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Mirrors > Home > MPE Home > Th. List > oppne3 | Structured version Visualization version GIF version |
Description: Points lying on opposite sides of a line cannot be equal. (Contributed by Thierry Arnoux, 3-Aug-2020.) |
Ref | Expression |
---|---|
hpg.p | ⊢ 𝑃 = (Base‘𝐺) |
hpg.d | ⊢ − = (dist‘𝐺) |
hpg.i | ⊢ 𝐼 = (Itv‘𝐺) |
hpg.o | ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} |
opphl.l | ⊢ 𝐿 = (LineG‘𝐺) |
opphl.d | ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) |
opphl.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
oppcom.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
oppcom.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
oppcom.o | ⊢ (𝜑 → 𝐴𝑂𝐵) |
Ref | Expression |
---|---|
oppne3 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hpg.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
2 | hpg.d | . . . 4 ⊢ − = (dist‘𝐺) | |
3 | hpg.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | hpg.o | . . . 4 ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} | |
5 | opphl.l | . . . 4 ⊢ 𝐿 = (LineG‘𝐺) | |
6 | opphl.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) | |
7 | opphl.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
8 | oppcom.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
9 | oppcom.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
10 | oppcom.o | . . . 4 ⊢ (𝜑 → 𝐴𝑂𝐵) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | oppne1 27098 | . . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) |
12 | 7 | ad3antrrr 727 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG) |
13 | 8 | ad3antrrr 727 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴 ∈ 𝑃) |
14 | 6 | ad3antrrr 727 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐷 ∈ ran 𝐿) |
15 | simplr 766 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ 𝐷) | |
16 | 1, 5, 3, 12, 14, 15 | tglnpt 26906 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ 𝑃) |
17 | simpr 485 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐴𝐼𝐵)) | |
18 | simpllr 773 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴 = 𝐵) | |
19 | 18 | oveq2d 7285 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → (𝐴𝐼𝐴) = (𝐴𝐼𝐵)) |
20 | 17, 19 | eleqtrrd 2844 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐴𝐼𝐴)) |
21 | 1, 2, 3, 12, 13, 16, 20 | axtgbtwnid 26823 | . . . . 5 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴 = 𝑡) |
22 | 21, 15 | eqeltrd 2841 | . . . 4 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴 ∈ 𝐷) |
23 | 1, 2, 3, 4, 8, 9 | islnopp 27096 | . . . . . . 7 ⊢ (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)))) |
24 | 10, 23 | mpbid 231 | . . . . . 6 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))) |
25 | 24 | simprd 496 | . . . . 5 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)) |
26 | 25 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)) |
27 | 22, 26 | r19.29a 3220 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐷) |
28 | 11, 27 | mtand 813 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) |
29 | 28 | neqned 2952 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ∃wrex 3067 ∖ cdif 3889 class class class wbr 5079 {copab 5141 ran crn 5590 ‘cfv 6431 (class class class)co 7269 Basecbs 16908 distcds 16967 TarskiGcstrkg 26784 Itvcitv 26790 LineGclng 26791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-sep 5227 ax-nul 5234 ax-pr 5356 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-rab 3075 df-v 3433 df-sbc 3721 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-br 5080 df-opab 5142 df-cnv 5597 df-dm 5599 df-rn 5600 df-iota 6389 df-fv 6439 df-ov 7272 df-oprab 7273 df-mpo 7274 df-trkgb 26806 df-trkg 26810 |
This theorem is referenced by: colopp 27126 trgcopyeulem 27162 |
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