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Theorem clwlkclwwlkf1lem2 29523
Description: Lemma 2 for clwlkclwwlkf1 29528. (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 29522 . . . 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 29522 . . . 4 (π‘Š ∈ 𝐢 β†’ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•))
84, 7anim12i 611 . . 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 2730 . . . . . . 7 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
109wlkpwrd 29139 . . . . . 6 (𝐴(Walksβ€˜πΊ)𝐡 β†’ 𝐡 ∈ Word (Vtxβ€˜πΊ))
11103ad2ant1 1131 . . . . 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 29139 . . . . . 6 (𝐷(Walksβ€˜πΊ)𝐸 β†’ 𝐸 ∈ Word (Vtxβ€˜πΊ))
13123ad2ant1 1131 . . . . 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 611 . . . 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 12485 . . . . . 6 ((β™―β€˜π΄) ∈ β„• β†’ (β™―β€˜π΄) ∈ β„•0)
16153ad2ant3 1133 . . . . 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 12485 . . . . . 6 ((β™―β€˜π·) ∈ β„• β†’ (β™―β€˜π·) ∈ β„•0)
18173ad2ant3 1133 . . . . 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 611 . . . 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 29140 . . . . . . . 8 (𝐴(Walksβ€˜πΊ)𝐡 β†’ (β™―β€˜π΅) = ((β™―β€˜π΄) + 1))
21 nnre 12225 . . . . . . . . . 10 ((β™―β€˜π΄) ∈ β„• β†’ (β™―β€˜π΄) ∈ ℝ)
2221lep1d 12151 . . . . . . . . 9 ((β™―β€˜π΄) ∈ β„• β†’ (β™―β€˜π΄) ≀ ((β™―β€˜π΄) + 1))
23 breq2 5153 . . . . . . . . 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 1109 . . . . 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 29140 . . . . . . . 8 (𝐷(Walksβ€˜πΊ)𝐸 β†’ (β™―β€˜πΈ) = ((β™―β€˜π·) + 1))
29 nnre 12225 . . . . . . . . . 10 ((β™―β€˜π·) ∈ β„• β†’ (β™―β€˜π·) ∈ ℝ)
3029lep1d 12151 . . . . . . . . 9 ((β™―β€˜π·) ∈ β„• β†’ (β™―β€˜π·) ≀ ((β™―β€˜π·) + 1))
31 breq2 5153 . . . . . . . . 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 1109 . . . . 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 611 . . . 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 1126 . . 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 14652 . . 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 1467 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  {crab 3430   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  β„•cn 12218  β„•0cn0 12478  ..^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-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:  clwlkclwwlkf1lem3  29524  clwlkclwwlkf1  29528
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