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Theorem wlklnwwlkln2lem 29136
Description: Lemma for wlklnwwlkln2 29137 and wlklnwwlklnupgr2 29139. Formerly part of proof for wlklnwwlkln2 29137. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
Hypothesis
Ref Expression
wlklnwwlkln2lem.1 (πœ‘ β†’ (𝑃 ∈ (WWalksβ€˜πΊ) β†’ βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃))
Assertion
Ref Expression
wlklnwwlkln2lem (πœ‘ β†’ (𝑃 ∈ (𝑁 WWalksN 𝐺) β†’ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜π‘“) = 𝑁)))
Distinct variable groups:   𝑓,𝐺   𝑓,𝑁   𝑃,𝑓   πœ‘,𝑓

Proof of Theorem wlklnwwlkln2lem
StepHypRef Expression
1 eqid 2733 . . . 4 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
21wwlknbp 29096 . . 3 (𝑃 ∈ (𝑁 WWalksN 𝐺) β†’ (𝐺 ∈ V ∧ 𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)))
3 iswwlksn 29092 . . . . . 6 (𝑁 ∈ β„•0 β†’ (𝑃 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1))))
43adantr 482 . . . . 5 ((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ (𝑃 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1))))
5 lencl 14483 . . . . . . . . . . . . . 14 (𝑃 ∈ Word (Vtxβ€˜πΊ) β†’ (β™―β€˜π‘ƒ) ∈ β„•0)
65nn0cnd 12534 . . . . . . . . . . . . 13 (𝑃 ∈ Word (Vtxβ€˜πΊ) β†’ (β™―β€˜π‘ƒ) ∈ β„‚)
76adantl 483 . . . . . . . . . . . 12 ((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ (β™―β€˜π‘ƒ) ∈ β„‚)
8 1cnd 11209 . . . . . . . . . . . 12 ((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ 1 ∈ β„‚)
9 nn0cn 12482 . . . . . . . . . . . . 13 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ β„‚)
109adantr 482 . . . . . . . . . . . 12 ((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ 𝑁 ∈ β„‚)
117, 8, 10subadd2d 11590 . . . . . . . . . . 11 ((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ (((β™―β€˜π‘ƒ) βˆ’ 1) = 𝑁 ↔ (𝑁 + 1) = (β™―β€˜π‘ƒ)))
12 eqcom 2740 . . . . . . . . . . 11 ((𝑁 + 1) = (β™―β€˜π‘ƒ) ↔ (β™―β€˜π‘ƒ) = (𝑁 + 1))
1311, 12bitr2di 288 . . . . . . . . . 10 ((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ ((β™―β€˜π‘ƒ) = (𝑁 + 1) ↔ ((β™―β€˜π‘ƒ) βˆ’ 1) = 𝑁))
1413biimpcd 248 . . . . . . . . 9 ((β™―β€˜π‘ƒ) = (𝑁 + 1) β†’ ((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = 𝑁))
1514adantl 483 . . . . . . . 8 ((𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1)) β†’ ((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = 𝑁))
1615impcom 409 . . . . . . 7 (((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ (𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1))) β†’ ((β™―β€˜π‘ƒ) βˆ’ 1) = 𝑁)
17 wlklnwwlkln2lem.1 . . . . . . . . . . . . . 14 (πœ‘ β†’ (𝑃 ∈ (WWalksβ€˜πΊ) β†’ βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃))
1817com12 32 . . . . . . . . . . . . 13 (𝑃 ∈ (WWalksβ€˜πΊ) β†’ (πœ‘ β†’ βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃))
1918adantr 482 . . . . . . . . . . . 12 ((𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1)) β†’ (πœ‘ β†’ βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃))
2019adantl 483 . . . . . . . . . . 11 (((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ (𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1))) β†’ (πœ‘ β†’ βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃))
2120imp 408 . . . . . . . . . 10 ((((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ (𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1))) ∧ πœ‘) β†’ βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃)
22 simpr 486 . . . . . . . . . . . . 13 (((((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ (𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1))) ∧ πœ‘) ∧ 𝑓(Walksβ€˜πΊ)𝑃) β†’ 𝑓(Walksβ€˜πΊ)𝑃)
23 wlklenvm1 28879 . . . . . . . . . . . . 13 (𝑓(Walksβ€˜πΊ)𝑃 β†’ (β™―β€˜π‘“) = ((β™―β€˜π‘ƒ) βˆ’ 1))
2422, 23jccir 523 . . . . . . . . . . . 12 (((((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ (𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1))) ∧ πœ‘) ∧ 𝑓(Walksβ€˜πΊ)𝑃) β†’ (𝑓(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜π‘“) = ((β™―β€˜π‘ƒ) βˆ’ 1)))
2524ex 414 . . . . . . . . . . 11 ((((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ (𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1))) ∧ πœ‘) β†’ (𝑓(Walksβ€˜πΊ)𝑃 β†’ (𝑓(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜π‘“) = ((β™―β€˜π‘ƒ) βˆ’ 1))))
2625eximdv 1921 . . . . . . . . . 10 ((((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ (𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1))) ∧ πœ‘) β†’ (βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃 β†’ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜π‘“) = ((β™―β€˜π‘ƒ) βˆ’ 1))))
2721, 26mpd 15 . . . . . . . . 9 ((((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ (𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1))) ∧ πœ‘) β†’ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜π‘“) = ((β™―β€˜π‘ƒ) βˆ’ 1)))
28 eqeq2 2745 . . . . . . . . . . 11 (((β™―β€˜π‘ƒ) βˆ’ 1) = 𝑁 β†’ ((β™―β€˜π‘“) = ((β™―β€˜π‘ƒ) βˆ’ 1) ↔ (β™―β€˜π‘“) = 𝑁))
2928anbi2d 630 . . . . . . . . . 10 (((β™―β€˜π‘ƒ) βˆ’ 1) = 𝑁 β†’ ((𝑓(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜π‘“) = ((β™―β€˜π‘ƒ) βˆ’ 1)) ↔ (𝑓(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜π‘“) = 𝑁)))
3029exbidv 1925 . . . . . . . . 9 (((β™―β€˜π‘ƒ) βˆ’ 1) = 𝑁 β†’ (βˆƒπ‘“(𝑓(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜π‘“) = ((β™―β€˜π‘ƒ) βˆ’ 1)) ↔ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜π‘“) = 𝑁)))
3127, 30imbitrid 243 . . . . . . . 8 (((β™―β€˜π‘ƒ) βˆ’ 1) = 𝑁 β†’ ((((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ (𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1))) ∧ πœ‘) β†’ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜π‘“) = 𝑁)))
3231expd 417 . . . . . . 7 (((β™―β€˜π‘ƒ) βˆ’ 1) = 𝑁 β†’ (((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ (𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1))) β†’ (πœ‘ β†’ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜π‘“) = 𝑁))))
3316, 32mpcom 38 . . . . . 6 (((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) ∧ (𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1))) β†’ (πœ‘ β†’ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜π‘“) = 𝑁)))
3433ex 414 . . . . 5 ((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ ((𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1)) β†’ (πœ‘ β†’ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜π‘“) = 𝑁))))
354, 34sylbid 239 . . . 4 ((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ (𝑃 ∈ (𝑁 WWalksN 𝐺) β†’ (πœ‘ β†’ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜π‘“) = 𝑁))))
36353adant1 1131 . . 3 ((𝐺 ∈ V ∧ 𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ (𝑃 ∈ (𝑁 WWalksN 𝐺) β†’ (πœ‘ β†’ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜π‘“) = 𝑁))))
372, 36mpcom 38 . 2 (𝑃 ∈ (𝑁 WWalksN 𝐺) β†’ (πœ‘ β†’ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜π‘“) = 𝑁)))
3837com12 32 1 (πœ‘ β†’ (𝑃 ∈ (𝑁 WWalksN 𝐺) β†’ βˆƒπ‘“(𝑓(Walksβ€˜πΊ)𝑃 ∧ (β™―β€˜π‘“) = 𝑁)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  Vcvv 3475   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  β„‚cc 11108  1c1 11111   + caddc 11113   βˆ’ cmin 11444  β„•0cn0 12472  β™―chash 14290  Word cword 14464  Vtxcvtx 28256  Walkscwlks 28853  WWalkscwwlks 29079   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-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 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-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-wlks 28856  df-wwlks 29084  df-wwlksn 29085
This theorem is referenced by:  wlklnwwlkln2  29137  wlklnwwlklnupgr2  29139
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