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Theorem clwlkclwwlkf1lem2 28998
Description: Lemma 2 for clwlkclwwlkf1 29003. (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 28997 . . . 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 28997 . . . 4 (π‘Š ∈ 𝐢 β†’ (𝐷(Walksβ€˜πΊ)𝐸 ∧ (πΈβ€˜0) = (πΈβ€˜(β™―β€˜π·)) ∧ (β™―β€˜π·) ∈ β„•))
84, 7anim12i 614 . . 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 2733 . . . . . . 7 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
109wlkpwrd 28614 . . . . . 6 (𝐴(Walksβ€˜πΊ)𝐡 β†’ 𝐡 ∈ Word (Vtxβ€˜πΊ))
11103ad2ant1 1134 . . . . 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 28614 . . . . . 6 (𝐷(Walksβ€˜πΊ)𝐸 β†’ 𝐸 ∈ Word (Vtxβ€˜πΊ))
13123ad2ant1 1134 . . . . 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 614 . . . 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 12428 . . . . . 6 ((β™―β€˜π΄) ∈ β„• β†’ (β™―β€˜π΄) ∈ β„•0)
16153ad2ant3 1136 . . . . 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 12428 . . . . . 6 ((β™―β€˜π·) ∈ β„• β†’ (β™―β€˜π·) ∈ β„•0)
18173ad2ant3 1136 . . . . 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 614 . . . 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 28615 . . . . . . . 8 (𝐴(Walksβ€˜πΊ)𝐡 β†’ (β™―β€˜π΅) = ((β™―β€˜π΄) + 1))
21 nnre 12168 . . . . . . . . . 10 ((β™―β€˜π΄) ∈ β„• β†’ (β™―β€˜π΄) ∈ ℝ)
2221lep1d 12094 . . . . . . . . 9 ((β™―β€˜π΄) ∈ β„• β†’ (β™―β€˜π΄) ≀ ((β™―β€˜π΄) + 1))
23 breq2 5113 . . . . . . . . 9 ((β™―β€˜π΅) = ((β™―β€˜π΄) + 1) β†’ ((β™―β€˜π΄) ≀ (β™―β€˜π΅) ↔ (β™―β€˜π΄) ≀ ((β™―β€˜π΄) + 1)))
2422, 23syl5ibr 246 . . . . . . . 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 1112 . . . . 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 28615 . . . . . . . 8 (𝐷(Walksβ€˜πΊ)𝐸 β†’ (β™―β€˜πΈ) = ((β™―β€˜π·) + 1))
29 nnre 12168 . . . . . . . . . 10 ((β™―β€˜π·) ∈ β„• β†’ (β™―β€˜π·) ∈ ℝ)
3029lep1d 12094 . . . . . . . . 9 ((β™―β€˜π·) ∈ β„• β†’ (β™―β€˜π·) ≀ ((β™―β€˜π·) + 1))
31 breq2 5113 . . . . . . . . 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, 31syl5ibr 246 . . . . . . . 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 1112 . . . . 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 614 . . . 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 1129 . . 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 14593 . . 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 1470 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  {crab 3406   class class class wbr 5109  β€˜cfv 6500  (class class class)co 7361  1st c1st 7923  2nd c2nd 7924  0cc0 11059  1c1 11060   + caddc 11062   ≀ cle 11198  β„•cn 12161  β„•0cn0 12421  ..^cfzo 13576  β™―chash 14239  Word cword 14411   prefix cpfx 14567  Vtxcvtx 27996  Walkscwlks 28593  ClWalkscclwlks 28767
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 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-card 9883  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-n0 12422  df-z 12508  df-uz 12772  df-fz 13434  df-fzo 13577  df-hash 14240  df-word 14412  df-substr 14538  df-pfx 14568  df-wlks 28596  df-clwlks 28768
This theorem is referenced by:  clwlkclwwlkf1lem3  28999  clwlkclwwlkf1  29003
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