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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 | β’ π = (BaseβπΊ) |
tgbtwnconn.i | β’ πΌ = (ItvβπΊ) |
tgbtwnconn.g | β’ (π β πΊ β TarskiG) |
tgbtwnconn.a | β’ (π β π΄ β π) |
tgbtwnconn.b | β’ (π β π΅ β π) |
tgbtwnconn.c | β’ (π β πΆ β π) |
tgbtwnconn.d | β’ (π β π· β π) |
tgbtwnconn22.e | β’ (π β πΈ β π) |
tgbtwnconn22.1 | β’ (π β π΄ β π΅) |
tgbtwnconn22.2 | β’ (π β πΆ β π΅) |
tgbtwnconn22.3 | β’ (π β π΅ β (π΄πΌπΆ)) |
tgbtwnconn22.4 | β’ (π β π΅ β (π΄πΌπ·)) |
tgbtwnconn22.5 | β’ (π β π΅ β (πΆπΌπΈ)) |
Ref | Expression |
---|---|
tgbtwnconn22 | β’ (π β π΅ β (π·πΌπΈ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgbtwnconn.p | . . 3 β’ π = (BaseβπΊ) | |
2 | eqid 2725 | . . 3 β’ (distβπΊ) = (distβπΊ) | |
3 | tgbtwnconn.i | . . 3 β’ πΌ = (ItvβπΊ) | |
4 | tgbtwnconn.g | . . . 4 β’ (π β πΊ β TarskiG) | |
5 | 4 | adantr 479 | . . 3 β’ ((π β§ πΆ β (π΅πΌπ·)) β πΊ β TarskiG) |
6 | tgbtwnconn.d | . . . 4 β’ (π β π· β π) | |
7 | 6 | adantr 479 | . . 3 β’ ((π β§ πΆ β (π΅πΌπ·)) β π· β π) |
8 | tgbtwnconn.c | . . . 4 β’ (π β πΆ β π) | |
9 | 8 | adantr 479 | . . 3 β’ ((π β§ πΆ β (π΅πΌπ·)) β πΆ β π) |
10 | tgbtwnconn.b | . . . 4 β’ (π β π΅ β π) | |
11 | 10 | adantr 479 | . . 3 β’ ((π β§ πΆ β (π΅πΌπ·)) β π΅ β π) |
12 | tgbtwnconn22.e | . . . 4 β’ (π β πΈ β π) | |
13 | 12 | adantr 479 | . . 3 β’ ((π β§ πΆ β (π΅πΌπ·)) β πΈ β π) |
14 | tgbtwnconn22.2 | . . . 4 β’ (π β πΆ β π΅) | |
15 | 14 | adantr 479 | . . 3 β’ ((π β§ πΆ β (π΅πΌπ·)) β πΆ β π΅) |
16 | simpr 483 | . . . 4 β’ ((π β§ πΆ β (π΅πΌπ·)) β πΆ β (π΅πΌπ·)) | |
17 | 1, 2, 3, 5, 11, 9, 7, 16 | tgbtwncom 28331 | . . 3 β’ ((π β§ πΆ β (π΅πΌπ·)) β πΆ β (π·πΌπ΅)) |
18 | tgbtwnconn22.5 | . . . 4 β’ (π β π΅ β (πΆπΌπΈ)) | |
19 | 18 | adantr 479 | . . 3 β’ ((π β§ πΆ β (π΅πΌπ·)) β π΅ β (πΆπΌπΈ)) |
20 | 1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19 | tgbtwnouttr2 28338 | . 2 β’ ((π β§ πΆ β (π΅πΌπ·)) β π΅ β (π·πΌπΈ)) |
21 | 4 | adantr 479 | . . 3 β’ ((π β§ π· β (π΅πΌπΆ)) β πΊ β TarskiG) |
22 | 6 | adantr 479 | . . 3 β’ ((π β§ π· β (π΅πΌπΆ)) β π· β π) |
23 | 10 | adantr 479 | . . 3 β’ ((π β§ π· β (π΅πΌπΆ)) β π΅ β π) |
24 | 12 | adantr 479 | . . 3 β’ ((π β§ π· β (π΅πΌπΆ)) β πΈ β π) |
25 | 8 | adantr 479 | . . 3 β’ ((π β§ π· β (π΅πΌπΆ)) β πΆ β π) |
26 | simpr 483 | . . 3 β’ ((π β§ π· β (π΅πΌπΆ)) β π· β (π΅πΌπΆ)) | |
27 | 18 | adantr 479 | . . . 4 β’ ((π β§ π· β (π΅πΌπΆ)) β π΅ β (πΆπΌπΈ)) |
28 | 1, 2, 3, 21, 25, 23, 24, 27 | tgbtwncom 28331 | . . 3 β’ ((π β§ π· β (π΅πΌπΆ)) β π΅ β (πΈπΌπΆ)) |
29 | 1, 2, 3, 21, 22, 23, 24, 25, 26, 28 | tgbtwnintr 28336 | . 2 β’ ((π β§ π· β (π΅πΌπΆ)) β π΅ β (π·πΌπΈ)) |
30 | tgbtwnconn.a | . . 3 β’ (π β π΄ β π) | |
31 | tgbtwnconn22.1 | . . 3 β’ (π β π΄ β π΅) | |
32 | tgbtwnconn22.3 | . . 3 β’ (π β π΅ β (π΄πΌπΆ)) | |
33 | tgbtwnconn22.4 | . . 3 β’ (π β π΅ β (π΄πΌπ·)) | |
34 | 1, 3, 4, 30, 10, 8, 6, 31, 32, 33 | tgbtwnconn2 28419 | . 2 β’ (π β (πΆ β (π΅πΌπ·) β¨ π· β (π΅πΌπΆ))) |
35 | 20, 29, 34 | mpjaodan 956 | 1 β’ (π β π΅ β (π·πΌπΈ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2930 βcfv 6543 (class class class)co 7413 Basecbs 17174 distcds 17236 TarskiGcstrkg 28270 Itvcitv 28276 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3961 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8718 df-pm 8841 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-dju 9919 df-card 9957 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-n0 12498 df-xnn0 12570 df-z 12584 df-uz 12848 df-fz 13512 df-fzo 13655 df-hash 14317 df-word 14492 df-concat 14548 df-s1 14573 df-s2 14826 df-s3 14827 df-trkgc 28291 df-trkgb 28292 df-trkgcb 28293 df-trkg 28296 df-cgrg 28354 |
This theorem is referenced by: mideulem2 28577 flatcgra 28667 |
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