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Mirrors > Home > MPE Home > Th. List > tgelrnln | Structured version Visualization version GIF version |
Description: The property of being a proper line, generated by two distinct points. (Contributed by Thierry Arnoux, 25-May-2019.) |
Ref | Expression |
---|---|
tglineelsb2.p | ⊢ 𝐵 = (Base‘𝐺) |
tglineelsb2.i | ⊢ 𝐼 = (Itv‘𝐺) |
tglineelsb2.l | ⊢ 𝐿 = (LineG‘𝐺) |
tglineelsb2.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tgelrnln.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
tgelrnln.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
tgelrnln.d | ⊢ (𝜑 → 𝑋 ≠ 𝑌) |
Ref | Expression |
---|---|
tgelrnln | ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7161 | . 2 ⊢ (𝑋𝐿𝑌) = (𝐿‘〈𝑋, 𝑌〉) | |
2 | tglineelsb2.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
3 | tglineelsb2.p | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
4 | tglineelsb2.l | . . . . 5 ⊢ 𝐿 = (LineG‘𝐺) | |
5 | tglineelsb2.i | . . . . 5 ⊢ 𝐼 = (Itv‘𝐺) | |
6 | 3, 4, 5 | tglnfn 26335 | . . . 4 ⊢ (𝐺 ∈ TarskiG → 𝐿 Fn ((𝐵 × 𝐵) ∖ I )) |
7 | 2, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝐿 Fn ((𝐵 × 𝐵) ∖ I )) |
8 | tgelrnln.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
9 | tgelrnln.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
10 | 8, 9 | opelxpd 5595 | . . . 4 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ (𝐵 × 𝐵)) |
11 | tgelrnln.d | . . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑌) | |
12 | df-br 5069 | . . . . . . . 8 ⊢ (𝑋 I 𝑌 ↔ 〈𝑋, 𝑌〉 ∈ I ) | |
13 | ideqg 5724 | . . . . . . . 8 ⊢ (𝑌 ∈ 𝐵 → (𝑋 I 𝑌 ↔ 𝑋 = 𝑌)) | |
14 | 12, 13 | syl5bbr 287 | . . . . . . 7 ⊢ (𝑌 ∈ 𝐵 → (〈𝑋, 𝑌〉 ∈ I ↔ 𝑋 = 𝑌)) |
15 | 14 | necon3bbid 3055 | . . . . . 6 ⊢ (𝑌 ∈ 𝐵 → (¬ 〈𝑋, 𝑌〉 ∈ I ↔ 𝑋 ≠ 𝑌)) |
16 | 15 | biimpar 480 | . . . . 5 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑋 ≠ 𝑌) → ¬ 〈𝑋, 𝑌〉 ∈ I ) |
17 | 9, 11, 16 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ¬ 〈𝑋, 𝑌〉 ∈ I ) |
18 | 10, 17 | eldifd 3949 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝑌〉 ∈ ((𝐵 × 𝐵) ∖ I )) |
19 | fnfvelrn 6850 | . . 3 ⊢ ((𝐿 Fn ((𝐵 × 𝐵) ∖ I ) ∧ 〈𝑋, 𝑌〉 ∈ ((𝐵 × 𝐵) ∖ I )) → (𝐿‘〈𝑋, 𝑌〉) ∈ ran 𝐿) | |
20 | 7, 18, 19 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐿‘〈𝑋, 𝑌〉) ∈ ran 𝐿) |
21 | 1, 20 | eqeltrid 2919 | 1 ⊢ (𝜑 → (𝑋𝐿𝑌) ∈ ran 𝐿) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 〈cop 4575 class class class wbr 5068 I cid 5461 × cxp 5555 ran crn 5558 Fn wfn 6352 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 TarskiGcstrkg 26218 Itvcitv 26224 LineGclng 26225 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-trkg 26241 |
This theorem is referenced by: tghilberti1 26425 tglineinteq 26433 colline 26437 tglowdim2ln 26439 footexALT 26506 footexlem2 26508 foot 26510 perprag 26514 colperpexlem3 26520 mideulem2 26522 midex 26525 outpasch 26543 lnopp2hpgb 26551 colopp 26557 lmieu 26572 lmimid 26582 hypcgrlem1 26587 hypcgrlem2 26588 lnperpex 26591 trgcopy 26592 trgcopyeulem 26593 acopy 26621 acopyeu 26622 tgasa1 26646 |
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