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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 7354 | . . . . . 6 ⊢ (𝑋𝐿𝑌) = (𝐿‘〈𝑋, 𝑌〉) | |
3 | 1, 2 | eleqtrdi 2848 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝐿‘〈𝑋, 𝑌〉)) |
4 | elfvdm 6876 | . . . . 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 27317 | . . . . 5 ⊢ (𝐺 ∈ TarskiG → 𝐿 Fn ((𝑃 × 𝑃) ∖ I )) |
11 | fndm 6602 | . . . . 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 2840 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ ((𝑃 × 𝑃) ∖ I )) |
14 | 13 | eldifbd 3921 | . 2 ⊢ (𝜑 → ¬ 〈𝑋, 𝑌〉 ∈ I ) |
15 | df-br 5104 | . . . 4 ⊢ (𝑋 I 𝑌 ↔ 〈𝑋, 𝑌〉 ∈ I ) | |
16 | tglngval.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
17 | ideqg 5805 | . . . . 5 ⊢ (𝑌 ∈ 𝑃 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌)) | |
18 | 16, 17 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌)) |
19 | 15, 18 | bitr3id 284 | . . 3 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ I ↔ 𝑋 = 𝑌)) |
20 | 19 | necon3bbid 2979 | . 2 ⊢ (𝜑 → (¬ 〈𝑋, 𝑌〉 ∈ I ↔ 𝑋 ≠ 𝑌)) |
21 | 14, 20 | mpbid 231 | 1 ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 ≠ wne 2941 ∖ cdif 3905 〈cop 4590 class class class wbr 5103 I cid 5528 × cxp 5629 dom cdm 5631 Fn wfn 6488 ‘cfv 6493 (class class class)co 7351 Basecbs 17042 TarskiGcstrkg 27197 Itvcitv 27203 LineGclng 27204 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fv 6501 df-ov 7354 df-oprab 7355 df-mpo 7356 df-1st 7913 df-2nd 7914 df-trkg 27223 |
This theorem is referenced by: lnhl 27385 tglnne 27398 tglineneq 27414 tglineinteq 27415 ncolncol 27416 coltr 27417 coltr3 27418 perprag 27496 |
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