MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  numclwwlk2lem1lem Structured version   Visualization version   GIF version
Theorem numclwwlk2lem1lem 29595
Description: Lemma for numclwwlk2lem1 29629. (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Revised by AV, 27-May-2021.) (Revised by AV, 15-Mar-2022.)
Assertion
Ref Expression
numclwwlk2lem1lem ((𝑋 ∈ (Vtxβ€˜πΊ) ∧ π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0)) β†’ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)β€˜0) = (π‘Šβ€˜0) ∧ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)β€˜π‘) β‰  (π‘Šβ€˜0)))
Proof of Theorem numclwwlk2lem1lem
StepHypRef Expression
1 wwlknbp1 29098 . . 3 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)))
2 simpl2 1193 . . . . . 6 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ (𝑋 ∈ (Vtxβ€˜πΊ) ∧ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0))) β†’ π‘Š ∈ Word (Vtxβ€˜πΊ))
3 s1cl 14552 . . . . . . 7 (𝑋 ∈ (Vtxβ€˜πΊ) β†’ βŸ¨β€œπ‘‹β€βŸ© ∈ Word (Vtxβ€˜πΊ))
43ad2antrl 727 . . . . . 6 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ (𝑋 ∈ (Vtxβ€˜πΊ) ∧ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0))) β†’ βŸ¨β€œπ‘‹β€βŸ© ∈ Word (Vtxβ€˜πΊ))
5 nn0p1gt0 12501 . . . . . . . . 9 (𝑁 ∈ β„•0 β†’ 0 < (𝑁 + 1))
653ad2ant1 1134 . . . . . . . 8 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) β†’ 0 < (𝑁 + 1))
76adantr 482 . . . . . . 7 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ (𝑋 ∈ (Vtxβ€˜πΊ) ∧ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0))) β†’ 0 < (𝑁 + 1))
8 breq2 5153 . . . . . . . . 9 ((β™―β€˜π‘Š) = (𝑁 + 1) β†’ (0 < (β™―β€˜π‘Š) ↔ 0 < (𝑁 + 1)))
983ad2ant3 1136 . . . . . . . 8 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) β†’ (0 < (β™―β€˜π‘Š) ↔ 0 < (𝑁 + 1)))
109adantr 482 . . . . . . 7 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ (𝑋 ∈ (Vtxβ€˜πΊ) ∧ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0))) β†’ (0 < (β™―β€˜π‘Š) ↔ 0 < (𝑁 + 1)))
117, 10mpbird 257 . . . . . 6 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ (𝑋 ∈ (Vtxβ€˜πΊ) ∧ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0))) β†’ 0 < (β™―β€˜π‘Š))
12 ccatfv0 14533 . . . . . 6 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ βŸ¨β€œπ‘‹β€βŸ© ∈ Word (Vtxβ€˜πΊ) ∧ 0 < (β™―β€˜π‘Š)) β†’ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)β€˜0) = (π‘Šβ€˜0))
132, 4, 11, 12syl3anc 1372 . . . . 5 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ (𝑋 ∈ (Vtxβ€˜πΊ) ∧ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0))) β†’ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)β€˜0) = (π‘Šβ€˜0))
14 oveq1 7416 . . . . . . . . . . . . 13 ((β™―β€˜π‘Š) = (𝑁 + 1) β†’ ((β™―β€˜π‘Š) βˆ’ 1) = ((𝑁 + 1) βˆ’ 1))
15143ad2ant3 1136 . . . . . . . . . . . 12 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) β†’ ((β™―β€˜π‘Š) βˆ’ 1) = ((𝑁 + 1) βˆ’ 1))
16 nn0cn 12482 . . . . . . . . . . . . . 14 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ β„‚)
17 pncan1 11638 . . . . . . . . . . . . . 14 (𝑁 ∈ β„‚ β†’ ((𝑁 + 1) βˆ’ 1) = 𝑁)
1816, 17syl 17 . . . . . . . . . . . . 13 (𝑁 ∈ β„•0 β†’ ((𝑁 + 1) βˆ’ 1) = 𝑁)
19183ad2ant1 1134 . . . . . . . . . . . 12 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) β†’ ((𝑁 + 1) βˆ’ 1) = 𝑁)
2015, 19eqtr2d 2774 . . . . . . . . . . 11 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) β†’ 𝑁 = ((β™―β€˜π‘Š) βˆ’ 1))
2120adantr 482 . . . . . . . . . 10 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ 𝑋 ∈ (Vtxβ€˜πΊ)) β†’ 𝑁 = ((β™―β€˜π‘Š) βˆ’ 1))
2221fveq2d 6896 . . . . . . . . 9 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ 𝑋 ∈ (Vtxβ€˜πΊ)) β†’ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)β€˜π‘) = ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)β€˜((β™―β€˜π‘Š) βˆ’ 1)))
23 simpl2 1193 . . . . . . . . . 10 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ 𝑋 ∈ (Vtxβ€˜πΊ)) β†’ π‘Š ∈ Word (Vtxβ€˜πΊ))
243adantl 483 . . . . . . . . . 10 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ 𝑋 ∈ (Vtxβ€˜πΊ)) β†’ βŸ¨β€œπ‘‹β€βŸ© ∈ Word (Vtxβ€˜πΊ))
256adantr 482 . . . . . . . . . . . 12 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ 𝑋 ∈ (Vtxβ€˜πΊ)) β†’ 0 < (𝑁 + 1))
269adantr 482 . . . . . . . . . . . 12 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ 𝑋 ∈ (Vtxβ€˜πΊ)) β†’ (0 < (β™―β€˜π‘Š) ↔ 0 < (𝑁 + 1)))
2725, 26mpbird 257 . . . . . . . . . . 11 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ 𝑋 ∈ (Vtxβ€˜πΊ)) β†’ 0 < (β™―β€˜π‘Š))
28 hashneq0 14324 . . . . . . . . . . . . . 14 (π‘Š ∈ Word (Vtxβ€˜πΊ) β†’ (0 < (β™―β€˜π‘Š) ↔ π‘Š β‰  βˆ…))
2928bicomd 222 . . . . . . . . . . . . 13 (π‘Š ∈ Word (Vtxβ€˜πΊ) β†’ (π‘Š β‰  βˆ… ↔ 0 < (β™―β€˜π‘Š)))
30293ad2ant2 1135 . . . . . . . . . . . 12 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) β†’ (π‘Š β‰  βˆ… ↔ 0 < (β™―β€˜π‘Š)))
3130adantr 482 . . . . . . . . . . 11 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ 𝑋 ∈ (Vtxβ€˜πΊ)) β†’ (π‘Š β‰  βˆ… ↔ 0 < (β™―β€˜π‘Š)))
3227, 31mpbird 257 . . . . . . . . . 10 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ 𝑋 ∈ (Vtxβ€˜πΊ)) β†’ π‘Š β‰  βˆ…)
33 ccatval1lsw 14534 . . . . . . . . . 10 ((π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ βŸ¨β€œπ‘‹β€βŸ© ∈ Word (Vtxβ€˜πΊ) ∧ π‘Š β‰  βˆ…) β†’ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)β€˜((β™―β€˜π‘Š) βˆ’ 1)) = (lastSβ€˜π‘Š))
3423, 24, 32, 33syl3anc 1372 . . . . . . . . 9 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ 𝑋 ∈ (Vtxβ€˜πΊ)) β†’ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)β€˜((β™―β€˜π‘Š) βˆ’ 1)) = (lastSβ€˜π‘Š))
3522, 34eqtr2d 2774 . . . . . . . 8 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ 𝑋 ∈ (Vtxβ€˜πΊ)) β†’ (lastSβ€˜π‘Š) = ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)β€˜π‘))
3635neeq1d 3001 . . . . . . 7 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ 𝑋 ∈ (Vtxβ€˜πΊ)) β†’ ((lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0) ↔ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)β€˜π‘) β‰  (π‘Šβ€˜0)))
3736biimpd 228 . . . . . 6 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ 𝑋 ∈ (Vtxβ€˜πΊ)) β†’ ((lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0) β†’ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)β€˜π‘) β‰  (π‘Šβ€˜0)))
3837impr 456 . . . . 5 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ (𝑋 ∈ (Vtxβ€˜πΊ) ∧ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0))) β†’ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)β€˜π‘) β‰  (π‘Šβ€˜0))
3913, 38jca 513 . . . 4 (((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) ∧ (𝑋 ∈ (Vtxβ€˜πΊ) ∧ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0))) β†’ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)β€˜0) = (π‘Šβ€˜0) ∧ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)β€˜π‘) β‰  (π‘Šβ€˜0)))
4039exp32 422 . . 3 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) β†’ (𝑋 ∈ (Vtxβ€˜πΊ) β†’ ((lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0) β†’ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)β€˜0) = (π‘Šβ€˜0) ∧ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)β€˜π‘) β‰  (π‘Šβ€˜0)))))
411, 40syl 17 . 2 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (𝑋 ∈ (Vtxβ€˜πΊ) β†’ ((lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0) β†’ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)β€˜0) = (π‘Šβ€˜0) ∧ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)β€˜π‘) β‰  (π‘Šβ€˜0)))))
42413imp21 1115 1 ((𝑋 ∈ (Vtxβ€˜πΊ) ∧ π‘Š ∈ (𝑁 WWalksN 𝐺) ∧ (lastSβ€˜π‘Š) β‰  (π‘Šβ€˜0)) β†’ (((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)β€˜0) = (π‘Šβ€˜0) ∧ ((π‘Š ++ βŸ¨β€œπ‘‹β€βŸ©)β€˜π‘) β‰  (π‘Šβ€˜0)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ…c0 4323   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  β„‚cc 11108  0cc0 11110  1c1 11111   + caddc 11113   < clt 11248   βˆ’ cmin 11444  β„•0cn0 12472  β™―chash 14290  Word cword 14464  lastSclsw 14512   ++ cconcat 14520  βŸ¨β€œcs1 14545  Vtxcvtx 28256   WWalksN cwwlksn 29080
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  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 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-xnn0 12545  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-lsw 14513  df-concat 14521  df-s1 14546  df-wwlks 29084  df-wwlksn 29085
This theorem is referenced by:  numclwwlk2lem1  29629
  Copyright terms: Public domain W3C validator