Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > btwnhl1 | Structured version Visualization version GIF version |
Description: Deduce half-line from betweenness. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
Ref | Expression |
---|---|
ishlg.p | ⊢ 𝑃 = (Base‘𝐺) |
ishlg.i | ⊢ 𝐼 = (Itv‘𝐺) |
ishlg.k | ⊢ 𝐾 = (hlG‘𝐺) |
ishlg.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
ishlg.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
ishlg.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
hlln.1 | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
hltr.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
btwnhl1.1 | ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐵)) |
btwnhl1.2 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
btwnhl1.3 | ⊢ (𝜑 → 𝐶 ≠ 𝐴) |
Ref | Expression |
---|---|
btwnhl1 | ⊢ (𝜑 → 𝐶(𝐾‘𝐴)𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | btwnhl1.3 | . 2 ⊢ (𝜑 → 𝐶 ≠ 𝐴) | |
2 | btwnhl1.2 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
3 | 2 | necomd 3071 | . 2 ⊢ (𝜑 → 𝐵 ≠ 𝐴) |
4 | btwnhl1.1 | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐵)) | |
5 | 4 | orcd 869 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐵 ∈ (𝐴𝐼𝐶))) |
6 | ishlg.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
7 | ishlg.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
8 | ishlg.k | . . 3 ⊢ 𝐾 = (hlG‘𝐺) | |
9 | ishlg.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
10 | ishlg.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
11 | ishlg.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
12 | hlln.1 | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
13 | 6, 7, 8, 9, 10, 11, 12 | ishlg 26388 | . 2 ⊢ (𝜑 → (𝐶(𝐾‘𝐴)𝐵 ↔ (𝐶 ≠ 𝐴 ∧ 𝐵 ≠ 𝐴 ∧ (𝐶 ∈ (𝐴𝐼𝐵) ∨ 𝐵 ∈ (𝐴𝐼𝐶))))) |
14 | 1, 3, 5, 13 | mpbir3and 1338 | 1 ⊢ (𝜑 → 𝐶(𝐾‘𝐴)𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 class class class wbr 5066 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 TarskiGcstrkg 26216 Itvcitv 26222 hlGchlg 26386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-hlg 26387 |
This theorem is referenced by: outpasch 26541 hlpasch 26542 lnopp2hpgb 26549 dfcgra2 26616 |
Copyright terms: Public domain | W3C validator |