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Mirrors > Home > MPE Home > Th. List > tgcgrtriv | Structured version Visualization version GIF version |
Description: Degenerate segments are congruent. Theorem 2.8 of [Schwabhauser] p. 28. (Contributed by Thierry Arnoux, 23-Mar-2019.) |
Ref | Expression |
---|---|
tkgeom.p | ⊢ 𝑃 = (Base‘𝐺) |
tkgeom.d | ⊢ − = (dist‘𝐺) |
tkgeom.i | ⊢ 𝐼 = (Itv‘𝐺) |
tkgeom.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tgcgrtriv.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
tgcgrtriv.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
Ref | Expression |
---|---|
tgcgrtriv | ⊢ (𝜑 → (𝐴 − 𝐴) = (𝐵 − 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tkgeom.p | . . . . 5 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tkgeom.d | . . . . 5 ⊢ − = (dist‘𝐺) | |
3 | tkgeom.i | . . . . 5 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tkgeom.g | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | ad2antrr 725 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝐺 ∈ TarskiG) |
6 | tgcgrtriv.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | 6 | ad2antrr 725 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝐴 ∈ 𝑃) |
8 | simplr 768 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝑥 ∈ 𝑃) | |
9 | tgcgrtriv.2 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
10 | 9 | ad2antrr 725 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 ∈ 𝑃) |
11 | simprr 772 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → (𝐴 − 𝑥) = (𝐵 − 𝐵)) | |
12 | 1, 2, 3, 5, 7, 8, 10, 11 | axtgcgrid 27694 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝐴 = 𝑥) |
13 | 12 | oveq2d 7420 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → (𝐴 − 𝐴) = (𝐴 − 𝑥)) |
14 | 13, 11 | eqtrd 2773 | . 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → (𝐴 − 𝐴) = (𝐵 − 𝐵)) |
15 | 1, 2, 3, 4, 9, 6, 9, 9 | axtgsegcon 27695 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) |
16 | 14, 15 | r19.29a 3163 | 1 ⊢ (𝜑 → (𝐴 − 𝐴) = (𝐵 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ‘cfv 6540 (class class class)co 7404 Basecbs 17140 distcds 17202 TarskiGcstrkg 27658 Itvcitv 27664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-nul 5305 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6492 df-fv 6548 df-ov 7407 df-trkgc 27679 df-trkgcb 27681 df-trkg 27684 |
This theorem is referenced by: tgcgrextend 27716 tgcgrsub 27740 iscgrglt 27745 trgcgrg 27746 tgbtwnconn1lem3 27805 leg0 27823 |
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