈ 𝑃)
10		tgcgrcomlr.b	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃)
11	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐵 ∈ 𝑃)
12		tgcgrcomlr.6	. . . . 5 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐶 − 𝐷))
13	12	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 − 𝐵) = (𝐶 − 𝐷))
14		simpr 476	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 = 𝐵)
15	14	oveq1d 6564	. . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐴 − 𝐵) = (𝐵 − 𝐵))
16	13, 15	eqtr3d 2646	. . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → (𝐶 − 𝐷) = (𝐵 − 𝐵))
17	1, 2, 3, 5, 7, 9, 11, 16	axtgcgrid 25162	. 2 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐶 = 𝐷)
18	4	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐺 ∈ TarskiG)
19		tgcgrcomlr.a	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃)
20	19	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐴 ∈ 𝑃)
21	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐵 ∈ 𝑃)
22	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐷 ∈ 𝑃)
23	12	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐴 − 𝐵) = (𝐶 − 𝐷))
24		simpr 476	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐶 = 𝐷)
25	24	oveq1d 6564	. . . 4 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐶 − 𝐷) = (𝐷 − 𝐷))
26	23, 25	eqtrd 2644	. . 3 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → (𝐴 − 𝐵) = (𝐷 − 𝐷))
27	1, 2, 3, 18, 20, 21, 22, 26	axtgcgrid 25162	. 2 ⊢ ((𝜑 ∧ 𝐶 = 𝐷) → 𝐴 = 𝐵)
28	17, 27	impbida 873	1 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ 𝐶 = 𝐷))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  TarskiGcstrkg 25129  Itvcitv 25135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-trkgc 25147  df-trkg 25152

This theorem is referenced by:  tgcgreq  25177  tgcgrneq  25178