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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 728 | . . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝐺 ∈ TarskiG) |
6 | tglnne.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
7 | 6 | ad3antrrr 728 | . . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑋 ∈ 𝐵) |
8 | tglnne.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
9 | 8 | ad3antrrr 728 | . . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑌 ∈ 𝐵) |
10 | simpllr 774 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ 𝐵) | |
11 | simplr 767 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑦 ∈ 𝐵) | |
12 | simprr 771 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ≠ 𝑦) | |
13 | eqid 2821 | . . . . . 6 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
14 | 1, 13, 3, 5, 10, 11 | tgbtwntriv1 26271 | . . . . 5 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑥𝐼𝑦)) |
15 | 1, 3, 2, 5, 10, 11, 10, 12, 14 | btwnlng1 26399 | . . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑥𝐿𝑦)) |
16 | simprl 769 | . . . 4 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → (𝑋𝐿𝑌) = (𝑥𝐿𝑦)) | |
17 | 15, 16 | eleqtrrd 2916 | . . 3 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑦 ∈ 𝐵) ∧ ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) → 𝑥 ∈ (𝑋𝐿𝑌)) |
18 | 1, 2, 3, 5, 7, 9, 17 | tglngne 26330 | . 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 26407 | . 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑋𝐿𝑌) = (𝑥𝐿𝑦) ∧ 𝑥 ≠ 𝑦)) |
21 | 18, 20 | r19.29vva 3336 | 1 ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ran crn 5550 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 distcds 16568 TarskiGcstrkg 26210 Itvcitv 26216 LineGclng 26217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-trkgc 26228 df-trkgb 26229 df-trkgcb 26230 df-trkg 26233 |
This theorem is referenced by: footne 26503 footeq 26504 hlperpnel 26505 colperp 26509 mideulem2 26514 opphllem 26515 midex 26517 opphllem3 26529 opphllem6 26532 opphl 26534 lmieu 26564 lnperpex 26583 trgcopy 26584 |
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