Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  clwlksf1clwwlklem0 Structured version   Visualization version   GIF version

Theorem clwlksf1clwwlklem0 41271
 Description: Lemma 1 for clwlksf1clwwlklem 41275. (Contributed by AV, 3-May-2021.)
Hypotheses
Ref Expression
clwlksfclwwlk.1 𝐴 = (1st𝑐)
clwlksfclwwlk.2 𝐵 = (2nd𝑐)
clwlksfclwwlk.c 𝐶 = {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁}
clwlksfclwwlk.f 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))
Assertion
Ref Expression
clwlksf1clwwlklem0 (𝑊𝐶 → (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(#‘(1st𝑊)))) ∧ (#‘(1st𝑊)) = 𝑁))
Distinct variable groups:   𝐺,𝑐   𝑁,𝑐   𝑊,𝑐   𝐶,𝑐   𝐹,𝑐
Allowed substitution hints:   𝐴(𝑐)   𝐵(𝑐)

Proof of Theorem clwlksf1clwwlklem0
Dummy variable 𝑖 is distinct from all other variables.
StepHypRef Expression
1 clwlksfclwwlk.1 . . . . . 6 𝐴 = (1st𝑐)
2 fveq2 6103 . . . . . 6 (𝑐 = 𝑊 → (1st𝑐) = (1st𝑊))
31, 2syl5eq 2656 . . . . 5 (𝑐 = 𝑊𝐴 = (1st𝑊))
43fveq2d 6107 . . . 4 (𝑐 = 𝑊 → (#‘𝐴) = (#‘(1st𝑊)))
54eqeq1d 2612 . . 3 (𝑐 = 𝑊 → ((#‘𝐴) = 𝑁 ↔ (#‘(1st𝑊)) = 𝑁))
6 clwlksfclwwlk.c . . 3 𝐶 = {𝑐 ∈ (ClWalkS‘𝐺) ∣ (#‘𝐴) = 𝑁}
75, 6elrab2 3333 . 2 (𝑊𝐶 ↔ (𝑊 ∈ (ClWalkS‘𝐺) ∧ (#‘(1st𝑊)) = 𝑁))
8 eqid 2610 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
9 eqid 2610 . . . . 5 (iEdg‘𝐺) = (iEdg‘𝐺)
10 eqid 2610 . . . . 5 (1st𝑊) = (1st𝑊)
11 eqid 2610 . . . . 5 (2nd𝑊) = (2nd𝑊)
128, 9, 10, 11clWlkcompim 40987 . . . 4 (𝑊 ∈ (ClWalkS‘𝐺) → (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘(1st𝑊)))if-(((2nd𝑊)‘𝑖) = ((2nd𝑊)‘(𝑖 + 1)), ((iEdg‘𝐺)‘((1st𝑊)‘𝑖)) = {((2nd𝑊)‘𝑖)}, {((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ⊆ ((iEdg‘𝐺)‘((1st𝑊)‘𝑖))) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(#‘(1st𝑊))))))
13 simpr 476 . . . . . 6 ((∀𝑖 ∈ (0..^(#‘(1st𝑊)))if-(((2nd𝑊)‘𝑖) = ((2nd𝑊)‘(𝑖 + 1)), ((iEdg‘𝐺)‘((1st𝑊)‘𝑖)) = {((2nd𝑊)‘𝑖)}, {((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ⊆ ((iEdg‘𝐺)‘((1st𝑊)‘𝑖))) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(#‘(1st𝑊)))) → ((2nd𝑊)‘0) = ((2nd𝑊)‘(#‘(1st𝑊))))
1413anim2i 591 . . . . 5 ((((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘(1st𝑊)))if-(((2nd𝑊)‘𝑖) = ((2nd𝑊)‘(𝑖 + 1)), ((iEdg‘𝐺)‘((1st𝑊)‘𝑖)) = {((2nd𝑊)‘𝑖)}, {((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ⊆ ((iEdg‘𝐺)‘((1st𝑊)‘𝑖))) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(#‘(1st𝑊))))) → (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(#‘(1st𝑊)))))
15 df-3an 1033 . . . . 5 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(#‘(1st𝑊)))) ↔ (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(#‘(1st𝑊)))))
1614, 15sylibr 223 . . . 4 ((((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘(1st𝑊)))if-(((2nd𝑊)‘𝑖) = ((2nd𝑊)‘(𝑖 + 1)), ((iEdg‘𝐺)‘((1st𝑊)‘𝑖)) = {((2nd𝑊)‘𝑖)}, {((2nd𝑊)‘𝑖), ((2nd𝑊)‘(𝑖 + 1))} ⊆ ((iEdg‘𝐺)‘((1st𝑊)‘𝑖))) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(#‘(1st𝑊))))) → ((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(#‘(1st𝑊)))))
1712, 16syl 17 . . 3 (𝑊 ∈ (ClWalkS‘𝐺) → ((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(#‘(1st𝑊)))))
1817anim1i 590 . 2 ((𝑊 ∈ (ClWalkS‘𝐺) ∧ (#‘(1st𝑊)) = 𝑁) → (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(#‘(1st𝑊)))) ∧ (#‘(1st𝑊)) = 𝑁))
197, 18sylbi 206 1 (𝑊𝐶 → (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶(Vtx‘𝐺) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(#‘(1st𝑊)))) ∧ (#‘(1st𝑊)) = 𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  if-wif 1006   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ⊆ wss 3540  {csn 4125  {cpr 4127  ⟨cop 4131   ↦ cmpt 4643  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  0cc0 9815  1c1 9816   + caddc 9818  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   substr csubstr 13150  Vtxcvtx 25673  iEdgciedg 25674  ClWalkScclwlks 40976 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-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-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  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-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-1wlks 40800  df-clwlks 40977 This theorem is referenced by:  clwlksf1clwwlklem1  41272  clwlksf1clwwlklem2  41273  clwlksf1clwwlklem3  41274
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