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Mirrors > Home > MPE Home > Th. List > tglinerflx2 | Structured version Visualization version GIF version |
Description: Reflexivity law for line membership. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Thierry Arnoux, 17-May-2019.) |
Ref | Expression |
---|---|
tglineelsb2.p | ⊢ 𝐵 = (Base‘𝐺) |
tglineelsb2.i | ⊢ 𝐼 = (Itv‘𝐺) |
tglineelsb2.l | ⊢ 𝐿 = (LineG‘𝐺) |
tglineelsb2.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tglineelsb2.1 | ⊢ (𝜑 → 𝑃 ∈ 𝐵) |
tglineelsb2.2 | ⊢ (𝜑 → 𝑄 ∈ 𝐵) |
tglineelsb2.4 | ⊢ (𝜑 → 𝑃 ≠ 𝑄) |
Ref | Expression |
---|---|
tglinerflx2 | ⊢ (𝜑 → 𝑄 ∈ (𝑃𝐿𝑄)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglineelsb2.p | . 2 ⊢ 𝐵 = (Base‘𝐺) | |
2 | tglineelsb2.i | . 2 ⊢ 𝐼 = (Itv‘𝐺) | |
3 | tglineelsb2.l | . 2 ⊢ 𝐿 = (LineG‘𝐺) | |
4 | tglineelsb2.g | . 2 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | tglineelsb2.1 | . 2 ⊢ (𝜑 → 𝑃 ∈ 𝐵) | |
6 | tglineelsb2.2 | . 2 ⊢ (𝜑 → 𝑄 ∈ 𝐵) | |
7 | tglineelsb2.4 | . 2 ⊢ (𝜑 → 𝑃 ≠ 𝑄) | |
8 | eqid 2734 | . . 3 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
9 | 1, 8, 2, 4, 5, 6 | tgbtwntriv2 26550 | . 2 ⊢ (𝜑 → 𝑄 ∈ (𝑃𝐼𝑄)) |
10 | 1, 2, 3, 4, 5, 6, 6, 7, 9 | btwnlng1 26682 | 1 ⊢ (𝜑 → 𝑄 ∈ (𝑃𝐿𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 ‘cfv 6369 (class class class)co 7202 Basecbs 16684 distcds 16776 TarskiGcstrkg 26493 Itvcitv 26499 LineGclng 26500 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pr 5311 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3403 df-sbc 3688 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-nul 4228 df-if 4430 df-sn 4532 df-pr 4534 df-op 4538 df-uni 4810 df-br 5044 df-opab 5106 df-id 5444 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-iota 6327 df-fun 6371 df-fv 6377 df-ov 7205 df-oprab 7206 df-mpo 7207 df-trkgc 26511 df-trkgcb 26513 df-trkg 26516 |
This theorem is referenced by: tghilberti1 26700 tglnpt2 26704 colline 26712 footexALT 26781 footexlem2 26783 foot 26785 footne 26786 perprag 26789 colperpexlem3 26795 mideulem2 26797 opphllem 26798 opphllem5 26814 opphllem6 26815 opphl 26817 outpasch 26818 hlpasch 26819 lnopp2hpgb 26826 hypcgrlem1 26862 hypcgrlem2 26863 trgcopyeulem 26868 acopy 26896 acopyeu 26897 tgasa1 26921 |
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