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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 7434 | . 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 28555 | . . . 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 5724 | . . . 4 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
| 11 | tgelrnln.d | . . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑌) | |
| 12 | df-br 5144 | . . . . . . . 8 ⊢ (𝑋 I 𝑌 ↔ 〈𝑋, 𝑌〉 ∈ I ) | |
| 13 | ideqg 5862 | . . . . . . . 8 ⊢ (𝑌 ∈ 𝐵 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌)) | |
| 14 | 12, 13 | bitr3id 285 | . . . . . . 7 ⊢ (𝑌 ∈ 𝐵 → (〈𝑋, 𝑌〉 ∈ I ↔ 𝑋 = 𝑌)) |
| 15 | 14 | necon3bbid 2978 | . . . . . 6 ⊢ (𝑌 ∈ 𝐵 → (¬ 〈𝑋, 𝑌〉 ∈ I ↔ 𝑋 ≠ 𝑌)) |
| 16 | 15 | biimpar 477 | . . . . 5 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌) → ¬ 〈𝑋, 𝑌〉 ∈ I ) |
| 17 | 9, 11, 16 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ¬ 〈𝑋, 𝑌〉 ∈ I ) |
| 18 | 10, 17 | eldifd 3962 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ ((𝐵 × 𝐵) ∖ I )) |
| 19 | fnfvelrn 7100 | . . 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 2845 | 1 ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 〈cop 4632 class class class wbr 5143 I cid 5577 × cxp 5683 ran crn 5686 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 TarskiGcstrkg 28435 Itvcitv 28441 LineGclng 28442 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-trkg 28461 |
| This theorem is referenced by: tghilberti1 28645 tglineinteq 28653 colline 28657 tglowdim2ln 28659 footexALT 28726 footexlem2 28728 foot 28730 perprag 28734 colperpexlem3 28740 mideulem2 28742 midex 28745 outpasch 28763 lnopp2hpgb 28771 colopp 28777 lmieu 28792 lmimid 28802 hypcgrlem1 28807 hypcgrlem2 28808 lnperpex 28811 trgcopy 28812 trgcopyeulem 28813 acopy 28841 acopyeu 28842 tgasa1 28866 |
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