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Mirrors > Home > MPE Home > Th. List > Mathboxes > btwnconn2 | Structured version Visualization version GIF version |
Description: Another connectivity law for betweenness. Theorem 5.2 of [Schwabhauser] p. 41. (Contributed by Scott Fenton, 9-Oct-2013.) |
Ref | Expression |
---|---|
btwnconn2 | ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐴 ≠ 𝐵 ∧ 𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉) → (𝐶 Btwn 〈𝐵, 𝐷〉 ∨ 𝐷 Btwn 〈𝐵, 𝐶〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | btwnconn1 32506 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐴 ≠ 𝐵 ∧ 𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉) → (𝐶 Btwn 〈𝐴, 𝐷〉 ∨ 𝐷 Btwn 〈𝐴, 𝐶〉))) | |
2 | simpr2 1233 | . . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) → 𝐵 Btwn 〈𝐴, 𝐶〉) | |
3 | 2 | anim1i 593 | . . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) ∧ 𝐶 Btwn 〈𝐴, 𝐷〉) → (𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐶 Btwn 〈𝐴, 𝐷〉)) |
4 | btwnexch3 32425 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐶 Btwn 〈𝐴, 𝐷〉) → 𝐶 Btwn 〈𝐵, 𝐷〉)) | |
5 | 4 | ad2antrr 764 | . . . . . 6 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) ∧ 𝐶 Btwn 〈𝐴, 𝐷〉) → ((𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐶 Btwn 〈𝐴, 𝐷〉) → 𝐶 Btwn 〈𝐵, 𝐷〉)) |
6 | 3, 5 | mpd 15 | . . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) ∧ 𝐶 Btwn 〈𝐴, 𝐷〉) → 𝐶 Btwn 〈𝐵, 𝐷〉) |
7 | 6 | ex 449 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) → (𝐶 Btwn 〈𝐴, 𝐷〉 → 𝐶 Btwn 〈𝐵, 𝐷〉)) |
8 | simpr3 1235 | . . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) → 𝐵 Btwn 〈𝐴, 𝐷〉) | |
9 | simp3r 1242 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → 𝐷 ∈ (𝔼‘𝑁)) | |
10 | simp3l 1241 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → 𝐶 ∈ (𝔼‘𝑁)) | |
11 | 9, 10 | jca 555 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) |
12 | btwnexch3 32425 | . . . . . . 7 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn 〈𝐴, 𝐷〉 ∧ 𝐷 Btwn 〈𝐴, 𝐶〉) → 𝐷 Btwn 〈𝐵, 𝐶〉)) | |
13 | 11, 12 | syld3an3 1512 | . . . . . 6 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn 〈𝐴, 𝐷〉 ∧ 𝐷 Btwn 〈𝐴, 𝐶〉) → 𝐷 Btwn 〈𝐵, 𝐶〉)) |
14 | 13 | adantr 472 | . . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) → ((𝐵 Btwn 〈𝐴, 𝐷〉 ∧ 𝐷 Btwn 〈𝐴, 𝐶〉) → 𝐷 Btwn 〈𝐵, 𝐶〉)) |
15 | 8, 14 | mpand 713 | . . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) → (𝐷 Btwn 〈𝐴, 𝐶〉 → 𝐷 Btwn 〈𝐵, 𝐶〉)) |
16 | 7, 15 | orim12d 919 | . . 3 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) ∧ (𝐴 ≠ 𝐵 ∧ 𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉)) → ((𝐶 Btwn 〈𝐴, 𝐷〉 ∨ 𝐷 Btwn 〈𝐴, 𝐶〉) → (𝐶 Btwn 〈𝐵, 𝐷〉 ∨ 𝐷 Btwn 〈𝐵, 𝐶〉))) |
17 | 16 | ex 449 | . 2 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐴 ≠ 𝐵 ∧ 𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉) → ((𝐶 Btwn 〈𝐴, 𝐷〉 ∨ 𝐷 Btwn 〈𝐴, 𝐶〉) → (𝐶 Btwn 〈𝐵, 𝐷〉 ∨ 𝐷 Btwn 〈𝐵, 𝐶〉)))) |
18 | 1, 17 | mpdd 43 | 1 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐴 ≠ 𝐵 ∧ 𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐵 Btwn 〈𝐴, 𝐷〉) → (𝐶 Btwn 〈𝐵, 𝐷〉 ∨ 𝐷 Btwn 〈𝐵, 𝐶〉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 382 ∧ wa 383 ∧ w3a 1072 ∈ wcel 2131 ≠ wne 2924 〈cop 4319 class class class wbr 4796 ‘cfv 6041 ℕcn 11204 𝔼cee 25959 Btwn cbtwn 25960 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-rep 4915 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-inf2 8703 ax-cnex 10176 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 ax-pre-sup 10198 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-fal 1630 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rmo 3050 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-int 4620 df-iun 4666 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-se 5218 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-isom 6050 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-om 7223 df-1st 7325 df-2nd 7326 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-1o 7721 df-oadd 7725 df-er 7903 df-map 8017 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-sup 8505 df-oi 8572 df-card 8947 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-div 10869 df-nn 11205 df-2 11263 df-3 11264 df-n0 11477 df-z 11562 df-uz 11872 df-rp 12018 df-ico 12366 df-icc 12367 df-fz 12512 df-fzo 12652 df-seq 12988 df-exp 13047 df-hash 13304 df-cj 14030 df-re 14031 df-im 14032 df-sqrt 14166 df-abs 14167 df-clim 14410 df-sum 14608 df-ee 25962 df-btwn 25963 df-cgr 25964 df-ofs 32388 df-colinear 32444 df-ifs 32445 df-cgr3 32446 df-fs 32447 |
This theorem is referenced by: btwnconn3 32508 segcon2 32510 btwnoutside 32530 broutsideof3 32531 lineunray 32552 lineelsb2 32553 |
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