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Mirrors > Home > MPE Home > Th. List > tgbtwnconnln2 | Structured version Visualization version GIF version |
Description: Derive colinearity from betweenness. (Contributed by Thierry Arnoux, 17-May-2019.) |
Ref | Expression |
---|---|
tgbtwnconn.p | ⊢ 𝑃 = (Base‘𝐺) |
tgbtwnconn.i | ⊢ 𝐼 = (Itv‘𝐺) |
tgbtwnconn.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
tgbtwnconn.a | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
tgbtwnconn.b | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
tgbtwnconn.c | ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
tgbtwnconn.d | ⊢ (𝜑 → 𝐷 ∈ 𝑃) |
tgbtwnconnln1.l | ⊢ 𝐿 = (LineG‘𝐺) |
tgbtwnconnln1.1 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
tgbtwnconnln1.2 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) |
tgbtwnconnln1.3 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) |
Ref | Expression |
---|---|
tgbtwnconnln2 | ⊢ (𝜑 → (𝐵 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgbtwnconn.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | tgbtwnconnln1.l | . . 3 ⊢ 𝐿 = (LineG‘𝐺) | |
3 | tgbtwnconn.i | . . 3 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tgbtwnconn.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐺 ∈ TarskiG) |
6 | tgbtwnconn.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
7 | 6 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ 𝑃) |
8 | tgbtwnconn.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
9 | 8 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐷 ∈ 𝑃) |
10 | tgbtwnconn.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
11 | 10 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐵 ∈ 𝑃) |
12 | simpr 477 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ (𝐵𝐼𝐷)) | |
13 | 1, 2, 3, 5, 7, 9, 11, 12 | btwncolg2 26034 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → (𝐵 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷)) |
14 | 4 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐺 ∈ TarskiG) |
15 | 6 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐶 ∈ 𝑃) |
16 | 8 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐷 ∈ 𝑃) |
17 | 10 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐵 ∈ 𝑃) |
18 | eqid 2772 | . . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
19 | simpr 477 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐷 ∈ (𝐵𝐼𝐶)) | |
20 | 1, 18, 3, 14, 17, 16, 15, 19 | tgbtwncom 25966 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐷 ∈ (𝐶𝐼𝐵)) |
21 | 1, 2, 3, 14, 15, 16, 17, 20 | btwncolg3 26035 | . 2 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → (𝐵 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷)) |
22 | tgbtwnconn.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
23 | tgbtwnconnln1.1 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
24 | tgbtwnconnln1.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) | |
25 | tgbtwnconnln1.3 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) | |
26 | 1, 3, 4, 22, 10, 6, 8, 23, 24, 25 | tgbtwnconn2 26054 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶))) |
27 | 13, 21, 26 | mpjaodan 941 | 1 ⊢ (𝜑 → (𝐵 ∈ (𝐶𝐿𝐷) ∨ 𝐶 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∨ wo 833 = wceq 1507 ∈ wcel 2048 ≠ wne 2961 ‘cfv 6182 (class class class)co 6970 Basecbs 16329 distcds 16420 TarskiGcstrkg 25908 Itvcitv 25914 LineGclng 25915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7494 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-oadd 7901 df-er 8081 df-pm 8201 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-dju 9116 df-card 9154 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-3 11497 df-n0 11701 df-xnn0 11773 df-z 11787 df-uz 12052 df-fz 12702 df-fzo 12843 df-hash 13499 df-word 13663 df-concat 13724 df-s1 13749 df-s2 14062 df-s3 14063 df-trkgc 25926 df-trkgb 25927 df-trkgcb 25928 df-trkg 25931 df-cgrg 25989 |
This theorem is referenced by: tglineeltr 26109 |
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