Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frgrwopreglem4 Structured version   Visualization version   GIF version

Theorem frgrwopreglem4 41484
 Description: Lemma 4 for frgrwopreg 41486. In a friendship graph each vertex with degree K is connected with a vertex with degree other than K. This corresponds to statement 4 in [Huneke] p. 2: "By the first claim, every vertex in A is adjacent to every vertex in B.". (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 10-May-2021.)
Hypotheses
Ref Expression
frgrwopreg.v 𝑉 = (Vtx‘𝐺)
frgrwopreg.d 𝐷 = (VtxDeg‘𝐺)
frgrwopreg.a 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
frgrwopreg.b 𝐵 = (𝑉𝐴)
frgrwopreg.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
frgrwopreglem4 (𝐺 ∈ FriendGraph → ∀𝑎𝐴𝑏𝐵 {𝑎, 𝑏} ∈ 𝐸)
Distinct variable groups:   𝑥,𝑉   𝑥,𝐴   𝑥,𝐺   𝑥,𝐾   𝑥,𝐷   𝐴,𝑏   𝐺,𝑎,𝑏,𝑥
Allowed substitution hints:   𝐴(𝑎)   𝐵(𝑥,𝑎,𝑏)   𝐷(𝑎,𝑏)   𝐸(𝑥,𝑎,𝑏)   𝐾(𝑎,𝑏)   𝑉(𝑎,𝑏)

Proof of Theorem frgrwopreglem4
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 frgrwopreg.v . . . 4 𝑉 = (Vtx‘𝐺)
2 frgrwopreg.d . . . 4 𝐷 = (VtxDeg‘𝐺)
3 frgrwopreg.a . . . 4 𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}
4 frgrwopreg.b . . . 4 𝐵 = (𝑉𝐴)
51, 2, 3, 4frgrwopreglem3 41483 . . 3 ((𝑎𝐴𝑏𝐵) → (𝐷𝑎) ≠ (𝐷𝑏))
61, 2frgrncvvdeq 41480 . . . 4 (𝐺 ∈ FriendGraph → ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)))
7 elrabi 3328 . . . . . . . . 9 (𝑎 ∈ {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾} → 𝑎𝑉)
87, 3eleq2s 2706 . . . . . . . 8 (𝑎𝐴𝑎𝑉)
9 sneq 4135 . . . . . . . . . . 11 (𝑥 = 𝑎 → {𝑥} = {𝑎})
109difeq2d 3690 . . . . . . . . . 10 (𝑥 = 𝑎 → (𝑉 ∖ {𝑥}) = (𝑉 ∖ {𝑎}))
11 oveq2 6557 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑎))
12 neleq2 2889 . . . . . . . . . . . 12 ((𝐺 NeighbVtx 𝑥) = (𝐺 NeighbVtx 𝑎) → (𝑦 ∉ (𝐺 NeighbVtx 𝑥) ↔ 𝑦 ∉ (𝐺 NeighbVtx 𝑎)))
1311, 12syl 17 . . . . . . . . . . 11 (𝑥 = 𝑎 → (𝑦 ∉ (𝐺 NeighbVtx 𝑥) ↔ 𝑦 ∉ (𝐺 NeighbVtx 𝑎)))
14 fveq2 6103 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝐷𝑥) = (𝐷𝑎))
1514eqeq1d 2612 . . . . . . . . . . 11 (𝑥 = 𝑎 → ((𝐷𝑥) = (𝐷𝑦) ↔ (𝐷𝑎) = (𝐷𝑦)))
1613, 15imbi12d 333 . . . . . . . . . 10 (𝑥 = 𝑎 → ((𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) ↔ (𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑦))))
1710, 16raleqbidv 3129 . . . . . . . . 9 (𝑥 = 𝑎 → (∀𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) ↔ ∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑦))))
1817rspcv 3278 . . . . . . . 8 (𝑎𝑉 → (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) → ∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑦))))
198, 18syl 17 . . . . . . 7 (𝑎𝐴 → (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) → ∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑦))))
2019adantr 480 . . . . . 6 ((𝑎𝐴𝑏𝐵) → (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) → ∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑦))))
214eleq2i 2680 . . . . . . . . . 10 (𝑏𝐵𝑏 ∈ (𝑉𝐴))
22 eldif 3550 . . . . . . . . . 10 (𝑏 ∈ (𝑉𝐴) ↔ (𝑏𝑉 ∧ ¬ 𝑏𝐴))
2321, 22bitri 263 . . . . . . . . 9 (𝑏𝐵 ↔ (𝑏𝑉 ∧ ¬ 𝑏𝐴))
24 simpll 786 . . . . . . . . . . 11 (((𝑏𝑉 ∧ ¬ 𝑏𝐴) ∧ 𝑎𝐴) → 𝑏𝑉)
25 eleq1a 2683 . . . . . . . . . . . . . . 15 (𝑎𝐴 → (𝑏 = 𝑎𝑏𝐴))
2625con3rr3 150 . . . . . . . . . . . . . 14 𝑏𝐴 → (𝑎𝐴 → ¬ 𝑏 = 𝑎))
2726adantl 481 . . . . . . . . . . . . 13 ((𝑏𝑉 ∧ ¬ 𝑏𝐴) → (𝑎𝐴 → ¬ 𝑏 = 𝑎))
2827imp 444 . . . . . . . . . . . 12 (((𝑏𝑉 ∧ ¬ 𝑏𝐴) ∧ 𝑎𝐴) → ¬ 𝑏 = 𝑎)
29 velsn 4141 . . . . . . . . . . . 12 (𝑏 ∈ {𝑎} ↔ 𝑏 = 𝑎)
3028, 29sylnibr 318 . . . . . . . . . . 11 (((𝑏𝑉 ∧ ¬ 𝑏𝐴) ∧ 𝑎𝐴) → ¬ 𝑏 ∈ {𝑎})
3124, 30eldifd 3551 . . . . . . . . . 10 (((𝑏𝑉 ∧ ¬ 𝑏𝐴) ∧ 𝑎𝐴) → 𝑏 ∈ (𝑉 ∖ {𝑎}))
3231ex 449 . . . . . . . . 9 ((𝑏𝑉 ∧ ¬ 𝑏𝐴) → (𝑎𝐴𝑏 ∈ (𝑉 ∖ {𝑎})))
3323, 32sylbi 206 . . . . . . . 8 (𝑏𝐵 → (𝑎𝐴𝑏 ∈ (𝑉 ∖ {𝑎})))
3433impcom 445 . . . . . . 7 ((𝑎𝐴𝑏𝐵) → 𝑏 ∈ (𝑉 ∖ {𝑎}))
35 neleq1 2888 . . . . . . . . 9 (𝑦 = 𝑏 → (𝑦 ∉ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∉ (𝐺 NeighbVtx 𝑎)))
36 fveq2 6103 . . . . . . . . . 10 (𝑦 = 𝑏 → (𝐷𝑦) = (𝐷𝑏))
3736eqeq2d 2620 . . . . . . . . 9 (𝑦 = 𝑏 → ((𝐷𝑎) = (𝐷𝑦) ↔ (𝐷𝑎) = (𝐷𝑏)))
3835, 37imbi12d 333 . . . . . . . 8 (𝑦 = 𝑏 → ((𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑦)) ↔ (𝑏 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑏))))
3938rspcv 3278 . . . . . . 7 (𝑏 ∈ (𝑉 ∖ {𝑎}) → (∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑦)) → (𝑏 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑏))))
4034, 39syl 17 . . . . . 6 ((𝑎𝐴𝑏𝐵) → (∀𝑦 ∈ (𝑉 ∖ {𝑎})(𝑦 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑦)) → (𝑏 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑏))))
41 nnel 2892 . . . . . . . . 9 𝑏 ∉ (𝐺 NeighbVtx 𝑎) ↔ 𝑏 ∈ (𝐺 NeighbVtx 𝑎))
42 frgrusgr 41432 . . . . . . . . . . . . . . 15 (𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
43 frgrwopreg.e . . . . . . . . . . . . . . . 16 𝐸 = (Edg‘𝐺)
4443nbusgreledg 40575 . . . . . . . . . . . . . . 15 (𝐺 ∈ USGraph → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ {𝑏, 𝑎} ∈ 𝐸))
4542, 44syl 17 . . . . . . . . . . . . . 14 (𝐺 ∈ FriendGraph → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ {𝑏, 𝑎} ∈ 𝐸))
46 prcom 4211 . . . . . . . . . . . . . . 15 {𝑏, 𝑎} = {𝑎, 𝑏}
4746eleq1i 2679 . . . . . . . . . . . . . 14 ({𝑏, 𝑎} ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸)
4845, 47syl6bb 275 . . . . . . . . . . . . 13 (𝐺 ∈ FriendGraph → (𝑏 ∈ (𝐺 NeighbVtx 𝑎) ↔ {𝑎, 𝑏} ∈ 𝐸))
4948biimpa 500 . . . . . . . . . . . 12 ((𝐺 ∈ FriendGraph ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)) → {𝑎, 𝑏} ∈ 𝐸)
5049a1d 25 . . . . . . . . . . 11 ((𝐺 ∈ FriendGraph ∧ 𝑏 ∈ (𝐺 NeighbVtx 𝑎)) → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸))
5150expcom 450 . . . . . . . . . 10 (𝑏 ∈ (𝐺 NeighbVtx 𝑎) → (𝐺 ∈ FriendGraph → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
5251a1d 25 . . . . . . . . 9 (𝑏 ∈ (𝐺 NeighbVtx 𝑎) → ((𝑎𝐴𝑏𝐵) → (𝐺 ∈ FriendGraph → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
5341, 52sylbi 206 . . . . . . . 8 𝑏 ∉ (𝐺 NeighbVtx 𝑎) → ((𝑎𝐴𝑏𝐵) → (𝐺 ∈ FriendGraph → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
54 eqneqall 2793 . . . . . . . . 9 ((𝐷𝑎) = (𝐷𝑏) → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸))
55542a1d 26 . . . . . . . 8 ((𝐷𝑎) = (𝐷𝑏) → ((𝑎𝐴𝑏𝐵) → (𝐺 ∈ FriendGraph → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
5653, 55ja 172 . . . . . . 7 ((𝑏 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑏)) → ((𝑎𝐴𝑏𝐵) → (𝐺 ∈ FriendGraph → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
5756com12 32 . . . . . 6 ((𝑎𝐴𝑏𝐵) → ((𝑏 ∉ (𝐺 NeighbVtx 𝑎) → (𝐷𝑎) = (𝐷𝑏)) → (𝐺 ∈ FriendGraph → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
5820, 40, 573syld 58 . . . . 5 ((𝑎𝐴𝑏𝐵) → (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) → (𝐺 ∈ FriendGraph → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
5958com3l 87 . . . 4 (∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (𝐺 NeighbVtx 𝑥) → (𝐷𝑥) = (𝐷𝑦)) → (𝐺 ∈ FriendGraph → ((𝑎𝐴𝑏𝐵) → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸))))
606, 59mpcom 37 . . 3 (𝐺 ∈ FriendGraph → ((𝑎𝐴𝑏𝐵) → ((𝐷𝑎) ≠ (𝐷𝑏) → {𝑎, 𝑏} ∈ 𝐸)))
615, 60mpdi 44 . 2 (𝐺 ∈ FriendGraph → ((𝑎𝐴𝑏𝐵) → {𝑎, 𝑏} ∈ 𝐸))
6261ralrimivv 2953 1 (𝐺 ∈ FriendGraph → ∀𝑎𝐴𝑏𝐵 {𝑎, 𝑏} ∈ 𝐸)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  ∀wral 2896  {crab 2900   ∖ cdif 3537  {csn 4125  {cpr 4127  ‘cfv 5804  (class class class)co 6549  Vtxcvtx 25673  Edgcedga 25792   USGraph cusgr 40379   NeighbVtx cnbgr 40550  VtxDegcvtxdg 40681   FriendGraph cfrgr 41428 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-xadd 11823  df-fz 12198  df-hash 12980  df-uhgr 25724  df-ushgr 25725  df-upgr 25749  df-umgr 25750  df-edga 25793  df-uspgr 40380  df-usgr 40381  df-nbgr 40554  df-vtxdg 40682  df-frgr 41429 This theorem is referenced by:  frgrwopreglem5  41485  frgrwopreg1  41487  frgrwopreg2  41488
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