|   | Mathbox for Thierry Arnoux | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > Mathboxes > tg5segofs | Structured version Visualization version GIF version | ||
| Description: Rephrase axtg5seg 28473 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 34687 | . . . . 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 232 | . . . 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 1143 | . . 3 ⊢ (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ∧ 𝐹 ∈ (𝐸𝐼𝐻))) | 
| 19 | 18 | simpld 494 | . 2 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) | 
| 20 | 18 | simprd 495 | . 2 ⊢ (𝜑 → 𝐹 ∈ (𝐸𝐼𝐻)) | 
| 21 | 17 | simp2d 1144 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝐵) = (𝐸 − 𝐹) ∧ (𝐵 − 𝐶) = (𝐹 − 𝐻))) | 
| 22 | 21 | simpld 494 | . 2 ⊢ (𝜑 → (𝐴 − 𝐵) = (𝐸 − 𝐹)) | 
| 23 | 21 | simprd 495 | . 2 ⊢ (𝜑 → (𝐵 − 𝐶) = (𝐹 − 𝐻)) | 
| 24 | 17 | simp3d 1145 | . . 3 ⊢ (𝜑 → ((𝐴 − 𝐷) = (𝐸 − 𝐼) ∧ (𝐵 − 𝐷) = (𝐹 − 𝐼))) | 
| 25 | 24 | simpld 494 | . 2 ⊢ (𝜑 → (𝐴 − 𝐷) = (𝐸 − 𝐼)) | 
| 26 | 24 | simprd 495 | . 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 28473 | 1 ⊢ (𝜑 → (𝐶 − 𝐷) = (𝐻 − 𝐼)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 〈cop 4632 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 distcds 17306 TarskiGcstrkg 28435 Itvcitv 28441 AFScafs 34684 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-trkgcb 28458 df-trkg 28461 df-afs 34685 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |