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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 6552 | . . . . . 6 ⊢ (𝑋𝐿𝑌) = (𝐿‘〈𝑋, 𝑌〉) | |
3 | 1, 2 | syl6eleq 2698 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝐿‘〈𝑋, 𝑌〉)) |
4 | elfvdm 6130 | . . . . 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 25242 | . . . . 5 ⊢ (𝐺 ∈ TarskiG → 𝐿 Fn ((𝑃 × 𝑃) ∖ I )) |
11 | fndm 5904 | . . . . 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 2690 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ ((𝑃 × 𝑃) ∖ I )) |
14 | 13 | eldifbd 3553 | . 2 ⊢ (𝜑 → ¬ 〈𝑋, 𝑌〉 ∈ I ) |
15 | df-br 4584 | . . . 4 ⊢ (𝑋 I 𝑌 ↔ 〈𝑋, 𝑌〉 ∈ I ) | |
16 | tglngval.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
17 | ideqg 5195 | . . . . 5 ⊢ (𝑌 ∈ 𝑃 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌)) | |
18 | 16, 17 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌)) |
19 | 15, 18 | syl5bbr 273 | . . 3 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ I ↔ 𝑋 = 𝑌)) |
20 | 19 | necon3bbid 2819 | . 2 ⊢ (𝜑 → (¬ 〈𝑋, 𝑌〉 ∈ I ↔ 𝑋 ≠ 𝑌)) |
21 | 14, 20 | mpbid 221 | 1 ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∖ cdif 3537 〈cop 4131 class class class wbr 4583 I cid 4948 × cxp 5036 dom cdm 5038 Fn wfn 5799 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 TarskiGcstrkg 25129 Itvcitv 25135 LineGclng 25136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-trkg 25152 |
This theorem is referenced by: lnhl 25310 tglnne 25323 tglineneq 25339 tglineinteq 25340 ncolncol 25341 coltr 25342 coltr3 25343 perprag 25418 opphl 25446 hlpasch 25448 |
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