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Theorem clwlknf1oclwwlknlem1 29067
Description: Lemma 1 for clwlknf1oclwwlkn 29070. (Contributed by AV, 26-May-2022.) (Revised by AV, 1-Nov-2022.)
Assertion
Ref Expression
clwlknf1oclwwlknlem1 ((𝐢 ∈ (ClWalksβ€˜πΊ) ∧ 1 ≀ (β™―β€˜(1st β€˜πΆ))) β†’ (β™―β€˜((2nd β€˜πΆ) prefix ((β™―β€˜(2nd β€˜πΆ)) βˆ’ 1))) = (β™―β€˜(1st β€˜πΆ)))

Proof of Theorem clwlknf1oclwwlknlem1
StepHypRef Expression
1 clwlkwlk 28765 . . 3 (𝐢 ∈ (ClWalksβ€˜πΊ) β†’ 𝐢 ∈ (Walksβ€˜πΊ))
2 wlkcpr 28619 . . . 4 (𝐢 ∈ (Walksβ€˜πΊ) ↔ (1st β€˜πΆ)(Walksβ€˜πΊ)(2nd β€˜πΆ))
3 eqid 2737 . . . . . . . 8 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
43wlkpwrd 28607 . . . . . . 7 ((1st β€˜πΆ)(Walksβ€˜πΊ)(2nd β€˜πΆ) β†’ (2nd β€˜πΆ) ∈ Word (Vtxβ€˜πΊ))
5 lencl 14428 . . . . . . . . 9 ((2nd β€˜πΆ) ∈ Word (Vtxβ€˜πΊ) β†’ (β™―β€˜(2nd β€˜πΆ)) ∈ β„•0)
64, 5syl 17 . . . . . . . 8 ((1st β€˜πΆ)(Walksβ€˜πΊ)(2nd β€˜πΆ) β†’ (β™―β€˜(2nd β€˜πΆ)) ∈ β„•0)
7 wlklenvm1 28612 . . . . . . . . . . 11 ((1st β€˜πΆ)(Walksβ€˜πΊ)(2nd β€˜πΆ) β†’ (β™―β€˜(1st β€˜πΆ)) = ((β™―β€˜(2nd β€˜πΆ)) βˆ’ 1))
87breq2d 5122 . . . . . . . . . 10 ((1st β€˜πΆ)(Walksβ€˜πΊ)(2nd β€˜πΆ) β†’ (1 ≀ (β™―β€˜(1st β€˜πΆ)) ↔ 1 ≀ ((β™―β€˜(2nd β€˜πΆ)) βˆ’ 1)))
9 1red 11163 . . . . . . . . . . . . 13 ((β™―β€˜(2nd β€˜πΆ)) ∈ β„•0 β†’ 1 ∈ ℝ)
10 nn0re 12429 . . . . . . . . . . . . 13 ((β™―β€˜(2nd β€˜πΆ)) ∈ β„•0 β†’ (β™―β€˜(2nd β€˜πΆ)) ∈ ℝ)
119, 9, 10leaddsub2d 11764 . . . . . . . . . . . 12 ((β™―β€˜(2nd β€˜πΆ)) ∈ β„•0 β†’ ((1 + 1) ≀ (β™―β€˜(2nd β€˜πΆ)) ↔ 1 ≀ ((β™―β€˜(2nd β€˜πΆ)) βˆ’ 1)))
12 1p1e2 12285 . . . . . . . . . . . . . 14 (1 + 1) = 2
1312breq1i 5117 . . . . . . . . . . . . 13 ((1 + 1) ≀ (β™―β€˜(2nd β€˜πΆ)) ↔ 2 ≀ (β™―β€˜(2nd β€˜πΆ)))
1413biimpi 215 . . . . . . . . . . . 12 ((1 + 1) ≀ (β™―β€˜(2nd β€˜πΆ)) β†’ 2 ≀ (β™―β€˜(2nd β€˜πΆ)))
1511, 14syl6bir 254 . . . . . . . . . . 11 ((β™―β€˜(2nd β€˜πΆ)) ∈ β„•0 β†’ (1 ≀ ((β™―β€˜(2nd β€˜πΆ)) βˆ’ 1) β†’ 2 ≀ (β™―β€˜(2nd β€˜πΆ))))
164, 5, 153syl 18 . . . . . . . . . 10 ((1st β€˜πΆ)(Walksβ€˜πΊ)(2nd β€˜πΆ) β†’ (1 ≀ ((β™―β€˜(2nd β€˜πΆ)) βˆ’ 1) β†’ 2 ≀ (β™―β€˜(2nd β€˜πΆ))))
178, 16sylbid 239 . . . . . . . . 9 ((1st β€˜πΆ)(Walksβ€˜πΊ)(2nd β€˜πΆ) β†’ (1 ≀ (β™―β€˜(1st β€˜πΆ)) β†’ 2 ≀ (β™―β€˜(2nd β€˜πΆ))))
1817imp 408 . . . . . . . 8 (((1st β€˜πΆ)(Walksβ€˜πΊ)(2nd β€˜πΆ) ∧ 1 ≀ (β™―β€˜(1st β€˜πΆ))) β†’ 2 ≀ (β™―β€˜(2nd β€˜πΆ)))
19 ige2m1fz 13538 . . . . . . . 8 (((β™―β€˜(2nd β€˜πΆ)) ∈ β„•0 ∧ 2 ≀ (β™―β€˜(2nd β€˜πΆ))) β†’ ((β™―β€˜(2nd β€˜πΆ)) βˆ’ 1) ∈ (0...(β™―β€˜(2nd β€˜πΆ))))
206, 18, 19syl2an2r 684 . . . . . . 7 (((1st β€˜πΆ)(Walksβ€˜πΊ)(2nd β€˜πΆ) ∧ 1 ≀ (β™―β€˜(1st β€˜πΆ))) β†’ ((β™―β€˜(2nd β€˜πΆ)) βˆ’ 1) ∈ (0...(β™―β€˜(2nd β€˜πΆ))))
21 pfxlen 14578 . . . . . . 7 (((2nd β€˜πΆ) ∈ Word (Vtxβ€˜πΊ) ∧ ((β™―β€˜(2nd β€˜πΆ)) βˆ’ 1) ∈ (0...(β™―β€˜(2nd β€˜πΆ)))) β†’ (β™―β€˜((2nd β€˜πΆ) prefix ((β™―β€˜(2nd β€˜πΆ)) βˆ’ 1))) = ((β™―β€˜(2nd β€˜πΆ)) βˆ’ 1))
224, 20, 21syl2an2r 684 . . . . . 6 (((1st β€˜πΆ)(Walksβ€˜πΊ)(2nd β€˜πΆ) ∧ 1 ≀ (β™―β€˜(1st β€˜πΆ))) β†’ (β™―β€˜((2nd β€˜πΆ) prefix ((β™―β€˜(2nd β€˜πΆ)) βˆ’ 1))) = ((β™―β€˜(2nd β€˜πΆ)) βˆ’ 1))
237eqcomd 2743 . . . . . . 7 ((1st β€˜πΆ)(Walksβ€˜πΊ)(2nd β€˜πΆ) β†’ ((β™―β€˜(2nd β€˜πΆ)) βˆ’ 1) = (β™―β€˜(1st β€˜πΆ)))
2423adantr 482 . . . . . 6 (((1st β€˜πΆ)(Walksβ€˜πΊ)(2nd β€˜πΆ) ∧ 1 ≀ (β™―β€˜(1st β€˜πΆ))) β†’ ((β™―β€˜(2nd β€˜πΆ)) βˆ’ 1) = (β™―β€˜(1st β€˜πΆ)))
2522, 24eqtrd 2777 . . . . 5 (((1st β€˜πΆ)(Walksβ€˜πΊ)(2nd β€˜πΆ) ∧ 1 ≀ (β™―β€˜(1st β€˜πΆ))) β†’ (β™―β€˜((2nd β€˜πΆ) prefix ((β™―β€˜(2nd β€˜πΆ)) βˆ’ 1))) = (β™―β€˜(1st β€˜πΆ)))
2625ex 414 . . . 4 ((1st β€˜πΆ)(Walksβ€˜πΊ)(2nd β€˜πΆ) β†’ (1 ≀ (β™―β€˜(1st β€˜πΆ)) β†’ (β™―β€˜((2nd β€˜πΆ) prefix ((β™―β€˜(2nd β€˜πΆ)) βˆ’ 1))) = (β™―β€˜(1st β€˜πΆ))))
272, 26sylbi 216 . . 3 (𝐢 ∈ (Walksβ€˜πΊ) β†’ (1 ≀ (β™―β€˜(1st β€˜πΆ)) β†’ (β™―β€˜((2nd β€˜πΆ) prefix ((β™―β€˜(2nd β€˜πΆ)) βˆ’ 1))) = (β™―β€˜(1st β€˜πΆ))))
281, 27syl 17 . 2 (𝐢 ∈ (ClWalksβ€˜πΊ) β†’ (1 ≀ (β™―β€˜(1st β€˜πΆ)) β†’ (β™―β€˜((2nd β€˜πΆ) prefix ((β™―β€˜(2nd β€˜πΆ)) βˆ’ 1))) = (β™―β€˜(1st β€˜πΆ))))
2928imp 408 1 ((𝐢 ∈ (ClWalksβ€˜πΊ) ∧ 1 ≀ (β™―β€˜(1st β€˜πΆ))) β†’ (β™―β€˜((2nd β€˜πΆ) prefix ((β™―β€˜(2nd β€˜πΆ)) βˆ’ 1))) = (β™―β€˜(1st β€˜πΆ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   class class class wbr 5110  β€˜cfv 6501  (class class class)co 7362  1st c1st 7924  2nd c2nd 7925  0cc0 11058  1c1 11059   + caddc 11061   ≀ cle 11197   βˆ’ cmin 11392  2c2 12215  β„•0cn0 12420  ...cfz 13431  β™―chash 14237  Word cword 14409   prefix cpfx 14565  Vtxcvtx 27989  Walkscwlks 28586  ClWalkscclwlks 28760
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 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9882  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-nn 12161  df-2 12223  df-n0 12421  df-z 12507  df-uz 12771  df-fz 13432  df-fzo 13575  df-hash 14238  df-word 14410  df-substr 14536  df-pfx 14566  df-wlks 28589  df-clwlks 28761
This theorem is referenced by:  clwlknf1oclwwlkn  29070
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