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Mirrors > Home > MPE Home > Th. List > tgbtwnconn2 | Structured version Visualization version GIF version |
Description: Another connectivity law for betweenness. Theorem 5.2 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) |
tgbtwnconn2.1 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
tgbtwnconn2.2 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) |
tgbtwnconn2.3 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) |
Ref | Expression |
---|---|
tgbtwnconn2 | ⊢ (𝜑 → (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgbtwnconn.p | . . . 4 ⊢ 𝑃 = (Base‘𝐺) | |
2 | eqid 2736 | . . . 4 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
3 | tgbtwnconn.i | . . . 4 ⊢ 𝐼 = (Itv‘𝐺) | |
4 | tgbtwnconn.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
5 | 4 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → 𝐺 ∈ TarskiG) |
6 | tgbtwnconn.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
7 | 6 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → 𝐴 ∈ 𝑃) |
8 | tgbtwnconn.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
9 | 8 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → 𝐵 ∈ 𝑃) |
10 | tgbtwnconn.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
11 | 10 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → 𝐶 ∈ 𝑃) |
12 | tgbtwnconn.d | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
13 | 12 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → 𝐷 ∈ 𝑃) |
14 | tgbtwnconn2.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) | |
15 | 14 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → 𝐵 ∈ (𝐴𝐼𝐶)) |
16 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → 𝐶 ∈ (𝐴𝐼𝐷)) | |
17 | 1, 2, 3, 5, 7, 9, 11, 13, 15, 16 | tgbtwnexch3 27334 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → 𝐶 ∈ (𝐵𝐼𝐷)) |
18 | 17 | orcd 871 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶))) |
19 | 4 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐺 ∈ TarskiG) |
20 | 6 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐴 ∈ 𝑃) |
21 | 8 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ 𝑃) |
22 | 12 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ 𝑃) |
23 | 10 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐶 ∈ 𝑃) |
24 | tgbtwnconn2.3 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) | |
25 | 24 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐵 ∈ (𝐴𝐼𝐷)) |
26 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ (𝐴𝐼𝐶)) | |
27 | 1, 2, 3, 19, 20, 21, 22, 23, 25, 26 | tgbtwnexch3 27334 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ (𝐵𝐼𝐶)) |
28 | 27 | olcd 872 | . 2 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶))) |
29 | tgbtwnconn2.1 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
30 | 1, 3, 4, 6, 8, 10, 12, 29, 14, 24 | tgbtwnconn1 27415 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶))) |
31 | 18, 28, 30 | mpjaodan 957 | 1 ⊢ (𝜑 → (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2942 ‘cfv 6494 (class class class)co 7354 Basecbs 17080 distcds 17139 TarskiGcstrkg 27267 Itvcitv 27273 |
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 2707 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 |
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 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-om 7800 df-1st 7918 df-2nd 7919 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-1o 8409 df-oadd 8413 df-er 8645 df-pm 8765 df-en 8881 df-dom 8882 df-sdom 8883 df-fin 8884 df-dju 9834 df-card 9872 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-nn 12151 df-2 12213 df-3 12214 df-n0 12411 df-xnn0 12483 df-z 12497 df-uz 12761 df-fz 13422 df-fzo 13565 df-hash 14228 df-word 14400 df-concat 14456 df-s1 14481 df-s2 14734 df-s3 14735 df-trkgc 27288 df-trkgb 27289 df-trkgcb 27290 df-trkg 27293 df-cgrg 27351 |
This theorem is referenced by: tgbtwnconn3 27417 tgbtwnconn22 27419 tgbtwnconnln2 27421 legtrid 27431 hlcgrex 27456 mirbtwnhl 27520 mirhl2 27521 krippenlem 27530 lnopp2hpgb 27603 flatcgra 27664 |
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