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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 6552 | . 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 25242 | . . . 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 | opelxpi 5072 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) | |
11 | 8, 9, 10 | syl2anc 691 | . . . 4 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
12 | tgelrnln.d | . . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑌) | |
13 | df-br 4584 | . . . . . . . 8 ⊢ (𝑋 I 𝑌 ↔ 〈𝑋, 𝑌〉 ∈ I ) | |
14 | ideqg 5195 | . . . . . . . 8 ⊢ (𝑌 ∈ 𝐵 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌)) | |
15 | 13, 14 | syl5bbr 273 | . . . . . . 7 ⊢ (𝑌 ∈ 𝐵 → (〈𝑋, 𝑌〉 ∈ I ↔ 𝑋 = 𝑌)) |
16 | 15 | necon3bbid 2819 | . . . . . 6 ⊢ (𝑌 ∈ 𝐵 → (¬ 〈𝑋, 𝑌〉 ∈ I ↔ 𝑋 ≠ 𝑌)) |
17 | 16 | biimpar 501 | . . . . 5 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌) → ¬ 〈𝑋, 𝑌〉 ∈ I ) |
18 | 9, 12, 17 | syl2anc 691 | . . . 4 ⊢ (𝜑 → ¬ 〈𝑋, 𝑌〉 ∈ I ) |
19 | 11, 18 | eldifd 3551 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ ((𝐵 × 𝐵) ∖ I )) |
20 | fnfvelrn 6264 | . . 3 ⊢ ((𝐿 Fn ((𝐵 × 𝐵) ∖ I ) ∧ 〈𝑋, 𝑌〉 ∈ ((𝐵 × 𝐵) ∖ I )) → (𝐿‘〈𝑋, 𝑌〉) ∈ ran 𝐿) | |
21 | 7, 19, 20 | syl2anc 691 | . 2 ⊢ (𝜑 → (𝐿‘〈𝑋, 𝑌〉) ∈ ran 𝐿) |
22 | 1, 21 | syl5eqel 2692 | 1 ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 〈cop 4131 class class class wbr 4583 I cid 4948 × cxp 5036 ran crn 5039 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 TarskiGcstrkg 25129 Itvcitv 25135 LineGclng 25136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-trkg 25152 |
This theorem is referenced by: tghilberti1 25332 tglineinteq 25340 colline 25344 tglowdim2ln 25346 footex 25413 foot 25414 perprag 25418 colperpexlem3 25424 mideulem2 25426 midex 25429 opphllem5 25443 opphllem6 25444 outpasch 25447 lnopp2hpgb 25455 colopp 25461 lmieu 25476 lmimid 25486 hypcgrlem1 25491 hypcgrlem2 25492 lnperpex 25495 trgcopy 25496 trgcopyeulem 25497 acopy 25524 acopyeu 25525 tgasa1 25539 |
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