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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 | β’ π΅ = (BaseβπΊ) |
tglineelsb2.i | β’ πΌ = (ItvβπΊ) |
tglineelsb2.l | β’ πΏ = (LineGβπΊ) |
tglineelsb2.g | β’ (π β πΊ β TarskiG) |
tgelrnln.x | β’ (π β π β π΅) |
tgelrnln.y | β’ (π β π β π΅) |
tgelrnln.d | β’ (π β π β π) |
Ref | Expression |
---|---|
tgelrnln | β’ (π β (ππΏπ) β ran πΏ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 7429 | . 2 β’ (ππΏπ) = (πΏββ¨π, πβ©) | |
2 | tglineelsb2.g | . . . 4 β’ (π β πΊ β TarskiG) | |
3 | tglineelsb2.p | . . . . 5 β’ π΅ = (BaseβπΊ) | |
4 | tglineelsb2.l | . . . . 5 β’ πΏ = (LineGβπΊ) | |
5 | tglineelsb2.i | . . . . 5 β’ πΌ = (ItvβπΊ) | |
6 | 3, 4, 5 | tglnfn 28379 | . . . 4 β’ (πΊ β TarskiG β πΏ Fn ((π΅ Γ π΅) β I )) |
7 | 2, 6 | syl 17 | . . 3 β’ (π β πΏ Fn ((π΅ Γ π΅) β I )) |
8 | tgelrnln.x | . . . . 5 β’ (π β π β π΅) | |
9 | tgelrnln.y | . . . . 5 β’ (π β π β π΅) | |
10 | 8, 9 | opelxpd 5721 | . . . 4 β’ (π β β¨π, πβ© β (π΅ Γ π΅)) |
11 | tgelrnln.d | . . . . 5 β’ (π β π β π) | |
12 | df-br 5153 | . . . . . . . 8 β’ (π I π β β¨π, πβ© β I ) | |
13 | ideqg 5858 | . . . . . . . 8 β’ (π β π΅ β (π I π β π = π)) | |
14 | 12, 13 | bitr3id 284 | . . . . . . 7 β’ (π β π΅ β (β¨π, πβ© β I β π = π)) |
15 | 14 | necon3bbid 2975 | . . . . . 6 β’ (π β π΅ β (Β¬ β¨π, πβ© β I β π β π)) |
16 | 15 | biimpar 476 | . . . . 5 β’ ((π β π΅ β§ π β π) β Β¬ β¨π, πβ© β I ) |
17 | 9, 11, 16 | syl2anc 582 | . . . 4 β’ (π β Β¬ β¨π, πβ© β I ) |
18 | 10, 17 | eldifd 3960 | . . 3 β’ (π β β¨π, πβ© β ((π΅ Γ π΅) β I )) |
19 | fnfvelrn 7095 | . . 3 β’ ((πΏ Fn ((π΅ Γ π΅) β I ) β§ β¨π, πβ© β ((π΅ Γ π΅) β I )) β (πΏββ¨π, πβ©) β ran πΏ) | |
20 | 7, 18, 19 | syl2anc 582 | . 2 β’ (π β (πΏββ¨π, πβ©) β ran πΏ) |
21 | 1, 20 | eqeltrid 2833 | 1 β’ (π β (ππΏπ) β ran πΏ) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 = wceq 1533 β wcel 2098 β wne 2937 β cdif 3946 β¨cop 4638 class class class wbr 5152 I cid 5579 Γ cxp 5680 ran crn 5683 Fn wfn 6548 βcfv 6553 (class class class)co 7426 Basecbs 17189 TarskiGcstrkg 28259 Itvcitv 28265 LineGclng 28266 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 8001 df-2nd 8002 df-trkg 28285 |
This theorem is referenced by: tghilberti1 28469 tglineinteq 28477 colline 28481 tglowdim2ln 28483 footexALT 28550 footexlem2 28552 foot 28554 perprag 28558 colperpexlem3 28564 mideulem2 28566 midex 28569 outpasch 28587 lnopp2hpgb 28595 colopp 28601 lmieu 28616 lmimid 28626 hypcgrlem1 28631 hypcgrlem2 28632 lnperpex 28635 trgcopy 28636 trgcopyeulem 28637 acopy 28665 acopyeu 28666 tgasa1 28690 |
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