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| Mirrors > Home > MPE Home > Th. List > tglngne | Structured version Visualization version GIF version | ||
| Description: It takes two different points to form a line. (Contributed by Thierry Arnoux, 6-Aug-2019.) |
| Ref | Expression |
|---|---|
| tglngval.p | ⊢ 𝑃 = (Base‘𝐺) |
| tglngval.l | ⊢ 𝐿 = (LineG‘𝐺) |
| tglngval.i | ⊢ 𝐼 = (Itv‘𝐺) |
| tglngval.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
| tglngval.x | ⊢ (𝜑 → 𝑋 ∈ 𝑃) |
| tglngval.y | ⊢ (𝜑 → 𝑌 ∈ 𝑃) |
| tglngne.1 | ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) |
| Ref | Expression |
|---|---|
| tglngne | ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | tglngne.1 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ (𝑋𝐿𝑌)) | |
| 2 | df-ov 7434 | . . . . . 6 ⊢ (𝑋𝐿𝑌) = (𝐿‘〈𝑋, 𝑌〉) | |
| 3 | 1, 2 | eleqtrdi 2851 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝐿‘〈𝑋, 𝑌〉)) |
| 4 | elfvdm 6943 | . . . . 5 ⊢ (𝑍 ∈ (𝐿‘〈𝑋, 𝑌〉) → 〈𝑋, 𝑌〉 ∈ dom 𝐿) | |
| 5 | 3, 4 | syl 17 | . . . 4 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ dom 𝐿) |
| 6 | tglngval.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
| 7 | tglngval.p | . . . . . 6 ⊢ 𝑃 = (Base‘𝐺) | |
| 8 | tglngval.l | . . . . . 6 ⊢ 𝐿 = (LineG‘𝐺) | |
| 9 | tglngval.i | . . . . . 6 ⊢ 𝐼 = (Itv‘𝐺) | |
| 10 | 7, 8, 9 | tglnfn 28555 | . . . . 5 ⊢ (𝐺 ∈ TarskiG → 𝐿 Fn ((𝑃 × 𝑃) ∖ I )) |
| 11 | fndm 6671 | . . . . 5 ⊢ (𝐿 Fn ((𝑃 × 𝑃) ∖ I ) → dom 𝐿 = ((𝑃 × 𝑃) ∖ I )) | |
| 12 | 6, 10, 11 | 3syl 18 | . . . 4 ⊢ (𝜑 → dom 𝐿 = ((𝑃 × 𝑃) ∖ I )) |
| 13 | 5, 12 | eleqtrd 2843 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ ((𝑃 × 𝑃) ∖ I )) |
| 14 | 13 | eldifbd 3964 | . 2 ⊢ (𝜑 → ¬ 〈𝑋, 𝑌〉 ∈ I ) |
| 15 | df-br 5144 | . . . 4 ⊢ (𝑋 I 𝑌 ↔ 〈𝑋, 𝑌〉 ∈ I ) | |
| 16 | tglngval.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
| 17 | ideqg 5862 | . . . . 5 ⊢ (𝑌 ∈ 𝑃 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌)) | |
| 18 | 16, 17 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌)) |
| 19 | 15, 18 | bitr3id 285 | . . 3 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ I ↔ 𝑋 = 𝑌)) |
| 20 | 19 | necon3bbid 2978 | . 2 ⊢ (𝜑 → (¬ 〈𝑋, 𝑌〉 ∈ I ↔ 𝑋 ≠ 𝑌)) |
| 21 | 14, 20 | mpbid 232 | 1 ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 ∖ cdif 3948 〈cop 4632 class class class wbr 5143 I cid 5577 × cxp 5683 dom cdm 5685 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: lnhl 28623 tglnne 28636 tglineneq 28652 tglineinteq 28653 ncolncol 28654 coltr 28655 coltr3 28656 perprag 28734 |
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