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Theorem clwlksf1clwwlklem2OLD 27406
 Description: Obsolete as of 24-May-2022. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
clwlksfclwwlkOLD.1 𝐴 = (1st𝑐)
clwlksfclwwlkOLD.2 𝐵 = (2nd𝑐)
clwlksfclwwlkOLD.c 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘𝐴) = 𝑁}
clwlksfclwwlkOLD.f 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))
Assertion
Ref Expression
clwlksf1clwwlklem2OLD (𝑊𝐶 → ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁))
Distinct variable groups:   𝐺,𝑐   𝑁,𝑐   𝑊,𝑐   𝐶,𝑐   𝐹,𝑐
Allowed substitution hints:   𝐴(𝑐)   𝐵(𝑐)

Proof of Theorem clwlksf1clwwlklem2OLD
StepHypRef Expression
1 clwlksfclwwlkOLD.1 . . 3 𝐴 = (1st𝑐)
2 clwlksfclwwlkOLD.2 . . 3 𝐵 = (2nd𝑐)
3 clwlksfclwwlkOLD.c . . 3 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘𝐴) = 𝑁}
4 clwlksfclwwlkOLD.f . . 3 𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))
51, 2, 3, 4clwlksf1clwwlklem0OLD 27404 . 2 (𝑊𝐶 → (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(♯‘(1st𝑊)))) ∧ (♯‘(1st𝑊)) = 𝑁))
6 fveq2 6412 . . . . . 6 ((♯‘(1st𝑊)) = 𝑁 → ((2nd𝑊)‘(♯‘(1st𝑊))) = ((2nd𝑊)‘𝑁))
76eqeq2d 2810 . . . . 5 ((♯‘(1st𝑊)) = 𝑁 → (((2nd𝑊)‘0) = ((2nd𝑊)‘(♯‘(1st𝑊))) ↔ ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁)))
87biimpcd 241 . . . 4 (((2nd𝑊)‘0) = ((2nd𝑊)‘(♯‘(1st𝑊))) → ((♯‘(1st𝑊)) = 𝑁 → ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁)))
983ad2ant3 1166 . . 3 (((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(♯‘(1st𝑊)))) → ((♯‘(1st𝑊)) = 𝑁 → ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁)))
109imp 396 . 2 ((((1st𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd𝑊):(0...(♯‘(1st𝑊)))⟶(Vtx‘𝐺) ∧ ((2nd𝑊)‘0) = ((2nd𝑊)‘(♯‘(1st𝑊)))) ∧ (♯‘(1st𝑊)) = 𝑁) → ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁))
115, 10syl 17 1 (𝑊𝐶 → ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 385   ∧ w3a 1108   = wceq 1653   ∈ wcel 2157  {crab 3094  ⟨cop 4375   ↦ cmpt 4923  dom cdm 5313  ⟶wf 6098  ‘cfv 6102  (class class class)co 6879  1st c1st 7400  2nd c2nd 7401  0cc0 10225  ...cfz 12579  ♯chash 13369  Word cword 13533   substr csubstr 13663  Vtxcvtx 26230  iEdgciedg 26231  ClWalkscclwlks 27023 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-rep 4965  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098  ax-un 7184  ax-cnex 10281  ax-resscn 10282  ax-1cn 10283  ax-icn 10284  ax-addcl 10285  ax-addrcl 10286  ax-mulcl 10287  ax-mulrcl 10288  ax-mulcom 10289  ax-addass 10290  ax-mulass 10291  ax-distr 10292  ax-i2m1 10293  ax-1ne0 10294  ax-1rid 10295  ax-rnegex 10296  ax-rrecex 10297  ax-cnre 10298  ax-pre-lttri 10299  ax-pre-lttrn 10300  ax-pre-ltadd 10301  ax-pre-mulgt0 10302 This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-ifp 1087  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3388  df-sbc 3635  df-csb 3730  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-pss 3786  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-tp 4374  df-op 4376  df-uni 4630  df-int 4669  df-iun 4713  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5221  df-eprel 5226  df-po 5234  df-so 5235  df-fr 5272  df-we 5274  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-rn 5324  df-res 5325  df-ima 5326  df-pred 5899  df-ord 5945  df-on 5946  df-lim 5947  df-suc 5948  df-iota 6065  df-fun 6104  df-fn 6105  df-f 6106  df-f1 6107  df-fo 6108  df-f1o 6109  df-fv 6110  df-riota 6840  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-om 7301  df-1st 7402  df-2nd 7403  df-wrecs 7646  df-recs 7708  df-rdg 7746  df-1o 7800  df-er 7983  df-map 8098  df-pm 8099  df-en 8197  df-dom 8198  df-sdom 8199  df-fin 8200  df-card 9052  df-pnf 10366  df-mnf 10367  df-xr 10368  df-ltxr 10369  df-le 10370  df-sub 10559  df-neg 10560  df-nn 11314  df-n0 11580  df-z 11666  df-uz 11930  df-fz 12580  df-fzo 12720  df-hash 13370  df-word 13534  df-wlks 26848  df-clwlks 27024 This theorem is referenced by:  clwlksf1clwwlklemOLD  27408
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