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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 7159 | . . . . . 6 ⊢ (𝑋𝐿𝑌) = (𝐿‘〈𝑋, 𝑌〉) | |
3 | 1, 2 | eleqtrdi 2923 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝐿‘〈𝑋, 𝑌〉)) |
4 | elfvdm 6702 | . . . . 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 26333 | . . . . 5 ⊢ (𝐺 ∈ TarskiG → 𝐿 Fn ((𝑃 × 𝑃) ∖ I )) |
11 | fndm 6455 | . . . . 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 2915 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ ((𝑃 × 𝑃) ∖ I )) |
14 | 13 | eldifbd 3949 | . 2 ⊢ (𝜑 → ¬ 〈𝑋, 𝑌〉 ∈ I ) |
15 | df-br 5067 | . . . 4 ⊢ (𝑋 I 𝑌 ↔ 〈𝑋, 𝑌〉 ∈ I ) | |
16 | tglngval.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑃) | |
17 | ideqg 5722 | . . . . 5 ⊢ (𝑌 ∈ 𝑃 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌)) | |
18 | 16, 17 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌)) |
19 | 15, 18 | syl5bbr 287 | . . 3 ⊢ (𝜑 → (〈𝑋, 𝑌〉 ∈ I ↔ 𝑋 = 𝑌)) |
20 | 19 | necon3bbid 3053 | . 2 ⊢ (𝜑 → (¬ 〈𝑋, 𝑌〉 ∈ I ↔ 𝑋 ≠ 𝑌)) |
21 | 14, 20 | mpbid 234 | 1 ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∖ cdif 3933 〈cop 4573 class class class wbr 5066 I cid 5459 × cxp 5553 dom cdm 5555 Fn wfn 6350 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 TarskiGcstrkg 26216 Itvcitv 26222 LineGclng 26223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-trkg 26239 |
This theorem is referenced by: lnhl 26401 tglnne 26414 tglineneq 26430 tglineinteq 26431 ncolncol 26432 coltr 26433 coltr3 26434 perprag 26512 |
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