Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tg5segofs | Structured version Visualization version GIF version |
Description: Rephrase axtg5seg 25164 using the outer five segment predicate. Theorem 2.10 of [Schwabhauser] p. 28. (Contributed by Thierry Arnoux, 23-Mar-2019.) |
Ref | Expression |
---|---|
tg5segofs.p | ⊢ 𝑃 = (Base‘𝐺) |
tg5segofs.m | ⊢ − = (dist‘𝐺) |
tg5segofs.s | ⊢ 𝐼 = (Itv‘𝐺) |
tg5segofs.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tg5segofs.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
tg5segofs.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
tg5segofs.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
tg5segofs.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
tg5segofs.e | ⊢ (𝜑 → 𝐸 ∈ 𝑃) |
tg5segofs.f | ⊢ (𝜑 → 𝐹 ∈ 𝑃) |
tg5segofs.o | ⊢ 𝑂 = (AFS‘𝐺) |
tg5segofs.h | ⊢ (𝜑 → 𝐻 ∈ 𝑃) |
tg5segofs.i | ⊢ (𝜑 → 𝐼 ∈ 𝑃) |
tg5segofs.1 | ⊢ (𝜑 → 〈〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉〉𝑂〈〈𝐸, 𝐹〉, 〈𝐻, 𝐼〉〉) |
tg5segofs.2 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Ref | Expression |
---|---|
tg5segofs | ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐻 − 𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tg5segofs.p | . 2 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tg5segofs.m | . 2 ⊢ − = (dist‘𝐺) | |
3 | tg5segofs.s | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tg5segofs.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | tg5segofs.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
6 | tg5segofs.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | tg5segofs.c | . 2 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
8 | tg5segofs.e | . 2 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
9 | tg5segofs.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑃) | |
10 | tg5segofs.h | . 2 ⊢ (𝜑 → 𝐻 ∈ 𝑃) | |
11 | tg5segofs.d | . 2 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
12 | tg5segofs.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝑃) | |
13 | tg5segofs.2 | . 2 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
14 | tg5segofs.1 | . . . . 5 ⊢ (𝜑 → 〈〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉〉𝑂〈〈𝐸, 𝐹〉, 〈𝐻, 𝐼〉〉) | |
15 | tg5segofs.o | . . . . . 6 ⊢ 𝑂 = (AFS‘𝐺) | |
16 | 1, 2, 3, 4, 15, 5, 6, 7, 11, 8, 9, 10, 12 | brafs 30003 | . . . . 5 ⊢ (𝜑 → (〈〈𝐴, 𝐵〉, 〈𝐶, 𝐷〉〉𝑂〈〈𝐸, 𝐹〉, 〈𝐻, 𝐼〉〉 ↔ ((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝐹 ∈ (𝐸𝐼𝐻)) ∧ ((𝐴 − 𝐵) = (𝐸 − 𝐹) ∧ (𝐵 − 𝐶) = (𝐹 − 𝐻)) ∧ ((𝐴 − 𝐷) = (𝐸 − 𝐼) ∧ (𝐵 − 𝐷) = (𝐹 − 𝐼))))) |
17 | 14, 16 | mpbid 221 | . . . 4 ⊢ (𝜑 → ((𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝐹 ∈ (𝐸𝐼𝐻)) ∧ ((𝐴 − 𝐵) = (𝐸 − 𝐹) ∧ (𝐵 − 𝐶) = (𝐹 − 𝐻)) ∧ ((𝐴 − 𝐷) = (𝐸 − 𝐼) ∧ (𝐵 − 𝐷) = (𝐹 − 𝐼)))) |
18 | 17 | simp1d 1066 | . . 3 ⊢ (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝐹 ∈ (𝐸𝐼𝐻))) |
19 | 18 | simpld 474 | . 2 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) |
20 | 18 | simprd 478 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝐸𝐼𝐻)) |
21 | 17 | simp2d 1067 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐸 − 𝐹) ∧ (𝐵 − 𝐶) = (𝐹 − 𝐻))) |
22 | 21 | simpld 474 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐸 − 𝐹)) |
23 | 21 | simprd 478 | . 2 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐹 − 𝐻)) |
24 | 17 | simp3d 1068 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝐷) = (𝐸 − 𝐼) ∧ (𝐵 − 𝐷) = (𝐹 − 𝐼))) |
25 | 24 | simpld 474 | . 2 ⊢ (𝜑 → (𝐴 − 𝐷) = (𝐸 − 𝐼)) |
26 | 24 | simprd 478 | . 2 ⊢ (𝜑 → (𝐵 − 𝐷) = (𝐹 − 𝐼)) |
27 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 19, 20, 22, 23, 25, 26 | axtg5seg 25164 | 1 ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐻 − 𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 〈cop 4131 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 distcds 15777 TarskiGcstrkg 25129 Itvcitv 25135 AFScafs 30000 |
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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
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-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-trkgcb 25149 df-trkg 25152 df-afs 30001 |
This theorem is referenced by: (None) |
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