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Mirrors > Home > MPE Home > Th. List > tgbtwnconn22 | Structured version Visualization version GIF version |
Description: Double connectivity law for betweenness. (Contributed by Thierry Arnoux, 1-Dec-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) |
tgbtwnconn22.e | ⊢ (𝜑 → 𝐸 ∈ 𝑃) |
tgbtwnconn22.1 | ⊢ (𝜑 → 𝐴 ≠ 𝐵) |
tgbtwnconn22.2 | ⊢ (𝜑 → 𝐶 ≠ 𝐵) |
tgbtwnconn22.3 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) |
tgbtwnconn22.4 | ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) |
tgbtwnconn22.5 | ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐸)) |
Ref | Expression |
---|---|
tgbtwnconn22 | ⊢ (𝜑 → 𝐵 ∈ (𝐷𝐼𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgbtwnconn.p | . . 3 ⊢ 𝑃 = (Base‘𝐺) | |
2 | eqid 2772 | . . 3 ⊢ (dist‘𝐺) = (dist‘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.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑃) | |
7 | 6 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐷 ∈ 𝑃) |
8 | tgbtwnconn.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑃) | |
9 | 8 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ 𝑃) |
10 | tgbtwnconn.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
11 | 10 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐵 ∈ 𝑃) |
12 | tgbtwnconn22.e | . . . 4 ⊢ (𝜑 → 𝐸 ∈ 𝑃) | |
13 | 12 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐸 ∈ 𝑃) |
14 | tgbtwnconn22.2 | . . . 4 ⊢ (𝜑 → 𝐶 ≠ 𝐵) | |
15 | 14 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ≠ 𝐵) |
16 | simpr 477 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ (𝐵𝐼𝐷)) | |
17 | 1, 2, 3, 5, 11, 9, 7, 16 | tgbtwncom 25988 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐶 ∈ (𝐷𝐼𝐵)) |
18 | tgbtwnconn22.5 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝐶𝐼𝐸)) | |
19 | 18 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐵 ∈ (𝐶𝐼𝐸)) |
20 | 1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19 | tgbtwnouttr2 25995 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ (𝐵𝐼𝐷)) → 𝐵 ∈ (𝐷𝐼𝐸)) |
21 | 4 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐺 ∈ TarskiG) |
22 | 6 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐷 ∈ 𝑃) |
23 | 10 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐵 ∈ 𝑃) |
24 | 12 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐸 ∈ 𝑃) |
25 | 8 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐶 ∈ 𝑃) |
26 | simpr 477 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐷 ∈ (𝐵𝐼𝐶)) | |
27 | 18 | adantr 473 | . . . 4 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐵 ∈ (𝐶𝐼𝐸)) |
28 | 1, 2, 3, 21, 25, 23, 24, 27 | tgbtwncom 25988 | . . 3 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐵 ∈ (𝐸𝐼𝐶)) |
29 | 1, 2, 3, 21, 22, 23, 24, 25, 26, 28 | tgbtwnintr 25993 | . 2 ⊢ ((𝜑 ∧ 𝐷 ∈ (𝐵𝐼𝐶)) → 𝐵 ∈ (𝐷𝐼𝐸)) |
30 | tgbtwnconn.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
31 | tgbtwnconn22.1 | . . 3 ⊢ (𝜑 → 𝐴 ≠ 𝐵) | |
32 | tgbtwnconn22.3 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐶)) | |
33 | tgbtwnconn22.4 | . . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴𝐼𝐷)) | |
34 | 1, 3, 4, 30, 10, 8, 6, 31, 32, 33 | tgbtwnconn2 26076 | . 2 ⊢ (𝜑 → (𝐶 ∈ (𝐵𝐼𝐷) ∨ 𝐷 ∈ (𝐵𝐼𝐶))) |
35 | 20, 29, 34 | mpjaodan 941 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝐷𝐼𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ≠ wne 2961 ‘cfv 6185 (class class class)co 6974 Basecbs 16337 distcds 16428 TarskiGcstrkg 25930 Itvcitv 25936 |
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 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 |
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 2016 df-mo 2547 df-eu 2584 df-clab 2753 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 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-pm 8207 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-dju 9122 df-card 9160 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-nn 11438 df-2 11501 df-3 11502 df-n0 11706 df-xnn0 11778 df-z 11792 df-uz 12057 df-fz 12707 df-fzo 12848 df-hash 13504 df-word 13671 df-concat 13732 df-s1 13757 df-s2 14070 df-s3 14071 df-trkgc 25948 df-trkgb 25949 df-trkgcb 25950 df-trkg 25953 df-cgrg 26011 |
This theorem is referenced by: mideulem2 26234 flatcgra 26324 |
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