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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 7194 | . 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 26592 | . . . 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 5574 | . . . 4 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
11 | tgelrnln.d | . . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑌) | |
12 | df-br 5040 | . . . . . . . 8 ⊢ (𝑋 I 𝑌 ↔ 〈𝑋, 𝑌〉 ∈ I ) | |
13 | ideqg 5705 | . . . . . . . 8 ⊢ (𝑌 ∈ 𝐵 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌)) | |
14 | 12, 13 | bitr3id 288 | . . . . . . 7 ⊢ (𝑌 ∈ 𝐵 → (〈𝑋, 𝑌〉 ∈ I ↔ 𝑋 = 𝑌)) |
15 | 14 | necon3bbid 2969 | . . . . . 6 ⊢ (𝑌 ∈ 𝐵 → (¬ 〈𝑋, 𝑌〉 ∈ I ↔ 𝑋 ≠ 𝑌)) |
16 | 15 | biimpar 481 | . . . . 5 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌) → ¬ 〈𝑋, 𝑌〉 ∈ I ) |
17 | 9, 11, 16 | syl2anc 587 | . . . 4 ⊢ (𝜑 → ¬ 〈𝑋, 𝑌〉 ∈ I ) |
18 | 10, 17 | eldifd 3864 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ ((𝐵 × 𝐵) ∖ I )) |
19 | fnfvelrn 6879 | . . 3 ⊢ ((𝐿 Fn ((𝐵 × 𝐵) ∖ I ) ∧ 〈𝑋, 𝑌〉 ∈ ((𝐵 × 𝐵) ∖ I )) → (𝐿‘〈𝑋, 𝑌〉) ∈ ran 𝐿) | |
20 | 7, 18, 19 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝐿‘〈𝑋, 𝑌〉) ∈ ran 𝐿) |
21 | 1, 20 | eqeltrid 2835 | 1 ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∖ cdif 3850 〈cop 4533 class class class wbr 5039 I cid 5439 × cxp 5534 ran crn 5537 Fn wfn 6353 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 TarskiGcstrkg 26475 Itvcitv 26481 LineGclng 26482 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-1st 7739 df-2nd 7740 df-trkg 26498 |
This theorem is referenced by: tghilberti1 26682 tglineinteq 26690 colline 26694 tglowdim2ln 26696 footexALT 26763 footexlem2 26765 foot 26767 perprag 26771 colperpexlem3 26777 mideulem2 26779 midex 26782 outpasch 26800 lnopp2hpgb 26808 colopp 26814 lmieu 26829 lmimid 26839 hypcgrlem1 26844 hypcgrlem2 26845 lnperpex 26848 trgcopy 26849 trgcopyeulem 26850 acopy 26878 acopyeu 26879 tgasa1 26903 |
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