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Mirrors > Home > MPE Home > Th. List > tgbtwnconn3 | Structured version Visualization version GIF version |
Description: Inner connectivity law for betweenness. Theorem 5.3 of [Schwabhauser] p. 41. (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) |
tgbtwnconn3.1 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) |
tgbtwnconn3.2 | ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷)) |
Ref | Expression |
---|---|
tgbtwnconn3 | ⊢ (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐴𝐼𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgbtwnconn.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
2 | eqid 2778 | . . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
3 | tgbtwnconn.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tgbtwnconn.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ (♯‘𝑃) = 1) → 𝐺 ∈ TarskiG) |
6 | tgbtwnconn.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
7 | 6 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ (♯‘𝑃) = 1) → 𝐵 ∈ 𝑃) |
8 | tgbtwnconn.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
9 | 8 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ (♯‘𝑃) = 1) → 𝐴 ∈ 𝑃) |
10 | tgbtwnconn.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
11 | 10 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ (♯‘𝑃) = 1) → 𝐶 ∈ 𝑃) |
12 | simpr 479 | . . . 4 ⊢ ((𝜑 ∧ (♯‘𝑃) = 1) → (♯‘𝑃) = 1) | |
13 | 1, 2, 3, 5, 7, 9, 11, 12 | tgldim0itv 25855 | . . 3 ⊢ ((𝜑 ∧ (♯‘𝑃) = 1) → 𝐵 ∈ (𝐴𝐼𝐶)) |
14 | 13 | orcd 862 | . 2 ⊢ ((𝜑 ∧ (♯‘𝑃) = 1) → (𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐴𝐼𝐵))) |
15 | 4 | ad3antrrr 720 | . . . 4 ⊢ ((((𝜑 ∧ 2 ≤ (♯‘𝑃)) ∧ 𝑝 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑝) ∧ 𝐴 ≠ 𝑝)) → 𝐺 ∈ TarskiG) |
16 | simplr 759 | . . . 4 ⊢ ((((𝜑 ∧ 2 ≤ (♯‘𝑃)) ∧ 𝑝 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑝) ∧ 𝐴 ≠ 𝑝)) → 𝑝 ∈ 𝑃) | |
17 | 8 | ad3antrrr 720 | . . . 4 ⊢ ((((𝜑 ∧ 2 ≤ (♯‘𝑃)) ∧ 𝑝 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑝) ∧ 𝐴 ≠ 𝑝)) → 𝐴 ∈ 𝑃) |
18 | 6 | ad3antrrr 720 | . . . 4 ⊢ ((((𝜑 ∧ 2 ≤ (♯‘𝑃)) ∧ 𝑝 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑝) ∧ 𝐴 ≠ 𝑝)) → 𝐵 ∈ 𝑃) |
19 | 10 | ad3antrrr 720 | . . . 4 ⊢ ((((𝜑 ∧ 2 ≤ (♯‘𝑃)) ∧ 𝑝 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑝) ∧ 𝐴 ≠ 𝑝)) → 𝐶 ∈ 𝑃) |
20 | simprr 763 | . . . . 5 ⊢ ((((𝜑 ∧ 2 ≤ (♯‘𝑃)) ∧ 𝑝 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑝) ∧ 𝐴 ≠ 𝑝)) → 𝐴 ≠ 𝑝) | |
21 | 20 | necomd 3024 | . . . 4 ⊢ ((((𝜑 ∧ 2 ≤ (♯‘𝑃)) ∧ 𝑝 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑝) ∧ 𝐴 ≠ 𝑝)) → 𝑝 ≠ 𝐴) |
22 | tgbtwnconn.d | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
23 | 22 | ad3antrrr 720 | . . . . . 6 ⊢ ((((𝜑 ∧ 2 ≤ (♯‘𝑃)) ∧ 𝑝 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑝) ∧ 𝐴 ≠ 𝑝)) → 𝐷 ∈ 𝑃) |
24 | tgbtwnconn3.1 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) | |
25 | 24 | ad3antrrr 720 | . . . . . 6 ⊢ ((((𝜑 ∧ 2 ≤ (♯‘𝑃)) ∧ 𝑝 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑝) ∧ 𝐴 ≠ 𝑝)) → 𝐵 ∈ (𝐴𝐼𝐷)) |
26 | simprl 761 | . . . . . . 7 ⊢ ((((𝜑 ∧ 2 ≤ (♯‘𝑃)) ∧ 𝑝 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑝) ∧ 𝐴 ≠ 𝑝)) → 𝐴 ∈ (𝐷𝐼𝑝)) | |
27 | 1, 2, 3, 15, 23, 17, 16, 26 | tgbtwncom 25839 | . . . . . 6 ⊢ ((((𝜑 ∧ 2 ≤ (♯‘𝑃)) ∧ 𝑝 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑝) ∧ 𝐴 ≠ 𝑝)) → 𝐴 ∈ (𝑝𝐼𝐷)) |
28 | 1, 2, 3, 15, 18, 17, 16, 23, 25, 27 | tgbtwnintr 25844 | . . . . 5 ⊢ ((((𝜑 ∧ 2 ≤ (♯‘𝑃)) ∧ 𝑝 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑝) ∧ 𝐴 ≠ 𝑝)) → 𝐴 ∈ (𝐵𝐼𝑝)) |
29 | 1, 2, 3, 15, 18, 17, 16, 28 | tgbtwncom 25839 | . . . 4 ⊢ ((((𝜑 ∧ 2 ≤ (♯‘𝑃)) ∧ 𝑝 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑝) ∧ 𝐴 ≠ 𝑝)) → 𝐴 ∈ (𝑝𝐼𝐵)) |
30 | tgbtwnconn3.2 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ (𝐴𝐼𝐷)) | |
31 | 30 | ad3antrrr 720 | . . . . . . 7 ⊢ ((((𝜑 ∧ 2 ≤ (♯‘𝑃)) ∧ 𝑝 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑝) ∧ 𝐴 ≠ 𝑝)) → 𝐶 ∈ (𝐴𝐼𝐷)) |
32 | 1, 2, 3, 15, 17, 19, 23, 31 | tgbtwncom 25839 | . . . . . 6 ⊢ ((((𝜑 ∧ 2 ≤ (♯‘𝑃)) ∧ 𝑝 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑝) ∧ 𝐴 ≠ 𝑝)) → 𝐶 ∈ (𝐷𝐼𝐴)) |
33 | 1, 2, 3, 15, 23, 19, 17, 16, 32, 26 | tgbtwnexch3 25845 | . . . . 5 ⊢ ((((𝜑 ∧ 2 ≤ (♯‘𝑃)) ∧ 𝑝 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑝) ∧ 𝐴 ≠ 𝑝)) → 𝐴 ∈ (𝐶𝐼𝑝)) |
34 | 1, 2, 3, 15, 19, 17, 16, 33 | tgbtwncom 25839 | . . . 4 ⊢ ((((𝜑 ∧ 2 ≤ (♯‘𝑃)) ∧ 𝑝 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑝) ∧ 𝐴 ≠ 𝑝)) → 𝐴 ∈ (𝑝𝐼𝐶)) |
35 | 1, 3, 15, 16, 17, 18, 19, 21, 29, 34 | tgbtwnconn2 25927 | . . 3 ⊢ ((((𝜑 ∧ 2 ≤ (♯‘𝑃)) ∧ 𝑝 ∈ 𝑃) ∧ (𝐴 ∈ (𝐷𝐼𝑝) ∧ 𝐴 ≠ 𝑝)) → (𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐴𝐼𝐵))) |
36 | 4 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 2 ≤ (♯‘𝑃)) → 𝐺 ∈ TarskiG) |
37 | 22 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 2 ≤ (♯‘𝑃)) → 𝐷 ∈ 𝑃) |
38 | 8 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ 2 ≤ (♯‘𝑃)) → 𝐴 ∈ 𝑃) |
39 | simpr 479 | . . . 4 ⊢ ((𝜑 ∧ 2 ≤ (♯‘𝑃)) → 2 ≤ (♯‘𝑃)) | |
40 | 1, 2, 3, 36, 37, 38, 39 | tgbtwndiff 25857 | . . 3 ⊢ ((𝜑 ∧ 2 ≤ (♯‘𝑃)) → ∃𝑝 ∈ 𝑃 (𝐴 ∈ (𝐷𝐼𝑝) ∧ 𝐴 ≠ 𝑝)) |
41 | 35, 40 | r19.29a 3264 | . 2 ⊢ ((𝜑 ∧ 2 ≤ (♯‘𝑃)) → (𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐴𝐼𝐵))) |
42 | 1, 8 | tgldimor 25853 | . 2 ⊢ (𝜑 → ((♯‘𝑃) = 1 ∨ 2 ≤ (♯‘𝑃))) |
43 | 14, 41, 42 | mpjaodan 944 | 1 ⊢ (𝜑 → (𝐵 ∈ (𝐴𝐼𝐶) ∨ 𝐶 ∈ (𝐴𝐼𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∨ wo 836 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 1c1 10273 ≤ cle 10412 2c2 11430 ♯chash 13435 Basecbs 16255 distcds 16347 TarskiGcstrkg 25781 Itvcitv 25787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-pm 8143 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-card 9098 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-xnn0 11715 df-z 11729 df-uz 11993 df-fz 12644 df-fzo 12785 df-hash 13436 df-word 13600 df-concat 13661 df-s1 13686 df-s2 13999 df-s3 14000 df-trkgc 25799 df-trkgb 25800 df-trkgcb 25801 df-trkg 25804 df-cgrg 25862 |
This theorem is referenced by: tgbtwnconnln3 25929 hltr 25961 hlbtwn 25962 hlpasch 26104 |
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