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Mirrors > Home > MPE Home > Th. List > fusgreg2wsp | Structured version Visualization version GIF version |
Description: In a finite simple graph, the set of all paths of length 2 is the union of all the paths of length 2 over the vertices which are in the middle of such a path. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 18-May-2021.) (Proof shortened by AV, 10-Jan-2022.) |
Ref | Expression |
---|---|
frgrhash2wsp.v | ⢠ð = (Vtxâðº) |
fusgreg2wsp.m | ⢠ð = (ð â ð ⊠{ð€ â (2 WSPathsN ðº) ⣠(ð€â1) = ð}) |
Ref | Expression |
---|---|
fusgreg2wsp | ⢠(ðº â FinUSGraph â (2 WSPathsN ðº) = ⪠ð¥ â ð (ðâð¥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wspthsswwlkn 29768 | . . . . . . . 8 ⢠(2 WSPathsN ðº) â (2 WWalksN ðº) | |
2 | 1 | sseli 3969 | . . . . . . 7 ⢠(ð â (2 WSPathsN ðº) â ð â (2 WWalksN ðº)) |
3 | frgrhash2wsp.v | . . . . . . . 8 ⢠ð = (Vtxâðº) | |
4 | 3 | midwwlks2s3 29802 | . . . . . . 7 ⢠(ð â (2 WWalksN ðº) â âð¥ â ð (ðâ1) = ð¥) |
5 | 2, 4 | syl 17 | . . . . . 6 ⢠(ð â (2 WSPathsN ðº) â âð¥ â ð (ðâ1) = ð¥) |
6 | 5 | a1i 11 | . . . . 5 ⢠(ðº â FinUSGraph â (ð â (2 WSPathsN ðº) â âð¥ â ð (ðâ1) = ð¥)) |
7 | 6 | pm4.71rd 561 | . . . 4 ⢠(ðº â FinUSGraph â (ð â (2 WSPathsN ðº) â (âð¥ â ð (ðâ1) = ð¥ â§ ð â (2 WSPathsN ðº)))) |
8 | ancom 459 | . . . . . . 7 ⢠((ð â (2 WSPathsN ðº) â§ (ðâ1) = ð¥) â ((ðâ1) = ð¥ â§ ð â (2 WSPathsN ðº))) | |
9 | 8 | rexbii 3084 | . . . . . 6 ⢠(âð¥ â ð (ð â (2 WSPathsN ðº) â§ (ðâ1) = ð¥) â âð¥ â ð ((ðâ1) = ð¥ â§ ð â (2 WSPathsN ðº))) |
10 | r19.41v 3179 | . . . . . 6 ⢠(âð¥ â ð ((ðâ1) = ð¥ â§ ð â (2 WSPathsN ðº)) â (âð¥ â ð (ðâ1) = ð¥ â§ ð â (2 WSPathsN ðº))) | |
11 | 9, 10 | bitr2i 275 | . . . . 5 ⢠((âð¥ â ð (ðâ1) = ð¥ â§ ð â (2 WSPathsN ðº)) â âð¥ â ð (ð â (2 WSPathsN ðº) â§ (ðâ1) = ð¥)) |
12 | 11 | a1i 11 | . . . 4 ⢠(ðº â FinUSGraph â ((âð¥ â ð (ðâ1) = ð¥ â§ ð â (2 WSPathsN ðº)) â âð¥ â ð (ð â (2 WSPathsN ðº) â§ (ðâ1) = ð¥))) |
13 | fusgreg2wsp.m | . . . . . . . 8 ⢠ð = (ð â ð ⊠{ð€ â (2 WSPathsN ðº) ⣠(ð€â1) = ð}) | |
14 | 3, 13 | fusgreg2wsplem 30182 | . . . . . . 7 ⢠(ð¥ â ð â (ð â (ðâð¥) â (ð â (2 WSPathsN ðº) â§ (ðâ1) = ð¥))) |
15 | 14 | bicomd 222 | . . . . . 6 ⢠(ð¥ â ð â ((ð â (2 WSPathsN ðº) â§ (ðâ1) = ð¥) â ð â (ðâð¥))) |
16 | 15 | adantl 480 | . . . . 5 ⢠((ðº â FinUSGraph â§ ð¥ â ð) â ((ð â (2 WSPathsN ðº) â§ (ðâ1) = ð¥) â ð â (ðâð¥))) |
17 | 16 | rexbidva 3167 | . . . 4 ⢠(ðº â FinUSGraph â (âð¥ â ð (ð â (2 WSPathsN ðº) â§ (ðâ1) = ð¥) â âð¥ â ð ð â (ðâð¥))) |
18 | 7, 12, 17 | 3bitrd 304 | . . 3 ⢠(ðº â FinUSGraph â (ð â (2 WSPathsN ðº) â âð¥ â ð ð â (ðâð¥))) |
19 | eliun 4996 | . . 3 ⢠(ð â ⪠ð¥ â ð (ðâð¥) â âð¥ â ð ð â (ðâð¥)) | |
20 | 18, 19 | bitr4di 288 | . 2 ⢠(ðº â FinUSGraph â (ð â (2 WSPathsN ðº) â ð â ⪠ð¥ â ð (ðâð¥))) |
21 | 20 | eqrdv 2723 | 1 ⢠(ðº â FinUSGraph â (2 WSPathsN ðº) = ⪠ð¥ â ð (ðâð¥)) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 â wb 205 â§ wa 394 = wceq 1533 â wcel 2098 âwrex 3060 {crab 3419 ⪠ciun 4992 ⊠cmpt 5227 âcfv 6543 (class class class)co 7413 1c1 11134 2c2 12292 Vtxcvtx 28848 FinUSGraphcfusgr 29168 WWalksN cwwlksn 29676 WSPathsN cwwspthsn 29678 |
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-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 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-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-wwlks 29680 df-wwlksn 29681 df-wspthsn 29683 |
This theorem is referenced by: fusgreghash2wsp 30187 |
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