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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 26342 | . . 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 4830 | . . . 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 3916 | . 2 ⊢ (𝜑 → 𝑋 ∈ ∪ ran 𝐿) |
12 | 6, 11 | sseldd 3916 | 1 ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ∪ cuni 4800 ran crn 5520 ‘cfv 6324 Basecbs 16475 TarskiGcstrkg 26224 Itvcitv 26230 LineGclng 26231 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-cnv 5527 df-dm 5529 df-rn 5530 df-iota 6283 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-trkg 26247 |
This theorem is referenced by: mirln 26470 mirln2 26471 perpcom 26507 perpneq 26508 ragperp 26511 foot 26516 footne 26517 footeq 26518 hlperpnel 26519 perprag 26520 perpdragALT 26521 perpdrag 26522 colperpexlem3 26526 oppne3 26537 oppcom 26538 oppnid 26540 opphllem1 26541 opphllem2 26542 opphllem3 26543 opphllem4 26544 opphllem5 26545 opphllem6 26546 oppperpex 26547 opphl 26548 outpasch 26549 lnopp2hpgb 26557 hpgerlem 26559 colopp 26563 colhp 26564 lmieu 26578 lmimid 26588 lnperpex 26597 trgcopy 26598 trgcopyeulem 26599 |
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