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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 28424 | . . 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 4941 | . . . 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 3977 | . 2 ⊢ (𝜑 → 𝑋 ∈ ∪ ran 𝐿) |
12 | 6, 11 | sseldd 3977 | 1 ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⊆ wss 3944 ∪ cuni 4909 ran crn 5679 ‘cfv 6549 Basecbs 17183 TarskiGcstrkg 28303 Itvcitv 28309 LineGclng 28310 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-cnv 5686 df-dm 5688 df-rn 5689 df-iota 6501 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-trkg 28329 |
This theorem is referenced by: mirln 28552 mirln2 28553 perpcom 28589 perpneq 28590 ragperp 28593 foot 28598 footne 28599 footeq 28600 hlperpnel 28601 perprag 28602 perpdragALT 28603 perpdrag 28604 colperpexlem3 28608 oppne3 28619 oppcom 28620 oppnid 28622 opphllem1 28623 opphllem2 28624 opphllem3 28625 opphllem4 28626 opphllem5 28627 opphllem6 28628 oppperpex 28629 opphl 28630 outpasch 28631 lnopp2hpgb 28639 hpgerlem 28641 colopp 28645 colhp 28646 lmieu 28660 lmimid 28670 lnperpex 28679 trgcopy 28680 trgcopyeulem 28681 |
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