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Theorem wlkp1lem6 26706
Description: Lemma for wlkp1 26709. (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 26705 . . 3 (𝜑 → ∀𝑥 ∈ (0...𝑁)(𝑄𝑥) = (𝑃𝑥))
17 elfzofz 12600 . . . . . . 7 (𝑘 ∈ (0..^𝑁) → 𝑘 ∈ (0...𝑁))
1817adantl 473 . . . . . 6 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑘 ∈ (0...𝑁))
19 fveq2 6304 . . . . . . . 8 (𝑥 = 𝑘 → (𝑄𝑥) = (𝑄𝑘))
20 fveq2 6304 . . . . . . . 8 (𝑥 = 𝑘 → (𝑃𝑥) = (𝑃𝑘))
2119, 20eqeq12d 2739 . . . . . . 7 (𝑥 = 𝑘 → ((𝑄𝑥) = (𝑃𝑥) ↔ (𝑄𝑘) = (𝑃𝑘)))
2221rspcv 3409 . . . . . 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 12680 . . . . . . 7 (𝑘 ∈ (0..^𝑁) → (𝑘 + 1) ∈ (0...𝑁))
2625adantl 473 . . . . . 6 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝑘 + 1) ∈ (0...𝑁))
27 fveq2 6304 . . . . . . . 8 (𝑥 = (𝑘 + 1) → (𝑄𝑥) = (𝑄‘(𝑘 + 1)))
28 fveq2 6304 . . . . . . . 8 (𝑥 = (𝑘 + 1) → (𝑃𝑥) = (𝑃‘(𝑘 + 1)))
2927, 28eqeq12d 2739 . . . . . . 7 (𝑥 = (𝑘 + 1) → ((𝑄𝑥) = (𝑃𝑥) ↔ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1))))
3029rspcv 3409 . . . . . 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 472 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))
3413fveq1i 6305 . . . . . . . 8 (𝐻𝑘) = ((𝐹 ∪ {⟨𝑁, 𝐵⟩})‘𝑘)
35 fzonel 12598 . . . . . . . . . . . . . 14 ¬ 𝑁 ∈ (0..^𝑁)
36 eleq1 2791 . . . . . . . . . . . . . 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 2903 . . . . . . . . 9 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝑁𝑘)
42 fvunsn 6561 . . . . . . . . 9 (𝑁𝑘 → ((𝐹 ∪ {⟨𝑁, 𝐵⟩})‘𝑘) = (𝐹𝑘))
4341, 42syl 17 . . . . . . . 8 ((𝜑𝑘 ∈ (0..^𝑁)) → ((𝐹 ∪ {⟨𝑁, 𝐵⟩})‘𝑘) = (𝐹𝑘))
4434, 43syl5eq 2770 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → (𝐻𝑘) = (𝐹𝑘))
4533, 44fveq12d 6310 . . . . . 6 ((𝜑𝑘 ∈ (0..^𝑁)) → ((iEdg‘𝑆)‘(𝐻𝑘)) = ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘(𝐹𝑘)))
469oveq2i 6776 . . . . . . . . . . . . . . . 16 (0..^𝑁) = (0..^(♯‘𝐹))
4746eleq2i 2795 . . . . . . . . . . . . . . 15 (𝑘 ∈ (0..^𝑁) ↔ 𝑘 ∈ (0..^(♯‘𝐹)))
482wlkf 26641 . . . . . . . . . . . . . . . . 17 (𝐹(Walks‘𝐺)𝑃𝐹 ∈ Word dom 𝐼)
498, 48syl 17 . . . . . . . . . . . . . . . 16 (𝜑𝐹 ∈ Word dom 𝐼)
50 wrdsymbcl 13425 . . . . . . . . . . . . . . . . 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 2791 . . . . . . . . . . . . 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 2903 . . . . . . 7 ((𝜑𝑘 ∈ (0..^𝑁)) → 𝐵 ≠ (𝐹𝑘))
62 fvunsn 6561 . . . . . . 7 (𝐵 ≠ (𝐹𝑘) → ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘(𝐹𝑘)) = (𝐼‘(𝐹𝑘)))
6361, 62syl 17 . . . . . 6 ((𝜑𝑘 ∈ (0..^𝑁)) → ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘(𝐹𝑘)) = (𝐼‘(𝐹𝑘)))
6445, 63eqtrd 2758 . . . . 5 ((𝜑𝑘 ∈ (0..^𝑁)) → ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘)))
6564adantr 472 . . . 4 (((𝜑𝑘 ∈ (0..^𝑁)) ∧ ∀𝑥 ∈ (0...𝑁)(𝑄𝑥) = (𝑃𝑥)) → ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘)))
6624, 32, 653jca 1379 . . 3 (((𝜑𝑘 ∈ (0..^𝑁)) ∧ ∀𝑥 ∈ (0...𝑁)(𝑄𝑥) = (𝑃𝑥)) → ((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))))
6716, 66mpidan 707 . 2 ((𝜑𝑘 ∈ (0..^𝑁)) → ((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))))
6867ralrimiva 3068 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 1072   = wceq 1596  wcel 2103  wne 2896  wral 3014  Vcvv 3304  cun 3678  wss 3680  {csn 4285  {cpr 4287  cop 4291   class class class wbr 4760  dom cdm 5218  Fun wfun 5995  cfv 6001  (class class class)co 6765  Fincfn 8072  0cc0 10049  1c1 10050   + caddc 10052  ...cfz 12440  ..^cfzo 12580  chash 13232  Word cword 13398  Vtxcvtx 25994  iEdgciedg 25995  Edgcedg 26059  Walkscwlks 26623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-cnex 10105  ax-resscn 10106  ax-1cn 10107  ax-icn 10108  ax-addcl 10109  ax-addrcl 10110  ax-mulcl 10111  ax-mulrcl 10112  ax-mulcom 10113  ax-addass 10114  ax-mulass 10115  ax-distr 10116  ax-i2m1 10117  ax-1ne0 10118  ax-1rid 10119  ax-rnegex 10120  ax-rrecex 10121  ax-cnre 10122  ax-pre-lttri 10123  ax-pre-lttrn 10124  ax-pre-ltadd 10125  ax-pre-mulgt0 10126
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1051  df-3or 1073  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-nel 3000  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-pss 3696  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-tp 4290  df-op 4292  df-uni 4545  df-int 4584  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-tr 4861  df-id 5128  df-eprel 5133  df-po 5139  df-so 5140  df-fr 5177  df-we 5179  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-pred 5793  df-ord 5839  df-on 5840  df-lim 5841  df-suc 5842  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-ov 6768  df-oprab 6769  df-mpt2 6770  df-om 7183  df-1st 7285  df-2nd 7286  df-wrecs 7527  df-recs 7588  df-rdg 7626  df-1o 7680  df-er 7862  df-map 7976  df-pm 7977  df-en 8073  df-dom 8074  df-sdom 8075  df-fin 8076  df-card 8878  df-pnf 10189  df-mnf 10190  df-xr 10191  df-ltxr 10192  df-le 10193  df-sub 10381  df-neg 10382  df-nn 11134  df-n0 11406  df-z 11491  df-uz 11801  df-fz 12441  df-fzo 12581  df-hash 13233  df-word 13406  df-wlks 26626
This theorem is referenced by:  wlkp1lem8  26708
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