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Mirrors > Home > MPE Home > Th. List > tgbtwnne | Structured version Visualization version GIF version |
Description: Betweenness and inequality. (Contributed by Thierry Arnoux, 1-Dec-2019.) |
Ref | Expression |
---|---|
tkgeom.p | β’ π = (BaseβπΊ) |
tkgeom.d | β’ β = (distβπΊ) |
tkgeom.i | β’ πΌ = (ItvβπΊ) |
tkgeom.g | β’ (π β πΊ β TarskiG) |
tgbtwntriv2.1 | β’ (π β π΄ β π) |
tgbtwntriv2.2 | β’ (π β π΅ β π) |
tgbtwncomb.3 | β’ (π β πΆ β π) |
tgbtwnne.1 | β’ (π β π΅ β (π΄πΌπΆ)) |
tgbtwnne.2 | β’ (π β π΅ β π΄) |
Ref | Expression |
---|---|
tgbtwnne | β’ (π β π΄ β πΆ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tkgeom.p | . . . . 5 β’ π = (BaseβπΊ) | |
2 | tkgeom.d | . . . . 5 β’ β = (distβπΊ) | |
3 | tkgeom.i | . . . . 5 β’ πΌ = (ItvβπΊ) | |
4 | tkgeom.g | . . . . . 6 β’ (π β πΊ β TarskiG) | |
5 | 4 | adantr 479 | . . . . 5 β’ ((π β§ π΄ = πΆ) β πΊ β TarskiG) |
6 | tgbtwntriv2.1 | . . . . . 6 β’ (π β π΄ β π) | |
7 | 6 | adantr 479 | . . . . 5 β’ ((π β§ π΄ = πΆ) β π΄ β π) |
8 | tgbtwntriv2.2 | . . . . . 6 β’ (π β π΅ β π) | |
9 | 8 | adantr 479 | . . . . 5 β’ ((π β§ π΄ = πΆ) β π΅ β π) |
10 | tgbtwnne.1 | . . . . . . 7 β’ (π β π΅ β (π΄πΌπΆ)) | |
11 | 10 | adantr 479 | . . . . . 6 β’ ((π β§ π΄ = πΆ) β π΅ β (π΄πΌπΆ)) |
12 | simpr 483 | . . . . . . 7 β’ ((π β§ π΄ = πΆ) β π΄ = πΆ) | |
13 | 12 | oveq2d 7427 | . . . . . 6 β’ ((π β§ π΄ = πΆ) β (π΄πΌπ΄) = (π΄πΌπΆ)) |
14 | 11, 13 | eleqtrrd 2834 | . . . . 5 β’ ((π β§ π΄ = πΆ) β π΅ β (π΄πΌπ΄)) |
15 | 1, 2, 3, 5, 7, 9, 14 | axtgbtwnid 27984 | . . . 4 β’ ((π β§ π΄ = πΆ) β π΄ = π΅) |
16 | 15 | eqcomd 2736 | . . 3 β’ ((π β§ π΄ = πΆ) β π΅ = π΄) |
17 | tgbtwnne.2 | . . . . 5 β’ (π β π΅ β π΄) | |
18 | 17 | adantr 479 | . . . 4 β’ ((π β§ π΄ = πΆ) β π΅ β π΄) |
19 | 18 | neneqd 2943 | . . 3 β’ ((π β§ π΄ = πΆ) β Β¬ π΅ = π΄) |
20 | 16, 19 | pm2.65da 813 | . 2 β’ (π β Β¬ π΄ = πΆ) |
21 | 20 | neqned 2945 | 1 β’ (π β π΄ β πΆ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1539 β wcel 2104 β wne 2938 βcfv 6542 (class class class)co 7411 Basecbs 17148 distcds 17210 TarskiGcstrkg 27945 Itvcitv 27951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-nul 5305 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-iota 6494 df-fv 6550 df-ov 7414 df-trkgb 27967 df-trkg 27971 |
This theorem is referenced by: mideulem2 28252 opphllem 28253 outpasch 28273 lnopp2hpgb 28281 lmieu 28302 dfcgra2 28348 |
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