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Mirrors > Home > MPE Home > Th. List > tgelrnln | Structured version Visualization version GIF version |
Description: The property of being a proper line, generated by two distinct points. (Contributed by Thierry Arnoux, 25-May-2019.) |
Ref | Expression |
---|---|
tglineelsb2.p | ⊢ 𝐵 = (Base‘𝐺) |
tglineelsb2.i | ⊢ 𝐼 = (Itv‘𝐺) |
tglineelsb2.l | ⊢ 𝐿 = (LineG‘𝐺) |
tglineelsb2.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tgelrnln.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
tgelrnln.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
tgelrnln.d | ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
Ref | Expression |
---|---|
tgelrnln | ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7341 | . 2 ⊢ (𝑋𝐿𝑌) = (𝐿‘〈𝑋, 𝑌〉) | |
2 | tglineelsb2.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
3 | tglineelsb2.p | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
4 | tglineelsb2.l | . . . . 5 ⊢ 𝐿 = (LineG‘𝐺) | |
5 | tglineelsb2.i | . . . . 5 ⊢ 𝐼 = (Itv‘𝐺) | |
6 | 3, 4, 5 | tglnfn 27198 | . . . 4 ⊢ (𝐺 ∈ TarskiG → 𝐿 Fn ((𝐵 × 𝐵) ∖ I )) |
7 | 2, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝐿 Fn ((𝐵 × 𝐵) ∖ I )) |
8 | tgelrnln.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | tgelrnln.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
10 | 8, 9 | opelxpd 5659 | . . . 4 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
11 | tgelrnln.d | . . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑌) | |
12 | df-br 5094 | . . . . . . . 8 ⊢ (𝑋 I 𝑌 ↔ 〈𝑋, 𝑌〉 ∈ I ) | |
13 | ideqg 5794 | . . . . . . . 8 ⊢ (𝑌 ∈ 𝐵 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌)) | |
14 | 12, 13 | bitr3id 284 | . . . . . . 7 ⊢ (𝑌 ∈ 𝐵 → (〈𝑋, 𝑌〉 ∈ I ↔ 𝑋 = 𝑌)) |
15 | 14 | necon3bbid 2978 | . . . . . 6 ⊢ (𝑌 ∈ 𝐵 → (¬ 〈𝑋, 𝑌〉 ∈ I ↔ 𝑋 ≠ 𝑌)) |
16 | 15 | biimpar 478 | . . . . 5 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌) → ¬ 〈𝑋, 𝑌〉 ∈ I ) |
17 | 9, 11, 16 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ¬ 〈𝑋, 𝑌〉 ∈ I ) |
18 | 10, 17 | eldifd 3909 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ ((𝐵 × 𝐵) ∖ I )) |
19 | fnfvelrn 7015 | . . 3 ⊢ ((𝐿 Fn ((𝐵 × 𝐵) ∖ I ) ∧ 〈𝑋, 𝑌〉 ∈ ((𝐵 × 𝐵) ∖ I )) → (𝐿‘〈𝑋, 𝑌〉) ∈ ran 𝐿) | |
20 | 7, 18, 19 | syl2anc 584 | . 2 ⊢ (𝜑 → (𝐿‘〈𝑋, 𝑌〉) ∈ ran 𝐿) |
21 | 1, 20 | eqeltrid 2841 | 1 ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 ∖ cdif 3895 〈cop 4580 class class class wbr 5093 I cid 5518 × cxp 5619 ran crn 5622 Fn wfn 6475 ‘cfv 6480 (class class class)co 7338 Basecbs 17010 TarskiGcstrkg 27078 Itvcitv 27084 LineGclng 27085 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5244 ax-nul 5251 ax-pr 5373 ax-un 7651 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4271 df-if 4475 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4854 df-iun 4944 df-br 5094 df-opab 5156 df-mpt 5177 df-id 5519 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-iota 6432 df-fun 6482 df-fn 6483 df-f 6484 df-fv 6488 df-ov 7341 df-oprab 7342 df-mpo 7343 df-1st 7900 df-2nd 7901 df-trkg 27104 |
This theorem is referenced by: tghilberti1 27288 tglineinteq 27296 colline 27300 tglowdim2ln 27302 footexALT 27369 footexlem2 27371 foot 27373 perprag 27377 colperpexlem3 27383 mideulem2 27385 midex 27388 outpasch 27406 lnopp2hpgb 27414 colopp 27420 lmieu 27435 lmimid 27445 hypcgrlem1 27450 hypcgrlem2 27451 lnperpex 27454 trgcopy 27455 trgcopyeulem 27456 acopy 27484 acopyeu 27485 tgasa1 27509 |
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