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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 29161 | . . . . . . . 8 ⢠(2 WSPathsN ðº) â (2 WWalksN ðº) | |
2 | 1 | sseli 3977 | . . . . . . 7 ⢠(ð â (2 WSPathsN ðº) â ð â (2 WWalksN ðº)) |
3 | frgrhash2wsp.v | . . . . . . . 8 ⢠ð = (Vtxâðº) | |
4 | 3 | midwwlks2s3 29195 | . . . . . . 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 563 | . . . 4 ⢠(ðº â FinUSGraph â (ð â (2 WSPathsN ðº) â (âð¥ â ð (ðâ1) = ð¥ ⧠ð â (2 WSPathsN ðº)))) |
8 | ancom 461 | . . . . . . 7 ⢠((ð â (2 WSPathsN ðº) ⧠(ðâ1) = ð¥) â ((ðâ1) = ð¥ ⧠ð â (2 WSPathsN ðº))) | |
9 | 8 | rexbii 3094 | . . . . . 6 ⢠(âð¥ â ð (ð â (2 WSPathsN ðº) ⧠(ðâ1) = ð¥) â âð¥ â ð ((ðâ1) = ð¥ ⧠ð â (2 WSPathsN ðº))) |
10 | r19.41v 3188 | . . . . . 6 ⢠(âð¥ â ð ((ðâ1) = ð¥ ⧠ð â (2 WSPathsN ðº)) â (âð¥ â ð (ðâ1) = ð¥ ⧠ð â (2 WSPathsN ðº))) | |
11 | 9, 10 | bitr2i 275 | . . . . 5 ⢠((âð¥ â ð (ðâ1) = ð¥ ⧠ð â (2 WSPathsN ðº)) â âð¥ â ð (ð â (2 WSPathsN ðº) ⧠(ðâ1) = ð¥)) |
12 | 11 | a1i 11 | . . . 4 ⢠(ðº â FinUSGraph â ((âð¥ â ð (ðâ1) = ð¥ ⧠ð â (2 WSPathsN ðº)) â âð¥ â ð (ð â (2 WSPathsN ðº) ⧠(ðâ1) = ð¥))) |
13 | fusgreg2wsp.m | . . . . . . . 8 ⢠ð = (ð â ð ⊠{ð€ â (2 WSPathsN ðº) ⣠(ð€â1) = ð}) | |
14 | 3, 13 | fusgreg2wsplem 29575 | . . . . . . 7 ⢠(ð¥ â ð â (ð â (ðâð¥) â (ð â (2 WSPathsN ðº) ⧠(ðâ1) = ð¥))) |
15 | 14 | bicomd 222 | . . . . . 6 ⢠(ð¥ â ð â ((ð â (2 WSPathsN ðº) ⧠(ðâ1) = ð¥) â ð â (ðâð¥))) |
16 | 15 | adantl 482 | . . . . 5 ⢠((ðº â FinUSGraph ⧠ð¥ â ð) â ((ð â (2 WSPathsN ðº) ⧠(ðâ1) = ð¥) â ð â (ðâð¥))) |
17 | 16 | rexbidva 3176 | . . . 4 ⢠(ðº â FinUSGraph â (âð¥ â ð (ð â (2 WSPathsN ðº) ⧠(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 5000 | . . 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 2730 | 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 ⧠wa 396 = wceq 1541 â wcel 2106 âwrex 3070 {crab 3432 ⪠ciun 4996 ⊠cmpt 5230 âcfv 6540 (class class class)co 7405 1c1 11107 2c2 12263 Vtxcvtx 28245 FinUSGraphcfusgr 28562 WWalksN cwwlksn 29069 WSPathsN cwwspthsn 29071 |
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 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-concat 14517 df-s1 14542 df-s2 14795 df-s3 14796 df-wwlks 29073 df-wwlksn 29074 df-wspthsn 29076 |
This theorem is referenced by: fusgreghash2wsp 29580 |
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