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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 28556 | . . 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 4937 | . . . 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 3984 | . 2 ⊢ (𝜑 → 𝑋 ∈ ∪ ran 𝐿) |
| 12 | 6, 11 | sseldd 3984 | 1 ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ∪ cuni 4907 ran crn 5686 ‘cfv 6561 Basecbs 17247 TarskiGcstrkg 28435 Itvcitv 28441 LineGclng 28442 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 df-iota 6514 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-trkg 28461 |
| This theorem is referenced by: mirln 28684 mirln2 28685 perpcom 28721 perpneq 28722 ragperp 28725 foot 28730 footne 28731 footeq 28732 hlperpnel 28733 perprag 28734 perpdragALT 28735 perpdrag 28736 colperpexlem3 28740 oppne3 28751 oppcom 28752 oppnid 28754 opphllem1 28755 opphllem2 28756 opphllem3 28757 opphllem4 28758 opphllem5 28759 opphllem6 28760 oppperpex 28761 opphl 28762 outpasch 28763 lnopp2hpgb 28771 hpgerlem 28773 colopp 28777 colhp 28778 lmieu 28792 lmimid 28802 lnperpex 28811 trgcopy 28812 trgcopyeulem 28813 |
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