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Theorem usgreghash2spotv 25042
 Description: According to the proof of the third claim in the proof of the friendship theorem in [Huneke] p. 2: "For each vertex v, there are exactly ( k 2 ) paths with length two having v in the middle, ..." in a finite k-regular graph. For simple paths of length 2 represented by ordered triples, we have again k*(k-1) such paths. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
Hypothesis
Ref Expression
usgreghash2spot.m 2SPathOnOt
Assertion
Ref Expression
usgreghash2spotv USGrph VDeg
Distinct variable groups:   ,,   ,,   ,,,   ,,
Allowed substitution hints:   (,,)   (,,)

Proof of Theorem usgreghash2spotv
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgreghash2spot.m . . . . . . . . 9 2SPathOnOt
21usg2spot2nb 25041 . . . . . . . 8 USGrph Neighbors Neighbors
323expa 1197 . . . . . . 7 USGrph Neighbors Neighbors
43fveq2d 5860 . . . . . 6 USGrph Neighbors Neighbors
5 nbfiusgrafi 24425 . . . . . . . 8 USGrph Neighbors
653expa 1197 . . . . . . 7 USGrph Neighbors
7 diffi 7753 . . . . . . . . . . 11 Neighbors Neighbors
85, 7syl 16 . . . . . . . . . 10 USGrph Neighbors
983expa 1197 . . . . . . . . 9 USGrph Neighbors
109adantr 465 . . . . . . . 8 USGrph Neighbors Neighbors
11 snfi 7598 . . . . . . . . . 10
1211a1i 11 . . . . . . . . 9 USGrph Neighbors Neighbors
1312ralrimiva 2857 . . . . . . . 8 USGrph Neighbors Neighbors
14 iunfi 7810 . . . . . . . 8 Neighbors Neighbors Neighbors
1510, 13, 14syl2anc 661 . . . . . . 7 USGrph Neighbors Neighbors
16 nbgrassvt 24409 . . . . . . . . . . . 12 USGrph Neighbors
1716ad3antrrr 729 . . . . . . . . . . 11 USGrph Neighbors Neighbors
1817ssdifd 3625 . . . . . . . . . 10 USGrph Neighbors Neighbors
19 iunss1 4327 . . . . . . . . . 10 Neighbors Neighbors
2018, 19syl 16 . . . . . . . . 9 USGrph Neighbors Neighbors
2120ralrimiva 2857 . . . . . . . 8 USGrph Neighbors Neighbors
22 simpr 461 . . . . . . . . 9 USGrph
23 otiunsndisj 4743 . . . . . . . . 9 Disj Neighbors
2422, 23syl 16 . . . . . . . 8 USGrph Disj Neighbors
25 disjss2 4410 . . . . . . . 8 Neighbors Neighbors Disj Neighbors Disj Neighbors Neighbors
2621, 24, 25sylc 60 . . . . . . 7 USGrph Disj Neighbors Neighbors
276, 15, 26hashiun 13617 . . . . . 6 USGrph Neighbors Neighbors Neighbors Neighbors
284, 27eqtrd 2484 . . . . 5 USGrph Neighbors Neighbors
2928adantr 465 . . . 4 USGrph VDeg Neighbors Neighbors
309ad2antrr 725 . . . . . . 7 USGrph VDeg Neighbors Neighbors
3111a1i 11 . . . . . . 7 USGrph VDeg Neighbors Neighbors
32 nbgraisvtx 24407 . . . . . . . . . 10 USGrph Neighbors
3332ad3antrrr 729 . . . . . . . . 9 USGrph VDeg Neighbors
3433imp 429 . . . . . . . 8 USGrph VDeg Neighbors
3522ad2antrr 725 . . . . . . . 8 USGrph VDeg Neighbors
36 otsndisj 4742 . . . . . . . 8 Disj Neighbors
3734, 35, 36syl2anc 661 . . . . . . 7 USGrph VDeg Neighbors Disj Neighbors
3830, 31, 37hashiun 13617 . . . . . 6 USGrph VDeg Neighbors Neighbors Neighbors
39 otex 4702 . . . . . . . 8
40 hashsng 12419 . . . . . . . 8
4139, 40mp1i 12 . . . . . . 7 USGrph VDeg Neighbors Neighbors
4241sumeq2dv 13506 . . . . . 6 USGrph VDeg Neighbors Neighbors Neighbors
43 ax-1cn 9553 . . . . . . 7
44 fsumconst 13586 . . . . . . 7 Neighbors Neighbors Neighbors
4530, 43, 44sylancl 662 . . . . . 6 USGrph VDeg Neighbors Neighbors Neighbors
4638, 42, 453eqtrd 2488 . . . . 5 USGrph VDeg Neighbors Neighbors Neighbors
4746sumeq2dv 13506 . . . 4 USGrph VDeg Neighbors Neighbors Neighbors Neighbors
486adantr 465 . . . . . . . . 9 USGrph VDeg Neighbors
49 hashdifsn 12458 . . . . . . . . 9 Neighbors Neighbors Neighbors Neighbors
5048, 49sylan 471 . . . . . . . 8 USGrph VDeg Neighbors Neighbors Neighbors
5150oveq1d 6296 . . . . . . 7 USGrph VDeg Neighbors Neighbors Neighbors
52 hashcl 12409 . . . . . . . . . . 11 Neighbors Neighbors
536, 52syl 16 . . . . . . . . . 10 USGrph Neighbors
5453nn0red 10860 . . . . . . . . 9 USGrph Neighbors
55 peano2rem 9891 . . . . . . . . 9 Neighbors Neighbors
56 ax-1rid 9565 . . . . . . . . 9 Neighbors Neighbors Neighbors
5754, 55, 563syl 20 . . . . . . . 8 USGrph Neighbors Neighbors
5857ad2antrr 725 . . . . . . 7 USGrph VDeg Neighbors Neighbors Neighbors
5951, 58eqtrd 2484 . . . . . 6 USGrph VDeg Neighbors Neighbors Neighbors
6059sumeq2dv 13506 . . . . 5 USGrph VDeg Neighbors Neighbors Neighbors Neighbors
6153nn0cnd 10861 . . . . . . . 8 USGrph Neighbors
6243a1i 11 . . . . . . . 8 USGrph
6361, 62subcld 9936 . . . . . . 7 USGrph Neighbors
6463adantr 465 . . . . . 6 USGrph VDeg Neighbors
65 fsumconst 13586 . . . . . 6 Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors
6648, 64, 65syl2anc 661 . . . . 5 USGrph VDeg Neighbors Neighbors Neighbors Neighbors
67 hashnbgravdg 24889 . . . . . . . 8 USGrph Neighbors VDeg
68 eqeq1 2447 . . . . . . . . . 10 VDeg Neighbors VDeg Neighbors
6968eqcoms 2455 . . . . . . . . 9 Neighbors VDeg VDeg Neighbors
70 id 22 . . . . . . . . . 10 Neighbors Neighbors
71 oveq1 6288 . . . . . . . . . 10 Neighbors Neighbors
7270, 71oveq12d 6299 . . . . . . . . 9 Neighbors Neighbors Neighbors
7369, 72syl6bi 228 . . . . . . . 8 Neighbors VDeg VDeg Neighbors Neighbors
7467, 73syl 16 . . . . . . 7 USGrph VDeg Neighbors Neighbors
7574adantlr 714 . . . . . 6 USGrph VDeg Neighbors Neighbors
7675imp 429 . . . . 5 USGrph VDeg Neighbors Neighbors
7760, 66, 763eqtrd 2488 . . . 4 USGrph VDeg Neighbors Neighbors
7829, 47, 773eqtrd 2488 . . 3 USGrph VDeg
7978ex 434 . 2 USGrph VDeg
8079ralrimiva 2857 1 USGrph VDeg
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   w3a 974   wceq 1383   wcel 1804  wral 2793  crab 2797  cvv 3095   cdif 3458   wss 3461  csn 4014  cop 4020  cotp 4022  ciun 4315  Disj wdisj 4407   class class class wbr 4437   cmpt 4495   cxp 4987  cfv 5578  (class class class)co 6281  c1st 6783  c2nd 6784  cfn 7518  cc 9493  cr 9494  c1 9496   cmul 9500   cmin 9810  cn0 10802  chash 12386  csu 13489   USGrph cusg 24306   Neighbors cnbgra 24393   2SPathOnOt c2spthot 24832   VDeg cvdg 24869 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-ot 4023  df-uni 4235  df-int 4272  df-iun 4317  df-disj 4408  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-n0 10803  df-z 10872  df-uz 11092  df-rp 11231  df-xadd 11329  df-fz 11683  df-fzo 11806  df-seq 12089  df-exp 12148  df-hash 12387  df-word 12523  df-cj 12913  df-re 12914  df-im 12915  df-sqrt 13049  df-abs 13050  df-clim 13292  df-sum 13490  df-usgra 24309  df-nbgra 24396  df-wlk 24484  df-trail 24485  df-pth 24486  df-spth 24487  df-wlkon 24490  df-spthon 24493  df-2wlkonot 24834  df-2spthonot 24836  df-2spthsot 24837  df-vdgr 24870 This theorem is referenced by:  usgreghash2spot  25045
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