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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 28749 | . . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐷) | 
| 12 | 7 | ad3antrrr 730 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐺 ∈ TarskiG) | 
| 13 | 8 | ad3antrrr 730 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴 ∈ 𝑃) | 
| 14 | 6 | ad3antrrr 730 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐷 ∈ ran 𝐿) | 
| 15 | simplr 769 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ 𝐷) | |
| 16 | 1, 5, 3, 12, 14, 15 | tglnpt 28557 | . . . . . 6 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ 𝑃) | 
| 17 | simpr 484 | . . . . . . 7 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝑡 ∈ (𝐴𝐼𝐵)) | |
| 18 | simpllr 776 | . . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐴 = 𝐵) ∧ 𝑡 ∈ 𝐷) ∧ 𝑡 ∈ (𝐴𝐼𝐵)) → 𝐴 = 𝐵) | |
| 19 | 18 | oveq2d 7447 | . . . . . . 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 28474 | . . . . 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 28747 | . . . . . . 7 ⊢ (𝜑 → (𝐴𝑂𝐵 ↔ ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)))) | 
| 24 | 10, 23 | mpbid 232 | . . . . . 6 ⊢ (𝜑 → ((¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵))) | 
| 25 | 24 | simprd 495 | . . . . 5 ⊢ (𝜑 → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)) | 
| 26 | 25 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝐴𝐼𝐵)) | 
| 27 | 22, 26 | r19.29a 3162 | . . 3 ⊢ ((𝜑 ∧ 𝐴 = 𝐵) → 𝐴 ∈ 𝐷) | 
| 28 | 11, 27 | mtand 816 | . 2 ⊢ (𝜑 → ¬ 𝐴 = 𝐵) | 
| 29 | 28 | neqned 2947 | 1 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 ∖ cdif 3948 class class class wbr 5143 {copab 5205 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 | 
| 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-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-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 df-iota 6514 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-trkgb 28457 df-trkg 28461 | 
| This theorem is referenced by: colopp 28777 trgcopyeulem 28813 | 
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