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Mirrors > Home > MPE Home > Th. List > tglinerflx1 | 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 |
---|---|
tglinerflx1 | ⊢ (𝜑 → 𝑃 ∈ (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 2798 | . . 3 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
9 | 1, 8, 2, 4, 5, 6 | tgbtwntriv1 26285 | . 2 ⊢ (𝜑 → 𝑃 ∈ (𝑃𝐼𝑄)) |
10 | 1, 2, 3, 4, 5, 6, 5, 7, 9 | btwnlng1 26413 | 1 ⊢ (𝜑 → 𝑃 ∈ (𝑃𝐿𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 distcds 16566 TarskiGcstrkg 26224 Itvcitv 26230 LineGclng 26231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-trkgc 26242 df-trkgb 26243 df-trkgcb 26244 df-trkg 26247 |
This theorem is referenced by: tghilberti1 26431 tglnne0 26434 tglnpt2 26435 tglineneq 26438 coltr 26441 colline 26443 footexALT 26512 footexlem1 26513 footexlem2 26514 foot 26516 footne 26517 perprag 26520 colperp 26523 colperpexlem3 26526 mideulem2 26528 outpasch 26549 hlpasch 26550 lnopp2hpgb 26557 colopp 26563 lmieu 26578 lmimid 26588 hypcgrlem1 26593 hypcgrlem2 26594 trgcopyeulem 26599 tgasa1 26652 |
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