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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 7408 | . 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 28306 | . . . 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 5708 | . . . 4 β’ (π β β¨π, πβ© β (π΅ Γ π΅)) |
11 | tgelrnln.d | . . . . 5 β’ (π β π β π) | |
12 | df-br 5142 | . . . . . . . 8 β’ (π I π β β¨π, πβ© β I ) | |
13 | ideqg 5845 | . . . . . . . 8 β’ (π β π΅ β (π I π β π = π)) | |
14 | 12, 13 | bitr3id 285 | . . . . . . 7 β’ (π β π΅ β (β¨π, πβ© β I β π = π)) |
15 | 14 | necon3bbid 2972 | . . . . . 6 β’ (π β π΅ β (Β¬ β¨π, πβ© β I β π β π)) |
16 | 15 | biimpar 477 | . . . . 5 β’ ((π β π΅ β§ π β π) β Β¬ β¨π, πβ© β I ) |
17 | 9, 11, 16 | syl2anc 583 | . . . 4 β’ (π β Β¬ β¨π, πβ© β I ) |
18 | 10, 17 | eldifd 3954 | . . 3 β’ (π β β¨π, πβ© β ((π΅ Γ π΅) β I )) |
19 | fnfvelrn 7076 | . . 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 583 | . 2 β’ (π β (πΏββ¨π, πβ©) β ran πΏ) |
21 | 1, 20 | eqeltrid 2831 | 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 2934 β cdif 3940 β¨cop 4629 class class class wbr 5141 I cid 5566 Γ cxp 5667 ran crn 5670 Fn wfn 6532 βcfv 6537 (class class class)co 7405 Basecbs 17153 TarskiGcstrkg 28186 Itvcitv 28192 LineGclng 28193 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-trkg 28212 |
This theorem is referenced by: tghilberti1 28396 tglineinteq 28404 colline 28408 tglowdim2ln 28410 footexALT 28477 footexlem2 28479 foot 28481 perprag 28485 colperpexlem3 28491 mideulem2 28493 midex 28496 outpasch 28514 lnopp2hpgb 28522 colopp 28528 lmieu 28543 lmimid 28553 hypcgrlem1 28558 hypcgrlem2 28559 lnperpex 28562 trgcopy 28563 trgcopyeulem 28564 acopy 28592 acopyeu 28593 tgasa1 28617 |
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