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 25364 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 30750 | . . . . 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 222 | . . . 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 1073 | . . 3 ⊢ (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝐹 ∈ (𝐸𝐼𝐻))) |
19 | 18 | simpld 475 | . 2 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) |
20 | 18 | simprd 479 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝐸𝐼𝐻)) |
21 | 17 | simp2d 1074 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐸 − 𝐹) ∧ (𝐵 − 𝐶) = (𝐹 − 𝐻))) |
22 | 21 | simpld 475 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐸 − 𝐹)) |
23 | 21 | simprd 479 | . 2 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐹 − 𝐻)) |
24 | 17 | simp3d 1075 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝐷) = (𝐸 − 𝐼) ∧ (𝐵 − 𝐷) = (𝐹 − 𝐼))) |
25 | 24 | simpld 475 | . 2 ⊢ (𝜑 → (𝐴 − 𝐷) = (𝐸 − 𝐼)) |
26 | 24 | simprd 479 | . 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 25364 | 1 ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐻 − 𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 〈cop 4183 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 distcds 15950 TarskiGcstrkg 25329 Itvcitv 25335 AFScafs 30747 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-trkgcb 25349 df-trkg 25352 df-afs 30748 |
This theorem is referenced by: (None) |
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