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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 26334 | . . 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 4868 | . . . 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 3968 | . 2 ⊢ (𝜑 → 𝑋 ∈ ∪ ran 𝐿) |
12 | 6, 11 | sseldd 3968 | 1 ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ∪ cuni 4838 ran crn 5556 ‘cfv 6355 Basecbs 16483 TarskiGcstrkg 26216 Itvcitv 26222 LineGclng 26223 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-cnv 5563 df-dm 5565 df-rn 5566 df-iota 6314 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-trkg 26239 |
This theorem is referenced by: mirln 26462 mirln2 26463 perpcom 26499 perpneq 26500 ragperp 26503 foot 26508 footne 26509 footeq 26510 hlperpnel 26511 perprag 26512 perpdragALT 26513 perpdrag 26514 colperpexlem3 26518 oppne3 26529 oppcom 26530 oppnid 26532 opphllem1 26533 opphllem2 26534 opphllem3 26535 opphllem4 26536 opphllem5 26537 opphllem6 26538 oppperpex 26539 opphl 26540 outpasch 26541 lnopp2hpgb 26549 hpgerlem 26551 colopp 26555 colhp 26556 lmieu 26570 lmimid 26580 lnperpex 26589 trgcopy 26590 trgcopyeulem 26591 |
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