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Mirrors > Home > MPE Home > Th. List > hlid | Structured version Visualization version GIF version |
Description: The half-line relation is reflexive. Theorem 6.5 of [Schwabhauser] p. 44. (Contributed by Thierry Arnoux, 21-Feb-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) |
hlid.1 | ⊢ (𝜑 → 𝐴 ≠ 𝐶) |
Ref | Expression |
---|---|
hlid | ⊢ (𝜑 → 𝐴(𝐾‘𝐶)𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlid.1 | . 2 ⊢ (𝜑 → 𝐴 ≠ 𝐶) | |
2 | ishlg.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
3 | eqid 2824 | . . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
4 | ishlg.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | hlln.1 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | ishlg.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
7 | ishlg.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
8 | 2, 3, 4, 5, 6, 7 | tgbtwntriv2 26276 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (𝐶𝐼𝐴)) |
9 | 8 | olcd 870 | . 2 ⊢ (𝜑 → (𝐴 ∈ (𝐶𝐼𝐴) ∨ 𝐴 ∈ (𝐶𝐼𝐴))) |
10 | ishlg.k | . . 3 ⊢ 𝐾 = (hlG‘𝐺) | |
11 | 2, 4, 10, 7, 7, 6, 5 | ishlg 26391 | . 2 ⊢ (𝜑 → (𝐴(𝐾‘𝐶)𝐴 ↔ (𝐴 ≠ 𝐶 ∧ 𝐴 ≠ 𝐶 ∧ (𝐴 ∈ (𝐶𝐼𝐴) ∨ 𝐴 ∈ (𝐶𝐼𝐴))))) |
12 | 1, 1, 9, 11 | mpbir3and 1338 | 1 ⊢ (𝜑 → 𝐴(𝐾‘𝐶)𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 843 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 distcds 16577 TarskiGcstrkg 26219 Itvcitv 26225 hlGchlg 26389 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-trkgc 26237 df-trkgcb 26239 df-trkg 26242 df-hlg 26390 |
This theorem is referenced by: opphl 26543 iscgra1 26599 cgraid 26608 cgrcgra 26610 dfcgra2 26619 tgsas1 26643 tgsas2 26645 tgsas3 26646 tgasa1 26647 |
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