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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 722 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝐺 ∈ TarskiG) |
6 | tgcgrtriv.1 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | 6 | ad2antrr 722 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝐴 ∈ 𝑃) |
8 | simplr 765 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝑥 ∈ 𝑃) | |
9 | tgcgrtriv.2 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
10 | 9 | ad2antrr 722 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝐵 ∈ 𝑃) |
11 | simprr 769 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → (𝐴 − 𝑥) = (𝐵 − 𝐵)) | |
12 | 1, 2, 3, 5, 7, 8, 10, 11 | axtgcgrid 26176 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → 𝐴 = 𝑥) |
13 | 12 | oveq2d 7161 | . . 3 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → (𝐴 − 𝐴) = (𝐴 − 𝑥)) |
14 | 13, 11 | eqtrd 2853 | . 2 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑃) ∧ (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) → (𝐴 − 𝐴) = (𝐵 − 𝐵)) |
15 | 1, 2, 3, 4, 9, 6, 9, 9 | axtgsegcon 26177 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑃 (𝐴 ∈ (𝐵𝐼𝑥) ∧ (𝐴 − 𝑥) = (𝐵 − 𝐵))) |
16 | 14, 15 | r19.29a 3286 | 1 ⊢ (𝜑 → (𝐴 − 𝐴) = (𝐵 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 distcds 16562 TarskiGcstrkg 26143 Itvcitv 26149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-trkgc 26161 df-trkgcb 26163 df-trkg 26166 |
This theorem is referenced by: tgcgrextend 26198 tgcgrsub 26222 iscgrglt 26227 trgcgrg 26228 tgbtwnconn1lem3 26287 leg0 26305 |
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