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Theorem dlwwlknondlwlknonf1olem1 29350
Description: Lemma 1 for dlwwlknondlwlknonf1o 29351. (Contributed by AV, 29-May-2022.) (Revised by AV, 1-Nov-2022.)
Assertion
Ref Expression
dlwwlknondlwlknonf1olem1 (((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ 𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))β€˜(𝑁 βˆ’ 2)) = ((2nd β€˜π‘)β€˜(𝑁 βˆ’ 2)))

Proof of Theorem dlwwlknondlwlknonf1olem1
StepHypRef Expression
1 clwlkwlk 28765 . . . . 5 (𝑐 ∈ (ClWalksβ€˜πΊ) β†’ 𝑐 ∈ (Walksβ€˜πΊ))
2 wlkcpr 28619 . . . . 5 (𝑐 ∈ (Walksβ€˜πΊ) ↔ (1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘))
31, 2sylib 217 . . . 4 (𝑐 ∈ (ClWalksβ€˜πΊ) β†’ (1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘))
4 eqid 2737 . . . . 5 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
54wlkpwrd 28607 . . . 4 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ (2nd β€˜π‘) ∈ Word (Vtxβ€˜πΊ))
63, 5syl 17 . . 3 (𝑐 ∈ (ClWalksβ€˜πΊ) β†’ (2nd β€˜π‘) ∈ Word (Vtxβ€˜πΊ))
763ad2ant2 1135 . 2 (((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ 𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (2nd β€˜π‘) ∈ Word (Vtxβ€˜πΊ))
8 eluzge2nn0 12819 . . . . . . 7 (𝑁 ∈ (β„€β‰₯β€˜2) β†’ 𝑁 ∈ β„•0)
983ad2ant3 1136 . . . . . 6 (((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ 𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ 𝑁 ∈ β„•0)
10 eleq1 2826 . . . . . . 7 ((β™―β€˜(1st β€˜π‘)) = 𝑁 β†’ ((β™―β€˜(1st β€˜π‘)) ∈ β„•0 ↔ 𝑁 ∈ β„•0))
11103ad2ant1 1134 . . . . . 6 (((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ 𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜(1st β€˜π‘)) ∈ β„•0 ↔ 𝑁 ∈ β„•0))
129, 11mpbird 257 . . . . 5 (((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ 𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (β™―β€˜(1st β€˜π‘)) ∈ β„•0)
13 nn0fz0 13546 . . . . 5 ((β™―β€˜(1st β€˜π‘)) ∈ β„•0 ↔ (β™―β€˜(1st β€˜π‘)) ∈ (0...(β™―β€˜(1st β€˜π‘))))
1412, 13sylib 217 . . . 4 (((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ 𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (β™―β€˜(1st β€˜π‘)) ∈ (0...(β™―β€˜(1st β€˜π‘))))
15 fzelp1 13500 . . . 4 ((β™―β€˜(1st β€˜π‘)) ∈ (0...(β™―β€˜(1st β€˜π‘))) β†’ (β™―β€˜(1st β€˜π‘)) ∈ (0...((β™―β€˜(1st β€˜π‘)) + 1)))
1614, 15syl 17 . . 3 (((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ 𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (β™―β€˜(1st β€˜π‘)) ∈ (0...((β™―β€˜(1st β€˜π‘)) + 1)))
17 wlklenvp1 28608 . . . . . . . 8 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ (β™―β€˜(2nd β€˜π‘)) = ((β™―β€˜(1st β€˜π‘)) + 1))
1817eqcomd 2743 . . . . . . 7 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ ((β™―β€˜(1st β€˜π‘)) + 1) = (β™―β€˜(2nd β€˜π‘)))
193, 18syl 17 . . . . . 6 (𝑐 ∈ (ClWalksβ€˜πΊ) β†’ ((β™―β€˜(1st β€˜π‘)) + 1) = (β™―β€˜(2nd β€˜π‘)))
2019oveq2d 7378 . . . . 5 (𝑐 ∈ (ClWalksβ€˜πΊ) β†’ (0...((β™―β€˜(1st β€˜π‘)) + 1)) = (0...(β™―β€˜(2nd β€˜π‘))))
2120eleq2d 2824 . . . 4 (𝑐 ∈ (ClWalksβ€˜πΊ) β†’ ((β™―β€˜(1st β€˜π‘)) ∈ (0...((β™―β€˜(1st β€˜π‘)) + 1)) ↔ (β™―β€˜(1st β€˜π‘)) ∈ (0...(β™―β€˜(2nd β€˜π‘)))))
22213ad2ant2 1135 . . 3 (((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ 𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ ((β™―β€˜(1st β€˜π‘)) ∈ (0...((β™―β€˜(1st β€˜π‘)) + 1)) ↔ (β™―β€˜(1st β€˜π‘)) ∈ (0...(β™―β€˜(2nd β€˜π‘)))))
2316, 22mpbid 231 . 2 (((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ 𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (β™―β€˜(1st β€˜π‘)) ∈ (0...(β™―β€˜(2nd β€˜π‘))))
24 2nn 12233 . . . . . . 7 2 ∈ β„•
2524a1i 11 . . . . . 6 (𝑁 ∈ (β„€β‰₯β€˜2) β†’ 2 ∈ β„•)
26 eluz2nn 12816 . . . . . 6 (𝑁 ∈ (β„€β‰₯β€˜2) β†’ 𝑁 ∈ β„•)
27 eluzle 12783 . . . . . 6 (𝑁 ∈ (β„€β‰₯β€˜2) β†’ 2 ≀ 𝑁)
28 elfz1b 13517 . . . . . 6 (2 ∈ (1...𝑁) ↔ (2 ∈ β„• ∧ 𝑁 ∈ β„• ∧ 2 ≀ 𝑁))
2925, 26, 27, 28syl3anbrc 1344 . . . . 5 (𝑁 ∈ (β„€β‰₯β€˜2) β†’ 2 ∈ (1...𝑁))
30 ubmelfzo 13644 . . . . 5 (2 ∈ (1...𝑁) β†’ (𝑁 βˆ’ 2) ∈ (0..^𝑁))
3129, 30syl 17 . . . 4 (𝑁 ∈ (β„€β‰₯β€˜2) β†’ (𝑁 βˆ’ 2) ∈ (0..^𝑁))
32313ad2ant3 1136 . . 3 (((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ 𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑁 βˆ’ 2) ∈ (0..^𝑁))
33 oveq2 7370 . . . . 5 ((β™―β€˜(1st β€˜π‘)) = 𝑁 β†’ (0..^(β™―β€˜(1st β€˜π‘))) = (0..^𝑁))
3433eleq2d 2824 . . . 4 ((β™―β€˜(1st β€˜π‘)) = 𝑁 β†’ ((𝑁 βˆ’ 2) ∈ (0..^(β™―β€˜(1st β€˜π‘))) ↔ (𝑁 βˆ’ 2) ∈ (0..^𝑁)))
35343ad2ant1 1134 . . 3 (((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ 𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ ((𝑁 βˆ’ 2) ∈ (0..^(β™―β€˜(1st β€˜π‘))) ↔ (𝑁 βˆ’ 2) ∈ (0..^𝑁)))
3632, 35mpbird 257 . 2 (((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ 𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑁 βˆ’ 2) ∈ (0..^(β™―β€˜(1st β€˜π‘))))
37 pfxfv 14577 . 2 (((2nd β€˜π‘) ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜(1st β€˜π‘)) ∈ (0...(β™―β€˜(2nd β€˜π‘))) ∧ (𝑁 βˆ’ 2) ∈ (0..^(β™―β€˜(1st β€˜π‘)))) β†’ (((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))β€˜(𝑁 βˆ’ 2)) = ((2nd β€˜π‘)β€˜(𝑁 βˆ’ 2)))
387, 23, 36, 37syl3anc 1372 1 (((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ 𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (((2nd β€˜π‘) prefix (β™―β€˜(1st β€˜π‘)))β€˜(𝑁 βˆ’ 2)) = ((2nd β€˜π‘)β€˜(𝑁 βˆ’ 2)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1088   = 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  β„•cn 12160  2c2 12215  β„•0cn0 12420  β„€β‰₯cuz 12770  ...cfz 13431  ..^cfzo 13574  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:  dlwwlknondlwlknonf1o  29351
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