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| Mirrors > Home > MPE Home > Th. List > tglnne | Structured version Visualization version GIF version | ||
| Description: It takes two different points to form a line. (Contributed by Thierry Arnoux, 27-Nov-2019.) |
| Ref | Expression |
|---|---|
| tglineelsb2.p | ⊢ 𝐵 = (Base‘𝐺) |
| tglineelsb2.i | ⊢ 𝐼 = (Itv‘𝐺) |
| tglineelsb2.l | ⊢ 𝐿 = (LineG‘𝐺) |
| tglineelsb2.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
| tglnne.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
| tglnne.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
| tglnne.1 | ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿) |
| Ref | Expression |
|---|---|
| tglnne | ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tglineelsb2.p | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
| 2 | tglineelsb2.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
| 3 | tglineelsb2.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
| 4 | tglineelsb2.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
| 5 | 4 | ad3antrrr 730 | . . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐺 ∈ TarskiG) |
| 6 | tglnne.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
| 7 | 6 | ad3antrrr 730 | . . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑋 ∈ 𝐵) |
| 8 | tglnne.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
| 9 | 8 | ad3antrrr 730 | . . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑌 ∈ 𝐵) |
| 10 | simpllr 776 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐵) | |
| 11 | simplr 769 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ 𝐵) | |
| 12 | simprr 773 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦) | |
| 13 | eqid 2737 | . . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
| 14 | 1, 13, 3, 5, 10, 11 | tgbtwntriv1 28499 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑥𝐼𝑦)) |
| 15 | 1, 3, 2, 5, 10, 11, 10, 12, 14 | btwnlng1 28627 | . . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑥𝐿𝑦)) |
| 16 | simprl 771 | . . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → (𝑋𝐿𝑌) = (𝑥𝐿𝑦)) | |
| 17 | 15, 16 | eleqtrrd 2844 | . . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑋𝐿𝑌)) |
| 18 | 1, 2, 3, 5, 7, 9, 17 | tglngne 28558 | . 2 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑋 ≠ 𝑌) |
| 19 | tglnne.1 | . . 3 ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿) | |
| 20 | 1, 3, 2, 4, 19 | tgisline 28635 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) |
| 21 | 18, 20 | r19.29vva 3216 | 1 ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ran crn 5686 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 distcds 17306 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-trkgc 28456 df-trkgb 28457 df-trkgcb 28458 df-trkg 28461 |
| This theorem is referenced by: footne 28731 footeq 28732 hlperpnel 28733 colperp 28737 mideulem2 28742 opphllem 28743 midex 28745 opphllem3 28757 opphllem6 28760 opphl 28762 lmieu 28792 lnperpex 28811 trgcopy 28812 |
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