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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > tg5segofs | Structured version Visualization version GIF version |
Description: Rephrase axtg5seg 27983 using the outer five segment predicate. Theorem 2.10 of [Schwabhauser] p. 28. (Contributed by Thierry Arnoux, 23-Mar-2019.) |
Ref | Expression |
---|---|
tg5segofs.p | β’ π = (BaseβπΊ) |
tg5segofs.m | β’ β = (distβπΊ) |
tg5segofs.s | β’ πΌ = (ItvβπΊ) |
tg5segofs.g | β’ (π β πΊ β TarskiG) |
tg5segofs.a | β’ (π β π΄ β π) |
tg5segofs.b | β’ (π β π΅ β π) |
tg5segofs.c | β’ (π β πΆ β π) |
tg5segofs.d | β’ (π β π· β π) |
tg5segofs.e | β’ (π β πΈ β π) |
tg5segofs.f | β’ (π β πΉ β π) |
tg5segofs.o | β’ π = (AFSβπΊ) |
tg5segofs.h | β’ (π β π» β π) |
tg5segofs.i | β’ (π β πΌ β π) |
tg5segofs.1 | β’ (π β β¨β¨π΄, π΅β©, β¨πΆ, π·β©β©πβ¨β¨πΈ, πΉβ©, β¨π», πΌβ©β©) |
tg5segofs.2 | β’ (π β π΄ β π΅) |
Ref | Expression |
---|---|
tg5segofs | β’ (π β (πΆ β π·) = (π» β πΌ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tg5segofs.p | . 2 β’ π = (BaseβπΊ) | |
2 | tg5segofs.m | . 2 β’ β = (distβπΊ) | |
3 | tg5segofs.s | . 2 β’ πΌ = (ItvβπΊ) | |
4 | tg5segofs.g | . 2 β’ (π β πΊ β TarskiG) | |
5 | tg5segofs.a | . 2 β’ (π β π΄ β π) | |
6 | tg5segofs.b | . 2 β’ (π β π΅ β π) | |
7 | tg5segofs.c | . 2 β’ (π β πΆ β π) | |
8 | tg5segofs.e | . 2 β’ (π β πΈ β π) | |
9 | tg5segofs.f | . 2 β’ (π β πΉ β π) | |
10 | tg5segofs.h | . 2 β’ (π β π» β π) | |
11 | tg5segofs.d | . 2 β’ (π β π· β π) | |
12 | tg5segofs.i | . 2 β’ (π β πΌ β π) | |
13 | tg5segofs.2 | . 2 β’ (π β π΄ β π΅) | |
14 | tg5segofs.1 | . . . . 5 β’ (π β β¨β¨π΄, π΅β©, β¨πΆ, π·β©β©πβ¨β¨πΈ, πΉβ©, β¨π», πΌβ©β©) | |
15 | tg5segofs.o | . . . . . 6 β’ π = (AFSβπΊ) | |
16 | 1, 2, 3, 4, 15, 5, 6, 7, 11, 8, 9, 10, 12 | brafs 33982 | . . . . 5 β’ (π β (β¨β¨π΄, π΅β©, β¨πΆ, π·β©β©πβ¨β¨πΈ, πΉβ©, β¨π», πΌβ©β© β ((π΅ β (π΄πΌπΆ) β§ πΉ β (πΈπΌπ»)) β§ ((π΄ β π΅) = (πΈ β πΉ) β§ (π΅ β πΆ) = (πΉ β π»)) β§ ((π΄ β π·) = (πΈ β πΌ) β§ (π΅ β π·) = (πΉ β πΌ))))) |
17 | 14, 16 | mpbid 231 | . . . 4 β’ (π β ((π΅ β (π΄πΌπΆ) β§ πΉ β (πΈπΌπ»)) β§ ((π΄ β π΅) = (πΈ β πΉ) β§ (π΅ β πΆ) = (πΉ β π»)) β§ ((π΄ β π·) = (πΈ β πΌ) β§ (π΅ β π·) = (πΉ β πΌ)))) |
18 | 17 | simp1d 1140 | . . 3 β’ (π β (π΅ β (π΄πΌπΆ) β§ πΉ β (πΈπΌπ»))) |
19 | 18 | simpld 493 | . 2 β’ (π β π΅ β (π΄πΌπΆ)) |
20 | 18 | simprd 494 | . 2 β’ (π β πΉ β (πΈπΌπ»)) |
21 | 17 | simp2d 1141 | . . 3 β’ (π β ((π΄ β π΅) = (πΈ β πΉ) β§ (π΅ β πΆ) = (πΉ β π»))) |
22 | 21 | simpld 493 | . 2 β’ (π β (π΄ β π΅) = (πΈ β πΉ)) |
23 | 21 | simprd 494 | . 2 β’ (π β (π΅ β πΆ) = (πΉ β π»)) |
24 | 17 | simp3d 1142 | . . 3 β’ (π β ((π΄ β π·) = (πΈ β πΌ) β§ (π΅ β π·) = (πΉ β πΌ))) |
25 | 24 | simpld 493 | . 2 β’ (π β (π΄ β π·) = (πΈ β πΌ)) |
26 | 24 | simprd 494 | . 2 β’ (π β (π΅ β π·) = (πΉ β πΌ)) |
27 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 19, 20, 22, 23, 25, 26 | axtg5seg 27983 | 1 β’ (π β (πΆ β π·) = (π» β πΌ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β§ w3a 1085 = wceq 1539 β wcel 2104 β wne 2938 β¨cop 4633 class class class wbr 5147 βcfv 6542 (class class class)co 7411 Basecbs 17148 distcds 17210 TarskiGcstrkg 27945 Itvcitv 27951 AFScafs 33979 |
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-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 |
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-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 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-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6494 df-fun 6544 df-fv 6550 df-ov 7414 df-trkgcb 27968 df-trkg 27971 df-afs 33980 |
This theorem is referenced by: (None) |
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