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| 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 | ⊢ 𝑃 = (Base‘𝐺) | 
| hpgid.i | ⊢ 𝐼 = (Itv‘𝐺) | 
| hpgid.l | ⊢ 𝐿 = (LineG‘𝐺) | 
| hpgid.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) | 
| hpgid.d | ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) | 
| hpgid.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) | 
| hpgid.o | ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} | 
| hpgcom.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) | 
| hpgcom.1 | ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐵) | 
| hpgtr.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) | 
| hpgtr.1 | ⊢ (𝜑 → 𝐵((hpG‘𝐺)‘𝐷)𝐶) | 
| Ref | Expression | 
|---|---|
| hpgtr | ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | hpgcom.1 | . . . 4 ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐵) | |
| 2 | hpgid.p | . . . . 5 ⊢ 𝑃 = (Base‘𝐺) | |
| 3 | hpgid.i | . . . . 5 ⊢ 𝐼 = (Itv‘𝐺) | |
| 4 | hpgid.l | . . . . 5 ⊢ 𝐿 = (LineG‘𝐺) | |
| 5 | hpgid.o | . . . . 5 ⊢ 𝑂 = {〈𝑎, 𝑏〉 ∣ ((𝑎 ∈ (𝑃 ∖ 𝐷) ∧ 𝑏 ∈ (𝑃 ∖ 𝐷)) ∧ ∃𝑡 ∈ 𝐷 𝑡 ∈ (𝑎𝐼𝑏))} | |
| 6 | hpgid.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
| 7 | hpgid.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ ran 𝐿) | |
| 8 | hpgid.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
| 9 | hpgcom.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
| 10 | 2, 3, 4, 5, 6, 7, 8, 9 | hpgbr 28769 | . . . 4 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐵 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐))) | 
| 11 | 1, 10 | mpbid 232 | . . 3 ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) | 
| 12 | simprl 770 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐴𝑂𝑐) | |
| 13 | hpgtr.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐵((hpG‘𝐺)‘𝐷)𝐶) | |
| 14 | 13 | ad2antrr 726 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐵((hpG‘𝐺)‘𝐷)𝐶) | 
| 15 | 6 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐺 ∈ TarskiG) | 
| 16 | 7 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐷 ∈ ran 𝐿) | 
| 17 | 9 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐵 ∈ 𝑃) | 
| 18 | hpgtr.c | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
| 19 | 18 | ad2antrr 726 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐶 ∈ 𝑃) | 
| 20 | simplr 768 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝑐 ∈ 𝑃) | |
| 21 | simprr 772 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐵𝑂𝑐) | |
| 22 | 2, 3, 4, 5, 15, 16, 17, 19, 20, 21 | lnopp2hpgb 28772 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → (𝐶𝑂𝑐 ↔ 𝐵((hpG‘𝐺)‘𝐷)𝐶)) | 
| 23 | 14, 22 | mpbird 257 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → 𝐶𝑂𝑐) | 
| 24 | 12, 23 | jca 511 | . . . . 5 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐)) → (𝐴𝑂𝑐 ∧ 𝐶𝑂𝑐)) | 
| 25 | 24 | ex 412 | . . . 4 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → ((𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐) → (𝐴𝑂𝑐 ∧ 𝐶𝑂𝑐))) | 
| 26 | 25 | reximdva 3167 | . . 3 ⊢ (𝜑 → (∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐵𝑂𝑐) → ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐶𝑂𝑐))) | 
| 27 | 11, 26 | mpd 15 | . 2 ⊢ (𝜑 → ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐶𝑂𝑐)) | 
| 28 | 2, 3, 4, 5, 6, 7, 8, 18 | hpgbr 28769 | . 2 ⊢ (𝜑 → (𝐴((hpG‘𝐺)‘𝐷)𝐶 ↔ ∃𝑐 ∈ 𝑃 (𝐴𝑂𝑐 ∧ 𝐶𝑂𝑐))) | 
| 29 | 27, 28 | mpbird 257 | 1 ⊢ (𝜑 → 𝐴((hpG‘𝐺)‘𝐷)𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∃wrex 3069 ∖ cdif 3947 class class class wbr 5142 {copab 5204 ran crn 5685 ‘cfv 6560 (class class class)co 7432 Basecbs 17248 TarskiGcstrkg 28436 Itvcitv 28442 LineGclng 28443 hpGchpg 28766 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-oadd 8511 df-er 8746 df-map 8869 df-pm 8870 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-dju 9942 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-xnn0 12602 df-z 12616 df-uz 12880 df-fz 13549 df-fzo 13696 df-hash 14371 df-word 14554 df-concat 14610 df-s1 14635 df-s2 14888 df-s3 14889 df-trkgc 28457 df-trkgb 28458 df-trkgcb 28459 df-trkgld 28461 df-trkg 28462 df-cgrg 28520 df-leg 28592 df-hlg 28610 df-mir 28662 df-rag 28703 df-perpg 28705 df-hpg 28767 | 
| This theorem is referenced by: trgcopy 28813 trgcopyeulem 28814 acopyeu 28843 | 
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