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Theorem usgreghash2spotv 28529
 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 28528 . . . . . . . 8 USGrph Neighbors Neighbors
323expa 1154 . . . . . . 7 USGrph Neighbors Neighbors
43fveq2d 5735 . . . . . 6 USGrph Neighbors Neighbors
5 nbfiusgrafi 28328 . . . . . . . 8 USGrph Neighbors
653expa 1154 . . . . . . 7 USGrph Neighbors
7 diffi 7342 . . . . . . . . . . 11 Neighbors Neighbors
85, 7syl 16 . . . . . . . . . 10 USGrph Neighbors
983expa 1154 . . . . . . . . 9 USGrph Neighbors
109adantr 453 . . . . . . . 8 USGrph Neighbors Neighbors
11 snfi 7190 . . . . . . . . . 10
1211a1i 11 . . . . . . . . 9 USGrph Neighbors Neighbors
1312ralrimiva 2791 . . . . . . . 8 USGrph Neighbors Neighbors
14 iunfi 7397 . . . . . . . 8 Neighbors Neighbors Neighbors
1510, 13, 14syl2anc 644 . . . . . . 7 USGrph Neighbors Neighbors
16 nbgrassvt 21450 . . . . . . . . . . . 12 USGrph Neighbors
1716ad3antrrr 712 . . . . . . . . . . 11 USGrph Neighbors Neighbors
1817ssdifd 3485 . . . . . . . . . 10 USGrph Neighbors Neighbors
19 iunss1 4106 . . . . . . . . . 10 Neighbors Neighbors
2018, 19syl 16 . . . . . . . . 9 USGrph Neighbors Neighbors
2120ralrimiva 2791 . . . . . . . 8 USGrph Neighbors Neighbors
22 simpr 449 . . . . . . . . 9 USGrph
23 otiunsndisj 28081 . . . . . . . . 9 Disj Neighbors
2422, 23syl 16 . . . . . . . 8 USGrph Disj Neighbors
25 disjss2 4188 . . . . . . . 8 Neighbors Neighbors Disj Neighbors Disj Neighbors Neighbors
2621, 24, 25sylc 59 . . . . . . 7 USGrph Disj Neighbors Neighbors
276, 15, 26hashiun 12606 . . . . . 6 USGrph Neighbors Neighbors Neighbors Neighbors
284, 27eqtrd 2470 . . . . 5 USGrph Neighbors Neighbors
2928adantr 453 . . . 4 USGrph VDeg Neighbors Neighbors
309ad2antrr 708 . . . . . . 7 USGrph VDeg Neighbors Neighbors
3111a1i 11 . . . . . . 7 USGrph VDeg Neighbors Neighbors
32 nbgraisvtx 21448 . . . . . . . . . 10 USGrph Neighbors
3332ad3antrrr 712 . . . . . . . . 9 USGrph VDeg Neighbors
3433imp 420 . . . . . . . 8 USGrph VDeg Neighbors
3522ad2antrr 708 . . . . . . . 8 USGrph VDeg Neighbors
36 otsndisj 28080 . . . . . . . 8 Disj Neighbors
3734, 35, 36syl2anc 644 . . . . . . 7 USGrph VDeg Neighbors Disj Neighbors
3830, 31, 37hashiun 12606 . . . . . 6 USGrph VDeg Neighbors Neighbors Neighbors
39 otex 4431 . . . . . . . 8
40 hashsng 11652 . . . . . . . 8
4139, 40mp1i 12 . . . . . . 7 USGrph VDeg Neighbors Neighbors
4241sumeq2dv 12502 . . . . . 6 USGrph VDeg Neighbors Neighbors Neighbors
43 ax-1cn 9053 . . . . . . 7
44 fsumconst 12578 . . . . . . 7 Neighbors Neighbors Neighbors
4530, 43, 44sylancl 645 . . . . . 6 USGrph VDeg Neighbors Neighbors Neighbors
4638, 42, 453eqtrd 2474 . . . . 5 USGrph VDeg Neighbors Neighbors Neighbors
4746sumeq2dv 12502 . . . 4 USGrph VDeg Neighbors Neighbors Neighbors Neighbors
486adantr 453 . . . . . . . . 9 USGrph VDeg Neighbors
49 hashdifsn 11684 . . . . . . . . 9 Neighbors Neighbors Neighbors Neighbors
5048, 49sylan 459 . . . . . . . 8 USGrph VDeg Neighbors Neighbors Neighbors
5150oveq1d 6099 . . . . . . 7 USGrph VDeg Neighbors Neighbors Neighbors
52 hashcl 11644 . . . . . . . . . . 11 Neighbors Neighbors
536, 52syl 16 . . . . . . . . . 10 USGrph Neighbors
5453nn0red 10280 . . . . . . . . 9 USGrph Neighbors
55 peano2rem 9372 . . . . . . . . 9 Neighbors Neighbors
56 ax-1rid 9065 . . . . . . . . 9 Neighbors Neighbors Neighbors
5754, 55, 563syl 19 . . . . . . . 8 USGrph Neighbors Neighbors
5857ad2antrr 708 . . . . . . 7 USGrph VDeg Neighbors Neighbors Neighbors
5951, 58eqtrd 2470 . . . . . 6 USGrph VDeg Neighbors Neighbors Neighbors
6059sumeq2dv 12502 . . . . 5 USGrph VDeg Neighbors Neighbors Neighbors Neighbors
6153nn0cnd 10281 . . . . . . . 8 USGrph Neighbors
6243a1i 11 . . . . . . . 8 USGrph
6361, 62subcld 9416 . . . . . . 7 USGrph Neighbors
6463adantr 453 . . . . . 6 USGrph VDeg Neighbors
65 fsumconst 12578 . . . . . 6 Neighbors Neighbors Neighbors Neighbors Neighbors Neighbors
6648, 64, 65syl2anc 644 . . . . 5 USGrph VDeg Neighbors Neighbors Neighbors Neighbors
67 hashnbgravdg 21687 . . . . . . . 8 USGrph Neighbors VDeg
68 eqeq1 2444 . . . . . . . . . 10 VDeg Neighbors VDeg Neighbors
6968eqcoms 2441 . . . . . . . . 9 Neighbors VDeg VDeg Neighbors
70 id 21 . . . . . . . . . 10 Neighbors Neighbors
71 oveq1 6091 . . . . . . . . . 10 Neighbors Neighbors
7270, 71oveq12d 6102 . . . . . . . . 9 Neighbors Neighbors Neighbors
7369, 72syl6bi 221 . . . . . . . 8 Neighbors VDeg VDeg Neighbors Neighbors
7467, 73syl 16 . . . . . . 7 USGrph VDeg Neighbors Neighbors
7574adantlr 697 . . . . . 6 USGrph VDeg Neighbors Neighbors
7675imp 420 . . . . 5 USGrph VDeg Neighbors Neighbors
7760, 66, 763eqtrd 2474 . . . 4 USGrph VDeg Neighbors Neighbors
7829, 47, 773eqtrd 2474 . . 3 USGrph VDeg
7978ex 425 . 2 USGrph VDeg
8079ralrimiva 2791 1 USGrph VDeg
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   w3a 937   wceq 1653   wcel 1726  wral 2707  crab 2711  cvv 2958   cdif 3319   wss 3322  csn 3816  cop 3819  cotp 3820  ciun 4095  Disj wdisj 4185   class class class wbr 4215   cmpt 4269   cxp 4879  cfv 5457  (class class class)co 6084  c1st 6350  c2nd 6351  cfn 7112  cc 8993  cr 8994  c1 8996   cmul 9000   cmin 9296  cn0 10226  chash 11623  csu 12484   USGrph cusg 21370   Neighbors cnbgra 21435   VDeg cvdg 21669   2SPathOnOt c2spthot 28388 This theorem is referenced by:  usgreghash2spot  28532 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704  ax-inf2 7599  ax-cnex 9051  ax-resscn 9052  ax-1cn 9053  ax-icn 9054  ax-addcl 9055  ax-addrcl 9056  ax-mulcl 9057  ax-mulrcl 9058  ax-mulcom 9059  ax-addass 9060  ax-mulass 9061  ax-distr 9062  ax-i2m1 9063  ax-1ne0 9064  ax-1rid 9065  ax-rnegex 9066  ax-rrecex 9067  ax-cnre 9068  ax-pre-lttri 9069  ax-pre-lttrn 9070  ax-pre-ltadd 9071  ax-pre-mulgt0 9072  ax-pre-sup 9073 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-ot 3826  df-uni 4018  df-int 4053  df-iun 4097  df-disj 4186  df-br 4216  df-opab 4270  df-mpt 4271  df-tr 4306  df-eprel 4497  df-id 4501  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-ord 4587  df-on 4588  df-lim 4589  df-suc 4590  df-om 4849  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-isom 5466  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-riota 6552  df-recs 6636  df-rdg 6671  df-1o 6727  df-2o 6728  df-oadd 6731  df-er 6908  df-map 7023  df-pm 7024  df-en 7113  df-dom 7114  df-sdom 7115  df-fin 7116  df-sup 7449  df-oi 7482  df-card 7831  df-cda 8053  df-pnf 9127  df-mnf 9128  df-xr 9129  df-ltxr 9130  df-le 9131  df-sub 9298  df-neg 9299  df-div 9683  df-nn 10006  df-2 10063  df-3 10064  df-n0 10227  df-z 10288  df-uz 10494  df-rp 10618  df-xadd 10716  df-fz 11049  df-fzo 11141  df-seq 11329  df-exp 11388  df-hash 11624  df-word 11728  df-cj 11909  df-re 11910  df-im 11911  df-sqr 12045  df-abs 12046  df-clim 12287  df-sum 12485  df-usgra 21372  df-nbgra 21438  df-wlk 21521  df-trail 21522  df-pth 21523  df-spth 21524  df-wlkon 21527  df-spthon 21530  df-vdgr 21670  df-2wlkonot 28390  df-2spthonot 28392  df-2spthsot 28393
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