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Mirrors > Home > MPE Home > Th. List > hpgtr | Structured version Visualization version GIF version |
Description: The half-plane relation is transitive. Theorem 9.13 of [Schwabhauser] p. 72. (Contributed by Thierry Arnoux, 4-Mar-2020.) |
Ref | Expression |
---|---|
hpgid.p | β’ π = (BaseβπΊ) |
hpgid.i | β’ πΌ = (ItvβπΊ) |
hpgid.l | β’ πΏ = (LineGβπΊ) |
hpgid.g | β’ (π β πΊ β TarskiG) |
hpgid.d | β’ (π β π· β ran πΏ) |
hpgid.a | β’ (π β π΄ β π) |
hpgid.o | β’ π = {β¨π, πβ© β£ ((π β (π β π·) β§ π β (π β π·)) β§ βπ‘ β π· π‘ β (ππΌπ))} |
hpgcom.b | β’ (π β π΅ β π) |
hpgcom.1 | β’ (π β π΄((hpGβπΊ)βπ·)π΅) |
hpgtr.c | β’ (π β πΆ β π) |
hpgtr.1 | β’ (π β π΅((hpGβπΊ)βπ·)πΆ) |
Ref | Expression |
---|---|
hpgtr | β’ (π β π΄((hpGβπΊ)βπ·)πΆ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hpgcom.1 | . . . 4 β’ (π β π΄((hpGβπΊ)βπ·)π΅) | |
2 | hpgid.p | . . . . 5 β’ π = (BaseβπΊ) | |
3 | hpgid.i | . . . . 5 β’ πΌ = (ItvβπΊ) | |
4 | hpgid.l | . . . . 5 β’ πΏ = (LineGβπΊ) | |
5 | hpgid.o | . . . . 5 β’ π = {β¨π, πβ© β£ ((π β (π β π·) β§ π β (π β π·)) β§ βπ‘ β π· π‘ β (ππΌπ))} | |
6 | hpgid.g | . . . . 5 β’ (π β πΊ β TarskiG) | |
7 | hpgid.d | . . . . 5 β’ (π β π· β ran πΏ) | |
8 | hpgid.a | . . . . 5 β’ (π β π΄ β π) | |
9 | hpgcom.b | . . . . 5 β’ (π β π΅ β π) | |
10 | 2, 3, 4, 5, 6, 7, 8, 9 | hpgbr 27799 | . . . 4 β’ (π β (π΄((hpGβπΊ)βπ·)π΅ β βπ β π (π΄ππ β§ π΅ππ))) |
11 | 1, 10 | mpbid 231 | . . 3 β’ (π β βπ β π (π΄ππ β§ π΅ππ)) |
12 | simprl 769 | . . . . . 6 β’ (((π β§ π β π) β§ (π΄ππ β§ π΅ππ)) β π΄ππ) | |
13 | hpgtr.1 | . . . . . . . 8 β’ (π β π΅((hpGβπΊ)βπ·)πΆ) | |
14 | 13 | ad2antrr 724 | . . . . . . 7 β’ (((π β§ π β π) β§ (π΄ππ β§ π΅ππ)) β π΅((hpGβπΊ)βπ·)πΆ) |
15 | 6 | ad2antrr 724 | . . . . . . . 8 β’ (((π β§ π β π) β§ (π΄ππ β§ π΅ππ)) β πΊ β TarskiG) |
16 | 7 | ad2antrr 724 | . . . . . . . 8 β’ (((π β§ π β π) β§ (π΄ππ β§ π΅ππ)) β π· β ran πΏ) |
17 | 9 | ad2antrr 724 | . . . . . . . 8 β’ (((π β§ π β π) β§ (π΄ππ β§ π΅ππ)) β π΅ β π) |
18 | hpgtr.c | . . . . . . . . 9 β’ (π β πΆ β π) | |
19 | 18 | ad2antrr 724 | . . . . . . . 8 β’ (((π β§ π β π) β§ (π΄ππ β§ π΅ππ)) β πΆ β π) |
20 | simplr 767 | . . . . . . . 8 β’ (((π β§ π β π) β§ (π΄ππ β§ π΅ππ)) β π β π) | |
21 | simprr 771 | . . . . . . . 8 β’ (((π β§ π β π) β§ (π΄ππ β§ π΅ππ)) β π΅ππ) | |
22 | 2, 3, 4, 5, 15, 16, 17, 19, 20, 21 | lnopp2hpgb 27802 | . . . . . . 7 β’ (((π β§ π β π) β§ (π΄ππ β§ π΅ππ)) β (πΆππ β π΅((hpGβπΊ)βπ·)πΆ)) |
23 | 14, 22 | mpbird 256 | . . . . . 6 β’ (((π β§ π β π) β§ (π΄ππ β§ π΅ππ)) β πΆππ) |
24 | 12, 23 | jca 512 | . . . . 5 β’ (((π β§ π β π) β§ (π΄ππ β§ π΅ππ)) β (π΄ππ β§ πΆππ)) |
25 | 24 | ex 413 | . . . 4 β’ ((π β§ π β π) β ((π΄ππ β§ π΅ππ) β (π΄ππ β§ πΆππ))) |
26 | 25 | reximdva 3167 | . . 3 β’ (π β (βπ β π (π΄ππ β§ π΅ππ) β βπ β π (π΄ππ β§ πΆππ))) |
27 | 11, 26 | mpd 15 | . 2 β’ (π β βπ β π (π΄ππ β§ πΆππ)) |
28 | 2, 3, 4, 5, 6, 7, 8, 18 | hpgbr 27799 | . 2 β’ (π β (π΄((hpGβπΊ)βπ·)πΆ β βπ β π (π΄ππ β§ πΆππ))) |
29 | 27, 28 | mpbird 256 | 1 β’ (π β π΄((hpGβπΊ)βπ·)πΆ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1541 β wcel 2106 βwrex 3069 β cdif 3925 class class class wbr 5125 {copab 5187 ran crn 5654 βcfv 6516 (class class class)co 7377 Basecbs 17109 TarskiGcstrkg 27466 Itvcitv 27472 LineGclng 27473 hpGchpg 27796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4886 df-int 4928 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-om 7823 df-1st 7941 df-2nd 7942 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-rdg 8376 df-1o 8432 df-oadd 8436 df-er 8670 df-map 8789 df-pm 8790 df-en 8906 df-dom 8907 df-sdom 8908 df-fin 8909 df-dju 9861 df-card 9899 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-nn 12178 df-2 12240 df-3 12241 df-n0 12438 df-xnn0 12510 df-z 12524 df-uz 12788 df-fz 13450 df-fzo 13593 df-hash 14256 df-word 14430 df-concat 14486 df-s1 14511 df-s2 14764 df-s3 14765 df-trkgc 27487 df-trkgb 27488 df-trkgcb 27489 df-trkgld 27491 df-trkg 27492 df-cgrg 27550 df-leg 27622 df-hlg 27640 df-mir 27692 df-rag 27733 df-perpg 27735 df-hpg 27797 |
This theorem is referenced by: trgcopy 27843 trgcopyeulem 27844 acopyeu 27873 |
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