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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 2622 | . . . 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 477 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → 𝐶 ∈ (𝐴𝐼𝐷)) | |
17 | 1, 2, 3, 5, 7, 9, 11, 13, 15, 16 | tgbtwnexch3 25389 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐴𝐼𝐷)) → 𝐶 ∈ (𝐵𝐼𝐷)) |
18 | 17 | orcd 407 | . 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 477 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ (𝐴𝐼𝐶)) | |
27 | 1, 2, 3, 19, 20, 21, 22, 23, 25, 26 | tgbtwnexch3 25389 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐴𝐼𝐶)) → 𝐷 ∈ (𝐵𝐼𝐶)) |
28 | 27 | olcd 408 | . 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 25470 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐴𝐼𝐷) ∨ 𝐷 ∈ (𝐴𝐼𝐶))) |
31 | 18, 28, 30 | mpjaodan 827 | 1 ⊢ (𝜑 → (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ wo 383 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 distcds 15950 TarskiGcstrkg 25329 Itvcitv 25335 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-s2 13593 df-s3 13594 df-trkgc 25347 df-trkgb 25348 df-trkgcb 25349 df-trkg 25352 df-cgrg 25406 |
This theorem is referenced by: tgbtwnconn3 25472 tgbtwnconn22 25474 tgbtwnconnln2 25476 legtrid 25486 hlcgrex 25511 mirbtwnhl 25575 mirhl2 25576 krippenlem 25585 lnopp2hpgb 25655 |
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