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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 2825 | . . 3 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
9 | 1, 8, 2, 4, 5, 6 | tgbtwntriv1 25803 | . 2 ⊢ (𝜑 → 𝑃 ∈ (𝑃𝐼𝑄)) |
10 | 1, 2, 3, 4, 5, 6, 5, 7, 9 | btwnlng1 25931 | 1 ⊢ (𝜑 → 𝑃 ∈ (𝑃𝐿𝑄)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 distcds 16314 TarskiGcstrkg 25742 Itvcitv 25748 LineGclng 25749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-iota 6086 df-fun 6125 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-trkgc 25760 df-trkgb 25761 df-trkgcb 25762 df-trkg 25765 |
This theorem is referenced by: tghilberti1 25949 tglnne0 25952 tglnpt2 25953 tglineneq 25956 coltr 25959 colline 25961 footex 26030 foot 26031 footne 26032 perprag 26035 colperp 26038 colperpexlem3 26041 mideulem2 26043 outpasch 26064 hlpasch 26065 lnopp2hpgb 26072 colopp 26078 lmieu 26093 lmimid 26103 hypcgrlem1 26108 hypcgrlem2 26109 trgcopyeulem 26114 tgasa1 26157 |
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