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Theorem clwlkclwwlkf1lem2 29255
Description: Lemma 2 for clwlkclwwlkf1 29260. (Contributed by AV, 24-May-2022.) (Revised by AV, 30-Oct-2022.)
Hypotheses
Ref Expression
clwlkclwwlkf.c 𝐢 = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))}
clwlkclwwlkf.a 𝐴 = (1st β€˜π‘ˆ)
clwlkclwwlkf.b 𝐡 = (2nd β€˜π‘ˆ)
clwlkclwwlkf.d 𝐷 = (1st β€˜π‘Š)
clwlkclwwlkf.e 𝐸 = (2nd β€˜π‘Š)
Assertion
Ref Expression
clwlkclwwlkf1lem2 ((π‘ˆ ∈ 𝐢 ∧ π‘Š ∈ 𝐢 ∧ (𝐡 prefix (β™―β€˜π΄)) = (𝐸 prefix (β™―β€˜π·))) β†’ ((β™―β€˜π΄) = (β™―β€˜π·) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–)))
Distinct variable groups:   𝑖,𝐺   𝑀,𝐺   𝑀,𝐴   𝑀,π‘ˆ   𝐴,𝑖   𝐡,𝑖   𝐷,𝑖   𝑀,𝐷   𝑖,𝐸   𝑀,π‘Š
Allowed substitution hints:   𝐡(𝑀)   𝐢(𝑀,𝑖)   π‘ˆ(𝑖)   𝐸(𝑀)   π‘Š(𝑖)

Proof of Theorem clwlkclwwlkf1lem2
StepHypRef Expression
1 clwlkclwwlkf.c . . . . 5 𝐢 = {𝑀 ∈ (ClWalksβ€˜πΊ) ∣ 1 ≀ (β™―β€˜(1st β€˜π‘€))}
2 clwlkclwwlkf.a . . . . 5 𝐴 = (1st β€˜π‘ˆ)
3 clwlkclwwlkf.b . . . . 5 𝐡 = (2nd β€˜π‘ˆ)
41, 2, 3clwlkclwwlkflem 29254 . . . 4 (π‘ˆ ∈ 𝐢 β†’ (𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•))
5 clwlkclwwlkf.d . . . . 5 𝐷 = (1st β€˜π‘Š)
6 clwlkclwwlkf.e . . . . 5 𝐸 = (2nd β€˜π‘Š)
71, 5, 6clwlkclwwlkflem 29254 . . . 4 (π‘Š ∈ 𝐢 β†’ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•))
84, 7anim12i 613 . . 3 ((π‘ˆ ∈ 𝐢 ∧ π‘Š ∈ 𝐢) β†’ ((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) ∧ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•)))
9 eqid 2732 . . . . . . 7 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
109wlkpwrd 28871 . . . . . 6 (𝐴(Walksβ€˜πΊ)𝐡 β†’ 𝐡 ∈ Word (Vtxβ€˜πΊ))
11103ad2ant1 1133 . . . . 5 ((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) β†’ 𝐡 ∈ Word (Vtxβ€˜πΊ))
129wlkpwrd 28871 . . . . . 6 (𝐷(Walksβ€˜πΊ)𝐸 β†’ 𝐸 ∈ Word (Vtxβ€˜πΊ))
13123ad2ant1 1133 . . . . 5 ((𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•) β†’ 𝐸 ∈ Word (Vtxβ€˜πΊ))
1411, 13anim12i 613 . . . 4 (((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) ∧ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•)) β†’ (𝐡 ∈ Word (Vtxβ€˜πΊ) ∧ 𝐸 ∈ Word (Vtxβ€˜πΊ)))
15 nnnn0 12478 . . . . . 6 ((β™―β€˜π΄) ∈ β„• β†’ (β™―β€˜π΄) ∈ β„•0)
16153ad2ant3 1135 . . . . 5 ((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) β†’ (β™―β€˜π΄) ∈ β„•0)
17 nnnn0 12478 . . . . . 6 ((β™―β€˜π·) ∈ β„• β†’ (β™―β€˜π·) ∈ β„•0)
18173ad2ant3 1135 . . . . 5 ((𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•) β†’ (β™―β€˜π·) ∈ β„•0)
1916, 18anim12i 613 . . . 4 (((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) ∧ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•)) β†’ ((β™―β€˜π΄) ∈ β„•0 ∧ (β™―β€˜π·) ∈ β„•0))
20 wlklenvp1 28872 . . . . . . . 8 (𝐴(Walksβ€˜πΊ)𝐡 β†’ (β™―β€˜π΅) = ((β™―β€˜π΄) + 1))
21 nnre 12218 . . . . . . . . . 10 ((β™―β€˜π΄) ∈ β„• β†’ (β™―β€˜π΄) ∈ ℝ)
2221lep1d 12144 . . . . . . . . 9 ((β™―β€˜π΄) ∈ β„• β†’ (β™―β€˜π΄) ≀ ((β™―β€˜π΄) + 1))
23 breq2 5152 . . . . . . . . 9 ((β™―β€˜π΅) = ((β™―β€˜π΄) + 1) β†’ ((β™―β€˜π΄) ≀ (β™―β€˜π΅) ↔ (β™―β€˜π΄) ≀ ((β™―β€˜π΄) + 1)))
2422, 23imbitrrid 245 . . . . . . . 8 ((β™―β€˜π΅) = ((β™―β€˜π΄) + 1) β†’ ((β™―β€˜π΄) ∈ β„• β†’ (β™―β€˜π΄) ≀ (β™―β€˜π΅)))
2520, 24syl 17 . . . . . . 7 (𝐴(Walksβ€˜πΊ)𝐡 β†’ ((β™―β€˜π΄) ∈ β„• β†’ (β™―β€˜π΄) ≀ (β™―β€˜π΅)))
2625a1d 25 . . . . . 6 (𝐴(Walksβ€˜πΊ)𝐡 β†’ ((π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) β†’ ((β™―β€˜π΄) ∈ β„• β†’ (β™―β€˜π΄) ≀ (β™―β€˜π΅))))
27263imp 1111 . . . . 5 ((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) β†’ (β™―β€˜π΄) ≀ (β™―β€˜π΅))
28 wlklenvp1 28872 . . . . . . . 8 (𝐷(Walksβ€˜πΊ)𝐸 β†’ (β™―β€˜πΈ) = ((β™―β€˜π·) + 1))
29 nnre 12218 . . . . . . . . . 10 ((β™―β€˜π·) ∈ β„• β†’ (β™―β€˜π·) ∈ ℝ)
3029lep1d 12144 . . . . . . . . 9 ((β™―β€˜π·) ∈ β„• β†’ (β™―β€˜π·) ≀ ((β™―β€˜π·) + 1))
31 breq2 5152 . . . . . . . . 9 ((β™―β€˜πΈ) = ((β™―β€˜π·) + 1) β†’ ((β™―β€˜π·) ≀ (β™―β€˜πΈ) ↔ (β™―β€˜π·) ≀ ((β™―β€˜π·) + 1)))
3230, 31imbitrrid 245 . . . . . . . 8 ((β™―β€˜πΈ) = ((β™―β€˜π·) + 1) β†’ ((β™―β€˜π·) ∈ β„• β†’ (β™―β€˜π·) ≀ (β™―β€˜πΈ)))
3328, 32syl 17 . . . . . . 7 (𝐷(Walksβ€˜πΊ)𝐸 β†’ ((β™―β€˜π·) ∈ β„• β†’ (β™―β€˜π·) ≀ (β™―β€˜πΈ)))
3433a1d 25 . . . . . 6 (𝐷(Walksβ€˜πΊ)𝐸 β†’ ((πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) β†’ ((β™―β€˜π·) ∈ β„• β†’ (β™―β€˜π·) ≀ (β™―β€˜πΈ))))
35343imp 1111 . . . . 5 ((𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•) β†’ (β™―β€˜π·) ≀ (β™―β€˜πΈ))
3627, 35anim12i 613 . . . 4 (((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) ∧ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•)) β†’ ((β™―β€˜π΄) ≀ (β™―β€˜π΅) ∧ (β™―β€˜π·) ≀ (β™―β€˜πΈ)))
3714, 19, 363jca 1128 . . 3 (((𝐴(Walksβ€˜πΊ)𝐡 ∧ (π΅β€˜0) = (π΅β€˜(β™―β€˜π΄)) ∧ (β™―β€˜π΄) ∈ β„•) ∧ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•)) β†’ ((𝐡 ∈ Word (Vtxβ€˜πΊ) ∧ 𝐸 ∈ Word (Vtxβ€˜πΊ)) ∧ ((β™―β€˜π΄) ∈ β„•0 ∧ (β™―β€˜π·) ∈ β„•0) ∧ ((β™―β€˜π΄) ≀ (β™―β€˜π΅) ∧ (β™―β€˜π·) ≀ (β™―β€˜πΈ))))
38 pfxeq 14645 . . 3 (((𝐡 ∈ Word (Vtxβ€˜πΊ) ∧ 𝐸 ∈ Word (Vtxβ€˜πΊ)) ∧ ((β™―β€˜π΄) ∈ β„•0 ∧ (β™―β€˜π·) ∈ β„•0) ∧ ((β™―β€˜π΄) ≀ (β™―β€˜π΅) ∧ (β™―β€˜π·) ≀ (β™―β€˜πΈ))) β†’ ((𝐡 prefix (β™―β€˜π΄)) = (𝐸 prefix (β™―β€˜π·)) ↔ ((β™―β€˜π΄) = (β™―β€˜π·) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–))))
398, 37, 383syl 18 . 2 ((π‘ˆ ∈ 𝐢 ∧ π‘Š ∈ 𝐢) β†’ ((𝐡 prefix (β™―β€˜π΄)) = (𝐸 prefix (β™―β€˜π·)) ↔ ((β™―β€˜π΄) = (β™―β€˜π·) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–))))
4039biimp3a 1469 1 ((π‘ˆ ∈ 𝐢 ∧ π‘Š ∈ 𝐢 ∧ (𝐡 prefix (β™―β€˜π΄)) = (𝐸 prefix (β™―β€˜π·))) β†’ ((β™―β€˜π΄) = (β™―β€˜π·) ∧ βˆ€π‘– ∈ (0..^(β™―β€˜π΄))(π΅β€˜π‘–) = (πΈβ€˜π‘–)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  {crab 3432   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7408  1st c1st 7972  2nd c2nd 7973  0cc0 11109  1c1 11110   + caddc 11112   ≀ cle 11248  β„•cn 12211  β„•0cn0 12471  ..^cfzo 13626  β™―chash 14289  Word cword 14463   prefix cpfx 14619  Vtxcvtx 28253  Walkscwlks 28850  ClWalkscclwlks 29024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-card 9933  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-n0 12472  df-z 12558  df-uz 12822  df-fz 13484  df-fzo 13627  df-hash 14290  df-word 14464  df-substr 14590  df-pfx 14620  df-wlks 28853  df-clwlks 29025
This theorem is referenced by:  clwlkclwwlkf1lem3  29256  clwlkclwwlkf1  29260
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