Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  upgr2wlk Structured version   Visualization version   GIF version

Theorem upgr2wlk 40876
 Description: Properties of a pair of functions to be a walk of length 2 in a pseudograph. Note that the vertices need not to be distinct and the edges can be loops or multiedges. (Contributed by Alexander van der Vekens, 16-Feb-2018.) (Revised by AV, 3-Jan-2021.)
Hypotheses
Ref Expression
upgr2wlk.v 𝑉 = (Vtx‘𝐺)
upgr2wlk.i 𝐼 = (iEdg‘𝐺)
Assertion
Ref Expression
upgr2wlk ((𝐺 ∈ UPGraph ∧ (𝐹𝑊𝑃𝑍)) → ((𝐹(1Walks‘𝐺)𝑃 ∧ (#‘𝐹) = 2) ↔ (𝐹:(0..^2)⟶dom 𝐼𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))

Proof of Theorem upgr2wlk
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 upgr2wlk.v . . . . 5 𝑉 = (Vtx‘𝐺)
2 upgr2wlk.i . . . . 5 𝐼 = (iEdg‘𝐺)
31, 2upgriswlk 40849 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝐹𝑊𝑃𝑍) → (𝐹(1Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
433expb 1258 . . 3 ((𝐺 ∈ UPGraph ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(1Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
54anbi1d 737 . 2 ((𝐺 ∈ UPGraph ∧ (𝐹𝑊𝑃𝑍)) → ((𝐹(1Walks‘𝐺)𝑃 ∧ (#‘𝐹) = 2) ↔ ((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ∧ (#‘𝐹) = 2)))
6 iswrdb 13166 . . . . . . . . 9 (𝐹 ∈ Word dom 𝐼𝐹:(0..^(#‘𝐹))⟶dom 𝐼)
7 oveq2 6557 . . . . . . . . . 10 ((#‘𝐹) = 2 → (0..^(#‘𝐹)) = (0..^2))
87feq2d 5944 . . . . . . . . 9 ((#‘𝐹) = 2 → (𝐹:(0..^(#‘𝐹))⟶dom 𝐼𝐹:(0..^2)⟶dom 𝐼))
96, 8syl5bb 271 . . . . . . . 8 ((#‘𝐹) = 2 → (𝐹 ∈ Word dom 𝐼𝐹:(0..^2)⟶dom 𝐼))
10 oveq2 6557 . . . . . . . . 9 ((#‘𝐹) = 2 → (0...(#‘𝐹)) = (0...2))
1110feq2d 5944 . . . . . . . 8 ((#‘𝐹) = 2 → (𝑃:(0...(#‘𝐹))⟶𝑉𝑃:(0...2)⟶𝑉))
12 fzo0to2pr 12420 . . . . . . . . . . 11 (0..^2) = {0, 1}
137, 12syl6eq 2660 . . . . . . . . . 10 ((#‘𝐹) = 2 → (0..^(#‘𝐹)) = {0, 1})
1413raleqdv 3121 . . . . . . . . 9 ((#‘𝐹) = 2 → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ {0, 1} (𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))
15 2Wlklem 40875 . . . . . . . . 9 (∀𝑘 ∈ {0, 1} (𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))
1614, 15syl6bb 275 . . . . . . . 8 ((#‘𝐹) = 2 → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))
179, 11, 163anbi123d 1391 . . . . . . 7 ((#‘𝐹) = 2 → ((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ↔ (𝐹:(0..^2)⟶dom 𝐼𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
1817adantl 481 . . . . . 6 (((𝐺 ∈ UPGraph ∧ (𝐹𝑊𝑃𝑍)) ∧ (#‘𝐹) = 2) → ((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ↔ (𝐹:(0..^2)⟶dom 𝐼𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
19 3anass 1035 . . . . . 6 ((𝐹:(0..^2)⟶dom 𝐼𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
2018, 19syl6bb 275 . . . . 5 (((𝐺 ∈ UPGraph ∧ (𝐹𝑊𝑃𝑍)) ∧ (#‘𝐹) = 2) → ((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})))))
2120ex 449 . . . 4 ((𝐺 ∈ UPGraph ∧ (𝐹𝑊𝑃𝑍)) → ((#‘𝐹) = 2 → ((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))))
2221pm5.32rd 670 . . 3 ((𝐺 ∈ UPGraph ∧ (𝐹𝑊𝑃𝑍)) → (((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ∧ (#‘𝐹) = 2) ↔ ((𝐹:(0..^2)⟶dom 𝐼 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) ∧ (#‘𝐹) = 2)))
23 3anass 1035 . . . 4 (((𝐹:(0..^2)⟶dom 𝐼 ∧ (#‘𝐹) = 2) ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ↔ ((𝐹:(0..^2)⟶dom 𝐼 ∧ (#‘𝐹) = 2) ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
24 an32 835 . . . 4 (((𝐹:(0..^2)⟶dom 𝐼 ∧ (#‘𝐹) = 2) ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) ↔ ((𝐹:(0..^2)⟶dom 𝐼 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) ∧ (#‘𝐹) = 2))
2523, 24bitri 263 . . 3 (((𝐹:(0..^2)⟶dom 𝐼 ∧ (#‘𝐹) = 2) ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ↔ ((𝐹:(0..^2)⟶dom 𝐼 ∧ (𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))) ∧ (#‘𝐹) = 2))
2622, 25syl6bbr 277 . 2 ((𝐺 ∈ UPGraph ∧ (𝐹𝑊𝑃𝑍)) → (((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}) ∧ (#‘𝐹) = 2) ↔ ((𝐹:(0..^2)⟶dom 𝐼 ∧ (#‘𝐹) = 2) ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
27 2nn0 11186 . . . . . . 7 2 ∈ ℕ0
28 fnfzo0hash 13091 . . . . . . 7 ((2 ∈ ℕ0𝐹:(0..^2)⟶dom 𝐼) → (#‘𝐹) = 2)
2927, 28mpan 702 . . . . . 6 (𝐹:(0..^2)⟶dom 𝐼 → (#‘𝐹) = 2)
3029pm4.71i 662 . . . . 5 (𝐹:(0..^2)⟶dom 𝐼 ↔ (𝐹:(0..^2)⟶dom 𝐼 ∧ (#‘𝐹) = 2))
3130bicomi 213 . . . 4 ((𝐹:(0..^2)⟶dom 𝐼 ∧ (#‘𝐹) = 2) ↔ 𝐹:(0..^2)⟶dom 𝐼)
3231a1i 11 . . 3 ((𝐺 ∈ UPGraph ∧ (𝐹𝑊𝑃𝑍)) → ((𝐹:(0..^2)⟶dom 𝐼 ∧ (#‘𝐹) = 2) ↔ 𝐹:(0..^2)⟶dom 𝐼))
33323anbi1d 1395 . 2 ((𝐺 ∈ UPGraph ∧ (𝐹𝑊𝑃𝑍)) → (((𝐹:(0..^2)⟶dom 𝐼 ∧ (#‘𝐹) = 2) ∧ 𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)})) ↔ (𝐹:(0..^2)⟶dom 𝐼𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
345, 26, 333bitrd 293 1 ((𝐺 ∈ UPGraph ∧ (𝐹𝑊𝑃𝑍)) → ((𝐹(1Walks‘𝐺)𝑃 ∧ (#‘𝐹) = 2) ↔ (𝐹:(0..^2)⟶dom 𝐼𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {cpr 4127   class class class wbr 4583  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  2c2 10947  ℕ0cn0 11169  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146  Vtxcvtx 25673  iEdgciedg 25674   UPGraph cupgr 25747  1Walksc1wlks 40796 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-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  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-map 7746  df-pm 7747  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-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-uhgr 25724  df-upgr 25749  df-edga 25793  df-1wlks 40800  df-wlks 40801 This theorem is referenced by:  umgrwwlks2on  41161
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