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Theorem dlwwlknondlwlknonf1olem1 29882
Description: Lemma 1 for dlwwlknondlwlknonf1o 29883. (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 29297 . . . . 5 (𝑐 ∈ (ClWalksβ€˜πΊ) β†’ 𝑐 ∈ (Walksβ€˜πΊ))
2 wlkcpr 29151 . . . . 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 2730 . . . . 5 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
54wlkpwrd 29139 . . . 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 1132 . 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 12877 . . . . . . 7 (𝑁 ∈ (β„€β‰₯β€˜2) β†’ 𝑁 ∈ β„•0)
983ad2ant3 1133 . . . . . 6 (((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ 𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ 𝑁 ∈ β„•0)
10 eleq1 2819 . . . . . . 7 ((β™―β€˜(1st β€˜π‘)) = 𝑁 β†’ ((β™―β€˜(1st β€˜π‘)) ∈ β„•0 ↔ 𝑁 ∈ β„•0))
11103ad2ant1 1131 . . . . . 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 256 . . . . 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 13605 . . . . 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 13559 . . . 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 29140 . . . . . . . 8 ((1st β€˜π‘)(Walksβ€˜πΊ)(2nd β€˜π‘) β†’ (β™―β€˜(2nd β€˜π‘)) = ((β™―β€˜(1st β€˜π‘)) + 1))
1817eqcomd 2736 . . . . . . 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 7429 . . . . 5 (𝑐 ∈ (ClWalksβ€˜πΊ) β†’ (0...((β™―β€˜(1st β€˜π‘)) + 1)) = (0...(β™―β€˜(2nd β€˜π‘))))
2120eleq2d 2817 . . . 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 1132 . . 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 12291 . . . . . . 7 2 ∈ β„•
2524a1i 11 . . . . . 6 (𝑁 ∈ (β„€β‰₯β€˜2) β†’ 2 ∈ β„•)
26 eluz2nn 12874 . . . . . 6 (𝑁 ∈ (β„€β‰₯β€˜2) β†’ 𝑁 ∈ β„•)
27 eluzle 12841 . . . . . 6 (𝑁 ∈ (β„€β‰₯β€˜2) β†’ 2 ≀ 𝑁)
28 elfz1b 13576 . . . . . 6 (2 ∈ (1...𝑁) ↔ (2 ∈ β„• ∧ 𝑁 ∈ β„• ∧ 2 ≀ 𝑁))
2925, 26, 27, 28syl3anbrc 1341 . . . . 5 (𝑁 ∈ (β„€β‰₯β€˜2) β†’ 2 ∈ (1...𝑁))
30 ubmelfzo 13703 . . . . 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 1133 . . 3 (((β™―β€˜(1st β€˜π‘)) = 𝑁 ∧ 𝑐 ∈ (ClWalksβ€˜πΊ) ∧ 𝑁 ∈ (β„€β‰₯β€˜2)) β†’ (𝑁 βˆ’ 2) ∈ (0..^𝑁))
33 oveq2 7421 . . . . 5 ((β™―β€˜(1st β€˜π‘)) = 𝑁 β†’ (0..^(β™―β€˜(1st β€˜π‘))) = (0..^𝑁))
3433eleq2d 2817 . . . 4 ((β™―β€˜(1st β€˜π‘)) = 𝑁 β†’ ((𝑁 βˆ’ 2) ∈ (0..^(β™―β€˜(1st β€˜π‘))) ↔ (𝑁 βˆ’ 2) ∈ (0..^𝑁)))
35343ad2ant1 1131 . . 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 256 . 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 14638 . 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 1369 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 1085   = wceq 1539   ∈ wcel 2104   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7413  1st c1st 7977  2nd c2nd 7978  0cc0 11114  1c1 11115   + caddc 11117   ≀ cle 11255   βˆ’ cmin 11450  β„•cn 12218  2c2 12273  β„•0cn0 12478  β„€β‰₯cuz 12828  ...cfz 13490  ..^cfzo 13633  β™―chash 14296  Word cword 14470   prefix cpfx 14626  Vtxcvtx 28521  Walkscwlks 29118  ClWalkscclwlks 29292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-card 9938  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11452  df-neg 11453  df-nn 12219  df-2 12281  df-n0 12479  df-z 12565  df-uz 12829  df-fz 13491  df-fzo 13634  df-hash 14297  df-word 14471  df-substr 14597  df-pfx 14627  df-wlks 29121  df-clwlks 29293
This theorem is referenced by:  dlwwlknondlwlknonf1o  29883
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