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Theorem eleclclwwlknlem2 29054
Description: Lemma 2 for eleclclwwlkn 29069. (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by AV, 30-Apr-2021.)
Hypothesis
Ref Expression
erclwwlkn1.w 𝑊 = (𝑁 ClWWalksN 𝐺)
Assertion
Ref Expression
eleclclwwlknlem2 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))
Distinct variable groups:   𝑚,𝑛,𝐺   𝑚,𝑁,𝑛   𝑚,𝑋,𝑛   𝑚,𝑌,𝑛   𝑘,𝑚,𝑛   𝑥,𝑚,𝑛
Allowed substitution hints:   𝐺(𝑥,𝑘)   𝑁(𝑥,𝑘)   𝑊(𝑥,𝑘,𝑚,𝑛)   𝑋(𝑥,𝑘)   𝑌(𝑥,𝑘)

Proof of Theorem eleclclwwlknlem2
StepHypRef Expression
1 simpl 484 . . . . 5 ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → 𝑘 ∈ (0...𝑁))
21anim1i 616 . . . 4 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑘 ∈ (0...𝑁) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)))
32adantr 482 . . 3 ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) ∧ ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚)) → (𝑘 ∈ (0...𝑁) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)))
4 simpr 486 . . . . 5 ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → 𝑋 = (𝑥 cyclShift 𝑘))
54adantr 482 . . . 4 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → 𝑋 = (𝑥 cyclShift 𝑘))
65anim1i 616 . . 3 ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) ∧ ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚)) → (𝑋 = (𝑥 cyclShift 𝑘) ∧ ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚)))
7 erclwwlkn1.w . . . 4 𝑊 = (𝑁 ClWWalksN 𝐺)
87eleclclwwlknlem1 29053 . . 3 ((𝑘 ∈ (0...𝑁) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → ((𝑋 = (𝑥 cyclShift 𝑘) ∧ ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))
93, 6, 8sylc 65 . 2 ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) ∧ ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))
10 eqid 2733 . . . . . . . . . . . 12 (Vtx‘𝐺) = (Vtx‘𝐺)
1110clwwlknbp 29028 . . . . . . . . . . 11 (𝑥 ∈ (𝑁 ClWWalksN 𝐺) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
1211, 7eleq2s 2852 . . . . . . . . . 10 (𝑥 ∈ 𝑊 → (𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁))
13 fznn0sub2 13557 . . . . . . . . . . . 12 (𝑘 ∈ (0...𝑁) → (𝑁 − 𝑘) ∈ (0...𝑁))
14 oveq1 7368 . . . . . . . . . . . . 13 ((♯‘𝑥) = 𝑁 → ((♯‘𝑥) − 𝑘) = (𝑁 − 𝑘))
1514eleq1d 2819 . . . . . . . . . . . 12 ((♯‘𝑥) = 𝑁 → (((♯‘𝑥) − 𝑘) ∈ (0...𝑁) ↔ (𝑁 − 𝑘) ∈ (0...𝑁)))
1613, 15syl5ibr 246 . . . . . . . . . . 11 ((♯‘𝑥) = 𝑁 → (𝑘 ∈ (0...𝑁) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁)))
1716adantl 483 . . . . . . . . . 10 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑘 ∈ (0...𝑁) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁)))
1812, 17syl 17 . . . . . . . . 9 (𝑥 ∈ 𝑊 → (𝑘 ∈ (0...𝑁) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁)))
1918adantl 483 . . . . . . . 8 ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → (𝑘 ∈ (0...𝑁) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁)))
2019com12 32 . . . . . . 7 (𝑘 ∈ (0...𝑁) → ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁)))
2120adantr 482 . . . . . 6 ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁)))
2221imp 408 . . . . 5 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁))
2322adantr 482 . . . 4 ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) → ((♯‘𝑥) − 𝑘) ∈ (0...𝑁))
24 simpr 486 . . . . . 6 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊))
2524ancomd 463 . . . . 5 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑥 ∈ 𝑊 ∧ 𝑋 ∈ 𝑊))
2625adantr 482 . . . 4 ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) → (𝑥 ∈ 𝑊 ∧ 𝑋 ∈ 𝑊))
2723, 26jca 513 . . 3 ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) → (((♯‘𝑥) − 𝑘) ∈ (0...𝑁) ∧ (𝑥 ∈ 𝑊 ∧ 𝑋 ∈ 𝑊)))
28 simpll 766 . . . . . . . . . . . . 13 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑘 ∈ (0...𝑁)) → 𝑥 ∈ Word (Vtx‘𝐺))
29 oveq2 7369 . . . . . . . . . . . . . . . . 17 (𝑁 = (♯‘𝑥) → (0...𝑁) = (0...(♯‘𝑥)))
3029eleq2d 2820 . . . . . . . . . . . . . . . 16 (𝑁 = (♯‘𝑥) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(♯‘𝑥))))
3130eqcoms 2741 . . . . . . . . . . . . . . 15 ((♯‘𝑥) = 𝑁 → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(♯‘𝑥))))
3231adantl 483 . . . . . . . . . . . . . 14 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑘 ∈ (0...𝑁) ↔ 𝑘 ∈ (0...(♯‘𝑥))))
3332biimpa 478 . . . . . . . . . . . . 13 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ (0...(♯‘𝑥)))
3428, 33jca 513 . . . . . . . . . . . 12 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) ∧ 𝑘 ∈ (0...𝑁)) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥))))
3534ex 414 . . . . . . . . . . 11 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑥) = 𝑁) → (𝑘 ∈ (0...𝑁) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥)))))
3612, 35syl 17 . . . . . . . . . 10 (𝑥 ∈ 𝑊 → (𝑘 ∈ (0...𝑁) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥)))))
3736adantl 483 . . . . . . . . 9 ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → (𝑘 ∈ (0...𝑁) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥)))))
3837com12 32 . . . . . . . 8 (𝑘 ∈ (0...𝑁) → ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥)))))
3938adantr 482 . . . . . . 7 ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → ((𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥)))))
4039imp 408 . . . . . 6 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥))))
414eqcomd 2739 . . . . . . 7 ((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) → (𝑥 cyclShift 𝑘) = 𝑋)
4241adantr 482 . . . . . 6 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑥 cyclShift 𝑘) = 𝑋)
43 oveq1 7368 . . . . . . . 8 (𝑋 = (𝑥 cyclShift 𝑘) → (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) = ((𝑥 cyclShift 𝑘) cyclShift ((♯‘𝑥) − 𝑘)))
4443eqcoms 2741 . . . . . . 7 ((𝑥 cyclShift 𝑘) = 𝑋 → (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) = ((𝑥 cyclShift 𝑘) cyclShift ((♯‘𝑥) − 𝑘)))
45 elfzelz 13450 . . . . . . . 8 (𝑘 ∈ (0...(♯‘𝑥)) → 𝑘 ∈ â„€)
46 2cshwid 14711 . . . . . . . 8 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ â„€) → ((𝑥 cyclShift 𝑘) cyclShift ((♯‘𝑥) − 𝑘)) = 𝑥)
4745, 46sylan2 594 . . . . . . 7 ((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥))) → ((𝑥 cyclShift 𝑘) cyclShift ((♯‘𝑥) − 𝑘)) = 𝑥)
4844, 47sylan9eqr 2795 . . . . . 6 (((𝑥 ∈ Word (Vtx‘𝐺) ∧ 𝑘 ∈ (0...(♯‘𝑥))) ∧ (𝑥 cyclShift 𝑘) = 𝑋) → (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) = 𝑥)
4940, 42, 48syl2anc 585 . . . . 5 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) = 𝑥)
5049eqcomd 2739 . . . 4 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → 𝑥 = (𝑋 cyclShift ((♯‘𝑥) − 𝑘)))
5150anim1i 616 . . 3 ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) → (𝑥 = (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))
527eleclclwwlknlem1 29053 . . 3 ((((♯‘𝑥) − 𝑘) ∈ (0...𝑁) ∧ (𝑥 ∈ 𝑊 ∧ 𝑋 ∈ 𝑊)) → ((𝑥 = (𝑋 cyclShift ((♯‘𝑥) − 𝑘)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚)))
5327, 51, 52sylc 65 . 2 ((((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)) → ∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚))
549, 53impbida 800 1 (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  âˆƒwrex 3070  â€˜cfv 6500  (class class class)co 7361  0cc0 11059   − cmin 11393  â„€cz 12507  ...cfz 13433  â™¯chash 14239  Word cword 14411   cyclShift ccsh 14685  Vtxcvtx 27996   ClWWalksN cclwwlkn 29017
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-sup 9386  df-inf 9387  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-n0 12422  df-z 12508  df-uz 12772  df-rp 12924  df-fz 13434  df-fzo 13577  df-fl 13706  df-mod 13784  df-hash 14240  df-word 14412  df-concat 14468  df-substr 14538  df-pfx 14568  df-csh 14686  df-clwwlk 28975  df-clwwlkn 29018
This theorem is referenced by:  eleclclwwlkn  29069
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