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Theorem wlkp1lem6 26631
 Description: Lemma for wlkp1 26634. (Contributed by AV, 6-Mar-2021.)
Hypotheses
Ref Expression
wlkp1.v 𝑉 = (Vtx‘𝐺)
wlkp1.i 𝐼 = (iEdg‘𝐺)
wlkp1.f (𝜑 → Fun 𝐼)
wlkp1.a (𝜑𝐼 ∈ Fin)
wlkp1.b (𝜑𝐵 ∈ V)
wlkp1.c (𝜑𝐶𝑉)
wlkp1.d (𝜑 → ¬ 𝐵 ∈ dom 𝐼)
wlkp1.w (𝜑𝐹(Walks‘𝐺)𝑃)
wlkp1.n 𝑁 = (#‘𝐹)
wlkp1.e (𝜑𝐸 ∈ (Edg‘𝐺))
wlkp1.x (𝜑 → {(𝑃𝑁), 𝐶} ⊆ 𝐸)
wlkp1.u (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
wlkp1.h 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})
wlkp1.q 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})
wlkp1.s (𝜑 → (Vtx‘𝑆) = 𝑉)
Assertion
Ref Expression
wlkp1lem6 (𝜑 → ∀𝑘 ∈ (0..^𝑁)((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))))
Distinct variable group:   𝜑,𝑘
Allowed substitution hints:   𝐵(𝑘)   𝐶(𝑘)   𝑃(𝑘)   𝑄(𝑘)   𝑆(𝑘)   𝐸(𝑘)   𝐹(𝑘)   𝐺(𝑘)   𝐻(𝑘)   𝐼(𝑘)   𝑁(𝑘)   𝑉(𝑘)

Proof of Theorem wlkp1lem6
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 wlkp1.v . . . 4 𝑉 = (Vtx‘𝐺)
2 wlkp1.i . . . 4 𝐼 = (iEdg‘𝐺)
3 wlkp1.f . . . 4 (𝜑 → Fun 𝐼)
4 wlkp1.a . . . 4 (𝜑𝐼 ∈ Fin)
5 wlkp1.b . . . 4 (𝜑𝐵 ∈ V)
6 wlkp1.c . . . 4 (𝜑𝐶𝑉)
7 wlkp1.d . . . 4 (𝜑 → ¬ 𝐵 ∈ dom 𝐼)
8 wlkp1.w . . . 4 (𝜑𝐹(Walks‘𝐺)𝑃)
9 wlkp1.n . . . 4 𝑁 = (#‘𝐹)
10 wlkp1.e . . . 4 (𝜑𝐸 ∈ (Edg‘𝐺))
11 wlkp1.x . . . 4 (𝜑 → {(𝑃𝑁), 𝐶} ⊆ 𝐸)
12 wlkp1.u . . . 4 (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
13 wlkp1.h . . . 4 𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})
14 wlkp1.q . . . 4 𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})
15 wlkp1.s . . . 4 (𝜑 → (Vtx‘𝑆) = 𝑉)
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15wlkp1lem5 26630 . . 3 (𝜑 → ∀𝑥 ∈ (0...𝑁)(𝑄𝑥) = (𝑃𝑥))
17 elfzofz 12524 . . . . . . 7 (𝑘 ∈ (0..^𝑁) → 𝑘 ∈ (0...𝑁))
1817adantl 481 . . . . . 6 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ (0...𝑁))
19 fveq2 6229 . . . . . . . 8 (𝑥 = 𝑘 → (𝑄𝑥) = (𝑄𝑘))
20 fveq2 6229 . . . . . . . 8 (𝑥 = 𝑘 → (𝑃𝑥) = (𝑃𝑘))
2119, 20eqeq12d 2666 . . . . . . 7 (𝑥 = 𝑘 → ((𝑄𝑥) = (𝑃𝑥) ↔ (𝑄𝑘) = (𝑃𝑘)))
2221rspcv 3336 . . . . . 6 (𝑘 ∈ (0...𝑁) → (∀𝑥 ∈ (0...𝑁)(𝑄𝑥) = (𝑃𝑥) → (𝑄𝑘) = (𝑃𝑘)))
2318, 22syl 17 . . . . 5 ((𝜑𝑘 ∈ (0..^𝑁)) → (∀𝑥 ∈ (0...𝑁)(𝑄𝑥) = (𝑃𝑥) → (𝑄𝑘) = (𝑃𝑘)))
2423imp 444 . . . 4 (((𝜑𝑘 ∈ (0..^𝑁)) ∧ ∀𝑥 ∈ (0...𝑁)(𝑄𝑥) = (𝑃𝑥)) → (𝑄𝑘) = (𝑃𝑘))
25 fzofzp1 12605 . . . . . . 7 (𝑘 ∈ (0..^𝑁) → (𝑘 + 1) ∈ (0...𝑁))
2625adantl 481 . . . . . 6 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 + 1) ∈ (0...𝑁))
27 fveq2 6229 . . . . . . . 8 (𝑥 = (𝑘 + 1) → (𝑄𝑥) = (𝑄‘(𝑘 + 1)))
28 fveq2 6229 . . . . . . . 8 (𝑥 = (𝑘 + 1) → (𝑃𝑥) = (𝑃‘(𝑘 + 1)))
2927, 28eqeq12d 2666 . . . . . . 7 (𝑥 = (𝑘 + 1) → ((𝑄𝑥) = (𝑃𝑥) ↔ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))))
3029rspcv 3336 . . . . . 6 ((𝑘 + 1) ∈ (0...𝑁) → (∀𝑥 ∈ (0...𝑁)(𝑄𝑥) = (𝑃𝑥) → (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))))
3126, 30syl 17 . . . . 5 ((𝜑𝑘 ∈ (0..^𝑁)) → (∀𝑥 ∈ (0...𝑁)(𝑄𝑥) = (𝑃𝑥) → (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))))
3231imp 444 . . . 4 (((𝜑𝑘 ∈ (0..^𝑁)) ∧ ∀𝑥 ∈ (0...𝑁)(𝑄𝑥) = (𝑃𝑥)) → (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)))
3312adantr 480 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
3413fveq1i 6230 . . . . . . . 8 (𝐻𝑘) = ((𝐹 ∪ {⟨𝑁, 𝐵⟩})‘𝑘)
35 fzonel 12522 . . . . . . . . . . . . . 14 ¬ 𝑁 ∈ (0..^𝑁)
36 eleq1 2718 . . . . . . . . . . . . . 14 (𝑁 = 𝑘 → (𝑁 ∈ (0..^𝑁) ↔ 𝑘 ∈ (0..^𝑁)))
3735, 36mtbii 315 . . . . . . . . . . . . 13 (𝑁 = 𝑘 → ¬ 𝑘 ∈ (0..^𝑁))
3837a1i 11 . . . . . . . . . . . 12 (𝜑 → (𝑁 = 𝑘 → ¬ 𝑘 ∈ (0..^𝑁)))
3938con2d 129 . . . . . . . . . . 11 (𝜑 → (𝑘 ∈ (0..^𝑁) → ¬ 𝑁 = 𝑘))
4039imp 444 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0..^𝑁)) → ¬ 𝑁 = 𝑘)
4140neqned 2830 . . . . . . . . 9 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑁𝑘)
42 fvunsn 6486 . . . . . . . . 9 (𝑁𝑘 → ((𝐹 ∪ {⟨𝑁, 𝐵⟩})‘𝑘) = (𝐹𝑘))
4341, 42syl 17 . . . . . . . 8 ((𝜑𝑘 ∈ (0..^𝑁)) → ((𝐹 ∪ {⟨𝑁, 𝐵⟩})‘𝑘) = (𝐹𝑘))
4434, 43syl5eq 2697 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝐻𝑘) = (𝐹𝑘))
4533, 44fveq12d 6235 . . . . . 6 ((𝜑𝑘 ∈ (0..^𝑁)) → ((iEdg‘𝑆)‘(𝐻𝑘)) = ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘(𝐹𝑘)))
469oveq2i 6701 . . . . . . . . . . . . . . . 16 (0..^𝑁) = (0..^(#‘𝐹))
4746eleq2i 2722 . . . . . . . . . . . . . . 15 (𝑘 ∈ (0..^𝑁) ↔ 𝑘 ∈ (0..^(#‘𝐹)))
482wlkf 26566 . . . . . . . . . . . . . . . . 17 (𝐹(Walks‘𝐺)𝑃𝐹 ∈ Word dom 𝐼)
498, 48syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐹 ∈ Word dom 𝐼)
50 wrdsymbcl 13350 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ Word dom 𝐼𝑘 ∈ (0..^(#‘𝐹))) → (𝐹𝑘) ∈ dom 𝐼)
5150ex 449 . . . . . . . . . . . . . . . 16 (𝐹 ∈ Word dom 𝐼 → (𝑘 ∈ (0..^(#‘𝐹)) → (𝐹𝑘) ∈ dom 𝐼))
5249, 51syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑘 ∈ (0..^(#‘𝐹)) → (𝐹𝑘) ∈ dom 𝐼))
5347, 52syl5bi 232 . . . . . . . . . . . . . 14 (𝜑 → (𝑘 ∈ (0..^𝑁) → (𝐹𝑘) ∈ dom 𝐼))
5453imp 444 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝐹𝑘) ∈ dom 𝐼)
55 eleq1 2718 . . . . . . . . . . . . 13 (𝐵 = (𝐹𝑘) → (𝐵 ∈ dom 𝐼 ↔ (𝐹𝑘) ∈ dom 𝐼))
5654, 55syl5ibrcom 237 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝐵 = (𝐹𝑘) → 𝐵 ∈ dom 𝐼))
5756con3d 148 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (0..^𝑁)) → (¬ 𝐵 ∈ dom 𝐼 → ¬ 𝐵 = (𝐹𝑘)))
5857ex 449 . . . . . . . . . 10 (𝜑 → (𝑘 ∈ (0..^𝑁) → (¬ 𝐵 ∈ dom 𝐼 → ¬ 𝐵 = (𝐹𝑘))))
597, 58mpid 44 . . . . . . . . 9 (𝜑 → (𝑘 ∈ (0..^𝑁) → ¬ 𝐵 = (𝐹𝑘)))
6059imp 444 . . . . . . . 8 ((𝜑𝑘 ∈ (0..^𝑁)) → ¬ 𝐵 = (𝐹𝑘))
6160neqned 2830 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝐵 ≠ (𝐹𝑘))
62 fvunsn 6486 . . . . . . 7 (𝐵 ≠ (𝐹𝑘) → ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘(𝐹𝑘)) = (𝐼‘(𝐹𝑘)))
6361, 62syl 17 . . . . . 6 ((𝜑𝑘 ∈ (0..^𝑁)) → ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘(𝐹𝑘)) = (𝐼‘(𝐹𝑘)))
6445, 63eqtrd 2685 . . . . 5 ((𝜑𝑘 ∈ (0..^𝑁)) → ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘)))
6564adantr 480 . . . 4 (((𝜑𝑘 ∈ (0..^𝑁)) ∧ ∀𝑥 ∈ (0...𝑁)(𝑄𝑥) = (𝑃𝑥)) → ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘)))
6624, 32, 653jca 1261 . . 3 (((𝜑𝑘 ∈ (0..^𝑁)) ∧ ∀𝑥 ∈ (0...𝑁)(𝑄𝑥) = (𝑃𝑥)) → ((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))))
6716, 66mpidan 705 . 2 ((𝜑𝑘 ∈ (0..^𝑁)) → ((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))))
6867ralrimiva 2995 1 (𝜑 → ∀𝑘 ∈ (0..^𝑁)((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∀wral 2941  Vcvv 3231   ∪ cun 3605   ⊆ wss 3607  {csn 4210  {cpr 4212  ⟨cop 4216   class class class wbr 4685  dom cdm 5143  Fun wfun 5920  ‘cfv 5926  (class class class)co 6690  Fincfn 7997  0cc0 9974  1c1 9975   + caddc 9977  ...cfz 12364  ..^cfzo 12504  #chash 13157  Word cword 13323  Vtxcvtx 25919  iEdgciedg 25920  Edgcedg 25984  Walkscwlks 26548 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-wlks 26551 This theorem is referenced by:  wlkp1lem8  26633
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