Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > tglnpt | Structured version Visualization version GIF version |
Description: Lines are sets of points. (Contributed by Thierry Arnoux, 17-Oct-2019.) |
Ref | Expression |
---|---|
tglng.p | ⊢ 𝑃 = (Base‘𝐺) |
tglng.l | ⊢ 𝐿 = (LineG‘𝐺) |
tglng.i | ⊢ 𝐼 = (Itv‘𝐺) |
tglnpt.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tglnpt.a | ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) |
tglnpt.x | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
Ref | Expression |
---|---|
tglnpt | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tglnpt.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
2 | tglng.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
3 | tglng.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
4 | tglng.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | 2, 3, 4 | tglnunirn 26262 | . . 3 ⊢ (𝐺 ∈ TarskiG → ∪ ran 𝐿 ⊆ 𝑃) |
6 | 1, 5 | syl 17 | . 2 ⊢ (𝜑 → ∪ ran 𝐿 ⊆ 𝑃) |
7 | tglnpt.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ran 𝐿) | |
8 | elssuni 4861 | . . . 4 ⊢ (𝐴 ∈ ran 𝐿 → 𝐴 ⊆ ∪ ran 𝐿) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴 ⊆ ∪ ran 𝐿) |
10 | tglnpt.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
11 | 9, 10 | sseldd 3967 | . 2 ⊢ (𝜑 → 𝑋 ∈ ∪ ran 𝐿) |
12 | 6, 11 | sseldd 3967 | 1 ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ⊆ wss 3935 ∪ cuni 4832 ran crn 5550 ‘cfv 6349 Basecbs 16473 TarskiGcstrkg 26144 Itvcitv 26150 LineGclng 26151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-cnv 5557 df-dm 5559 df-rn 5560 df-iota 6308 df-fv 6357 df-ov 7148 df-oprab 7149 df-mpo 7150 df-trkg 26167 |
This theorem is referenced by: mirln 26390 mirln2 26391 perpcom 26427 perpneq 26428 ragperp 26431 foot 26436 footne 26437 footeq 26438 hlperpnel 26439 perprag 26440 perpdragALT 26441 perpdrag 26442 colperpexlem3 26446 oppne3 26457 oppcom 26458 oppnid 26460 opphllem1 26461 opphllem2 26462 opphllem3 26463 opphllem4 26464 opphllem5 26465 opphllem6 26466 oppperpex 26467 opphl 26468 outpasch 26469 lnopp2hpgb 26477 hpgerlem 26479 colopp 26483 colhp 26484 lmieu 26498 lmimid 26508 lnperpex 26517 trgcopy 26518 trgcopyeulem 26519 |
Copyright terms: Public domain | W3C validator |