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Mirrors > Home > MPE Home > Th. List > frgr2wsp1 | Structured version Visualization version GIF version |
Description: In a friendship graph, there is exactly one simple path of length 2 between two different vertices. (Contributed by Alexander van der Vekens, 3-Mar-2018.) (Revised by AV, 13-May-2021.) |
Ref | Expression |
---|---|
frgr2wwlkeu.v | ⢠ð = (Vtxâðº) |
Ref | Expression |
---|---|
frgr2wsp1 | ⢠((ðº â FriendGraph â§ (ðŽ â ð â§ ðµ â ð) â§ ðŽ â ðµ) â (â¯â(ðŽ(2 WSPathsNOn ðº)ðµ)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frgrusgr 30115 | . . . . 5 ⢠(ðº â FriendGraph â ðº â USGraph) | |
2 | wpthswwlks2on 29816 | . . . . 5 ⢠((ðº â USGraph â§ ðŽ â ðµ) â (ðŽ(2 WSPathsNOn ðº)ðµ) = (ðŽ(2 WWalksNOn ðº)ðµ)) | |
3 | 1, 2 | sylan 578 | . . . 4 ⢠((ðº â FriendGraph â§ ðŽ â ðµ) â (ðŽ(2 WSPathsNOn ðº)ðµ) = (ðŽ(2 WWalksNOn ðº)ðµ)) |
4 | 3 | 3adant2 1128 | . . 3 ⢠((ðº â FriendGraph â§ (ðŽ â ð â§ ðµ â ð) â§ ðŽ â ðµ) â (ðŽ(2 WSPathsNOn ðº)ðµ) = (ðŽ(2 WWalksNOn ðº)ðµ)) |
5 | 4 | fveq2d 6896 | . 2 ⢠((ðº â FriendGraph â§ (ðŽ â ð â§ ðµ â ð) â§ ðŽ â ðµ) â (â¯â(ðŽ(2 WSPathsNOn ðº)ðµ)) = (â¯â(ðŽ(2 WWalksNOn ðº)ðµ))) |
6 | frgr2wwlkeu.v | . . 3 ⢠ð = (Vtxâðº) | |
7 | 6 | frgr2wwlk1 30183 | . 2 ⢠((ðº â FriendGraph â§ (ðŽ â ð â§ ðµ â ð) â§ ðŽ â ðµ) â (â¯â(ðŽ(2 WWalksNOn ðº)ðµ)) = 1) |
8 | 5, 7 | eqtrd 2765 | 1 ⢠((ðº â FriendGraph â§ (ðŽ â ð â§ ðµ â ð) â§ ðŽ â ðµ) â (â¯â(ðŽ(2 WSPathsNOn ðº)ðµ)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 â§ wa 394 â§ w3a 1084 = wceq 1533 â wcel 2098 â wne 2930 âcfv 6543 (class class class)co 7416 1c1 11139 2c2 12297 â¯chash 14321 Vtxcvtx 28853 USGraphcusgr 29006 WWalksNOn cwwlksnon 29682 WSPathsNOn cwwspthsnon 29684 FriendGraph cfrgr 30112 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-ac2 10486 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-ifp 1061 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-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-er 8723 df-map 8845 df-pm 8846 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-dju 9924 df-card 9962 df-ac 10139 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-xnn0 12575 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-hash 14322 df-word 14497 df-concat 14553 df-s1 14578 df-s2 14831 df-s3 14832 df-edg 28905 df-uhgr 28915 df-upgr 28939 df-umgr 28940 df-uspgr 29007 df-usgr 29008 df-wlks 29457 df-wlkson 29458 df-trls 29550 df-trlson 29551 df-pths 29574 df-spths 29575 df-pthson 29576 df-spthson 29577 df-wwlks 29685 df-wwlksn 29686 df-wwlksnon 29687 df-wspthsnon 29689 df-frgr 30113 |
This theorem is referenced by: frgrhash2wsp 30186 |
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