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Theorem numclwwlk2lem1lem 29860
Description: Lemma for numclwwlk2lem1 29894. (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 29363 . . 3 (π‘Š ∈ (𝑁 WWalksN 𝐺) β†’ (𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)))
2 simpl2 1190 . . . . . 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 14558 . . . . . . 7 (𝑋 ∈ (Vtxβ€˜πΊ) β†’ βŸ¨β€œπ‘‹β€βŸ© ∈ Word (Vtxβ€˜πΊ))
43ad2antrl 724 . . . . . 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 12507 . . . . . . . . 9 (𝑁 ∈ β„•0 β†’ 0 < (𝑁 + 1))
653ad2ant1 1131 . . . . . . . 8 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) β†’ 0 < (𝑁 + 1))
76adantr 479 . . . . . . 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 1133 . . . . . . . 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 479 . . . . . . 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 256 . . . . . 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 14539 . . . . . 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 1369 . . . . 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 7420 . . . . . . . . . . . . 13 ((β™―β€˜π‘Š) = (𝑁 + 1) β†’ ((β™―β€˜π‘Š) βˆ’ 1) = ((𝑁 + 1) βˆ’ 1))
15143ad2ant3 1133 . . . . . . . . . . . 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 12488 . . . . . . . . . . . . . 14 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ β„‚)
17 pncan1 11644 . . . . . . . . . . . . . 14 (𝑁 ∈ β„‚ β†’ ((𝑁 + 1) βˆ’ 1) = 𝑁)
1816, 17syl 17 . . . . . . . . . . . . 13 (𝑁 ∈ β„•0 β†’ ((𝑁 + 1) βˆ’ 1) = 𝑁)
19183ad2ant1 1131 . . . . . . . . . . . 12 ((𝑁 ∈ β„•0 ∧ π‘Š ∈ Word (Vtxβ€˜πΊ) ∧ (β™―β€˜π‘Š) = (𝑁 + 1)) β†’ ((𝑁 + 1) βˆ’ 1) = 𝑁)
2015, 19eqtr2d 2771 . . . . . . . . . . 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 479 . . . . . . . . . 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 1190 . . . . . . . . . 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 480 . . . . . . . . . 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 479 . . . . . . . . . . . 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 479 . . . . . . . . . . . 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 256 . . . . . . . . . . 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 14330 . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . 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 479 . . . . . . . . . . 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 256 . . . . . . . . . 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 14540 . . . . . . . . . 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 1369 . . . . . . . . 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 2771 . . . . . . . 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 2998 . . . . . . 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 453 . . . . 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 510 . . . 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 419 . . 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 1112 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  βˆ…c0 4323   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7413  β„‚cc 11112  0cc0 11114  1c1 11115   + caddc 11117   < clt 11254   βˆ’ cmin 11450  β„•0cn0 12478  β™―chash 14296  Word cword 14470  lastSclsw 14518   ++ cconcat 14526  βŸ¨β€œcs1 14551  Vtxcvtx 28521   WWalksN cwwlksn 29345
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-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-xnn0 12551  df-z 12565  df-uz 12829  df-fz 13491  df-fzo 13634  df-hash 14297  df-word 14471  df-lsw 14519  df-concat 14527  df-s1 14552  df-wwlks 29349  df-wwlksn 29350
This theorem is referenced by:  numclwwlk2lem1  29894
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