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Theorem wlkp1lem6 29710
Description: Lemma for wlkp1 29713. (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 (𝜑𝐵𝑊)
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)   𝑊(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 (𝜑𝐵𝑊)
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 29709 . . 3 (𝜑 → ∀𝑥 ∈ (0...𝑁)(𝑄𝑥) = (𝑃𝑥))
17 elfzofz 13711 . . . . . . 7 (𝑘 ∈ (0..^𝑁) → 𝑘 ∈ (0...𝑁))
1817adantl 481 . . . . . 6 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ (0...𝑁))
19 fveq2 6906 . . . . . . . 8 (𝑥 = 𝑘 → (𝑄𝑥) = (𝑄𝑘))
20 fveq2 6906 . . . . . . . 8 (𝑥 = 𝑘 → (𝑃𝑥) = (𝑃𝑘))
2119, 20eqeq12d 2750 . . . . . . 7 (𝑥 = 𝑘 → ((𝑄𝑥) = (𝑃𝑥) ↔ (𝑄𝑘) = (𝑃𝑘)))
2221rspcv 3617 . . . . . 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 406 . . . 4 (((𝜑𝑘 ∈ (0..^𝑁)) ∧ ∀𝑥 ∈ (0...𝑁)(𝑄𝑥) = (𝑃𝑥)) → (𝑄𝑘) = (𝑃𝑘))
25 fzofzp1 13799 . . . . . . 7 (𝑘 ∈ (0..^𝑁) → (𝑘 + 1) ∈ (0...𝑁))
2625adantl 481 . . . . . 6 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 + 1) ∈ (0...𝑁))
27 fveq2 6906 . . . . . . . 8 (𝑥 = (𝑘 + 1) → (𝑄𝑥) = (𝑄‘(𝑘 + 1)))
28 fveq2 6906 . . . . . . . 8 (𝑥 = (𝑘 + 1) → (𝑃𝑥) = (𝑃‘(𝑘 + 1)))
2927, 28eqeq12d 2750 . . . . . . 7 (𝑥 = (𝑘 + 1) → ((𝑄𝑥) = (𝑃𝑥) ↔ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))))
3029rspcv 3617 . . . . . 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 406 . . . 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 6907 . . . . . . . 8 (𝐻𝑘) = ((𝐹 ∪ {⟨𝑁, 𝐵⟩})‘𝑘)
35 fzonel 13709 . . . . . . . . . . . . . 14 ¬ 𝑁 ∈ (0..^𝑁)
36 eleq1 2826 . . . . . . . . . . . . . 14 (𝑁 = 𝑘 → (𝑁 ∈ (0..^𝑁) ↔ 𝑘 ∈ (0..^𝑁)))
3735, 36mtbii 326 . . . . . . . . . . . . 13 (𝑁 = 𝑘 → ¬ 𝑘 ∈ (0..^𝑁))
3837a1i 11 . . . . . . . . . . . 12 (𝜑 → (𝑁 = 𝑘 → ¬ 𝑘 ∈ (0..^𝑁)))
3938con2d 134 . . . . . . . . . . 11 (𝜑 → (𝑘 ∈ (0..^𝑁) → ¬ 𝑁 = 𝑘))
4039imp 406 . . . . . . . . . 10 ((𝜑𝑘 ∈ (0..^𝑁)) → ¬ 𝑁 = 𝑘)
4140neqned 2944 . . . . . . . . 9 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑁𝑘)
42 fvunsn 7198 . . . . . . . . 9 (𝑁𝑘 → ((𝐹 ∪ {⟨𝑁, 𝐵⟩})‘𝑘) = (𝐹𝑘))
4341, 42syl 17 . . . . . . . 8 ((𝜑𝑘 ∈ (0..^𝑁)) → ((𝐹 ∪ {⟨𝑁, 𝐵⟩})‘𝑘) = (𝐹𝑘))
4434, 43eqtrid 2786 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝐻𝑘) = (𝐹𝑘))
4533, 44fveq12d 6913 . . . . . 6 ((𝜑𝑘 ∈ (0..^𝑁)) → ((iEdg‘𝑆)‘(𝐻𝑘)) = ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘(𝐹𝑘)))
469oveq2i 7441 . . . . . . . . . . . . . . . 16 (0..^𝑁) = (0..^(♯‘𝐹))
4746eleq2i 2830 . . . . . . . . . . . . . . 15 (𝑘 ∈ (0..^𝑁) ↔ 𝑘 ∈ (0..^(♯‘𝐹)))
482wlkf 29646 . . . . . . . . . . . . . . . . 17 (𝐹(Walks‘𝐺)𝑃𝐹 ∈ Word dom 𝐼)
498, 48syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐹 ∈ Word dom 𝐼)
50 wrdsymbcl 14561 . . . . . . . . . . . . . . . . 17 ((𝐹 ∈ Word dom 𝐼𝑘 ∈ (0..^(♯‘𝐹))) → (𝐹𝑘) ∈ dom 𝐼)
5150ex 412 . . . . . . . . . . . . . . . 16 (𝐹 ∈ Word dom 𝐼 → (𝑘 ∈ (0..^(♯‘𝐹)) → (𝐹𝑘) ∈ dom 𝐼))
5249, 51syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑘 ∈ (0..^(♯‘𝐹)) → (𝐹𝑘) ∈ dom 𝐼))
5347, 52biimtrid 242 . . . . . . . . . . . . . 14 (𝜑 → (𝑘 ∈ (0..^𝑁) → (𝐹𝑘) ∈ dom 𝐼))
5453imp 406 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝐹𝑘) ∈ dom 𝐼)
55 eleq1 2826 . . . . . . . . . . . . 13 (𝐵 = (𝐹𝑘) → (𝐵 ∈ dom 𝐼 ↔ (𝐹𝑘) ∈ dom 𝐼))
5654, 55syl5ibrcom 247 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝐵 = (𝐹𝑘) → 𝐵 ∈ dom 𝐼))
5756con3d 152 . . . . . . . . . . 11 ((𝜑𝑘 ∈ (0..^𝑁)) → (¬ 𝐵 ∈ dom 𝐼 → ¬ 𝐵 = (𝐹𝑘)))
5857ex 412 . . . . . . . . . 10 (𝜑 → (𝑘 ∈ (0..^𝑁) → (¬ 𝐵 ∈ dom 𝐼 → ¬ 𝐵 = (𝐹𝑘))))
597, 58mpid 44 . . . . . . . . 9 (𝜑 → (𝑘 ∈ (0..^𝑁) → ¬ 𝐵 = (𝐹𝑘)))
6059imp 406 . . . . . . . 8 ((𝜑𝑘 ∈ (0..^𝑁)) → ¬ 𝐵 = (𝐹𝑘))
6160neqned 2944 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝐵 ≠ (𝐹𝑘))
62 fvunsn 7198 . . . . . . 7 (𝐵 ≠ (𝐹𝑘) → ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘(𝐹𝑘)) = (𝐼‘(𝐹𝑘)))
6361, 62syl 17 . . . . . 6 ((𝜑𝑘 ∈ (0..^𝑁)) → ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘(𝐹𝑘)) = (𝐼‘(𝐹𝑘)))
6445, 63eqtrd 2774 . . . . 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 1127 . . 3 (((𝜑𝑘 ∈ (0..^𝑁)) ∧ ∀𝑥 ∈ (0...𝑁)(𝑄𝑥) = (𝑃𝑥)) → ((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))))
6716, 66mpidan 689 . 2 ((𝜑𝑘 ∈ (0..^𝑁)) → ((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))))
6867ralrimiva 3143 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 395  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wral 3058  cun 3960  wss 3962  {csn 4630  {cpr 4632  cop 4636   class class class wbr 5147  dom cdm 5688  Fun wfun 6556  cfv 6562  (class class class)co 7430  Fincfn 8983  0cc0 11152  1c1 11153   + caddc 11155  ...cfz 13543  ..^cfzo 13690  chash 14365  Word cword 14548  Vtxcvtx 29027  iEdgciedg 29028  Edgcedg 29078  Walkscwlks 29628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-n0 12524  df-z 12611  df-uz 12876  df-fz 13544  df-fzo 13691  df-hash 14366  df-word 14549  df-wlks 29631
This theorem is referenced by:  wlkp1lem8  29712
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