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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 2849 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝐿‘〈𝑋, 𝑌〉)) |
4 | elfvdm 6944 | . . . . 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 28570 | . . . . 5 ⊢ (𝐺 ∈ TarskiG → 𝐿 Fn ((𝑃 × 𝑃) ∖ I )) |
11 | fndm 6672 | . . . . 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 2841 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ ((𝑃 × 𝑃) ∖ I )) |
14 | 13 | eldifbd 3976 | . 2 ⊢ (𝜑 → ¬ 〈𝑋, 𝑌〉 ∈ I ) |
15 | df-br 5149 | . . . 4 ⊢ (𝑋 I 𝑌 ↔ 〈𝑋, 𝑌〉 ∈ I ) | |
16 | tglngval.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
17 | ideqg 5865 | . . . . 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 2976 | . 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 1537 ∈ wcel 2106 ≠ wne 2938 ∖ cdif 3960 〈cop 4637 class class class wbr 5148 I cid 5582 × cxp 5687 dom cdm 5689 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 TarskiGcstrkg 28450 Itvcitv 28456 LineGclng 28457 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-trkg 28476 |
This theorem is referenced by: lnhl 28638 tglnne 28651 tglineneq 28667 tglineinteq 28668 ncolncol 28669 coltr 28670 coltr3 28671 perprag 28749 |
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