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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 26336 | . . 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 4870 | . . . 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 3970 | . 2 ⊢ (𝜑 → 𝑋 ∈ ∪ ran 𝐿) |
12 | 6, 11 | sseldd 3970 | 1 ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 ∪ cuni 4840 ran crn 5558 ‘cfv 6357 Basecbs 16485 TarskiGcstrkg 26218 Itvcitv 26224 LineGclng 26225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-cnv 5565 df-dm 5567 df-rn 5568 df-iota 6316 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-trkg 26241 |
This theorem is referenced by: mirln 26464 mirln2 26465 perpcom 26501 perpneq 26502 ragperp 26505 foot 26510 footne 26511 footeq 26512 hlperpnel 26513 perprag 26514 perpdragALT 26515 perpdrag 26516 colperpexlem3 26520 oppne3 26531 oppcom 26532 oppnid 26534 opphllem1 26535 opphllem2 26536 opphllem3 26537 opphllem4 26538 opphllem5 26539 opphllem6 26540 oppperpex 26541 opphl 26542 outpasch 26543 lnopp2hpgb 26551 hpgerlem 26553 colopp 26557 colhp 26558 lmieu 26572 lmimid 26582 lnperpex 26591 trgcopy 26592 trgcopyeulem 26593 |
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