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Theorem wlklnwwlkln2lem 28827
Description: Lemma for wlklnwwlkln2 28828 and wlklnwwlklnupgr2 28830. Formerly part of proof for wlklnwwlkln2 28828. (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 2736 . . . 4 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
21wwlknbp 28787 . . 3 (𝑃 ∈ (𝑁 WWalksN 𝐺) β†’ (𝐺 ∈ V ∧ 𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)))
3 iswwlksn 28783 . . . . . 6 (𝑁 ∈ β„•0 β†’ (𝑃 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1))))
43adantr 481 . . . . 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 14421 . . . . . . . . . . . . . 14 (𝑃 ∈ Word (Vtxβ€˜πΊ) β†’ (β™―β€˜π‘ƒ) ∈ β„•0)
65nn0cnd 12475 . . . . . . . . . . . . 13 (𝑃 ∈ Word (Vtxβ€˜πΊ) β†’ (β™―β€˜π‘ƒ) ∈ β„‚)
76adantl 482 . . . . . . . . . . . 12 ((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ (β™―β€˜π‘ƒ) ∈ β„‚)
8 1cnd 11150 . . . . . . . . . . . 12 ((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ 1 ∈ β„‚)
9 nn0cn 12423 . . . . . . . . . . . . 13 (𝑁 ∈ β„•0 β†’ 𝑁 ∈ β„‚)
109adantr 481 . . . . . . . . . . . 12 ((𝑁 ∈ β„•0 ∧ 𝑃 ∈ Word (Vtxβ€˜πΊ)) β†’ 𝑁 ∈ β„‚)
117, 8, 10subadd2d 11531 . . . . . . . . . . 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 2743 . . . . . . . . . . 11 ((𝑁 + 1) = (β™―β€˜π‘ƒ) ↔ (β™―β€˜π‘ƒ) = (𝑁 + 1))
1311, 12bitr2di 287 . . . . . . . . . 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 482 . . . . . . . 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 408 . . . . . . 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 481 . . . . . . . . . . . 12 ((𝑃 ∈ (WWalksβ€˜πΊ) ∧ (β™―β€˜π‘ƒ) = (𝑁 + 1)) β†’ (πœ‘ β†’ βˆƒπ‘“ 𝑓(Walksβ€˜πΊ)𝑃))
2019adantl 482 . . . . . . . . . . 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 407 . . . . . . . . . 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 485 . . . . . . . . . . . . 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 28570 . . . . . . . . . . . . 13 (𝑓(Walksβ€˜πΊ)𝑃 β†’ (β™―β€˜π‘“) = ((β™―β€˜π‘ƒ) βˆ’ 1))
2422, 23jccir 522 . . . . . . . . . . . 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 413 . . . . . . . . . . 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 1920 . . . . . . . . . 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 2748 . . . . . . . . . . 11 (((β™―β€˜π‘ƒ) βˆ’ 1) = 𝑁 β†’ ((β™―β€˜π‘“) = ((β™―β€˜π‘ƒ) βˆ’ 1) ↔ (β™―β€˜π‘“) = 𝑁))
2928anbi2d 629 . . . . . . . . . 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 1924 . . . . . . . . 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 416 . . . . . . 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 413 . . . . 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 1130 . . 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 396   ∧ w3a 1087   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  Vcvv 3445   class class class wbr 5105  β€˜cfv 6496  (class class class)co 7357  β„‚cc 11049  1c1 11052   + caddc 11054   βˆ’ cmin 11385  β„•0cn0 12413  β™―chash 14230  Word cword 14402  Vtxcvtx 27947  Walkscwlks 28544  WWalkscwwlks 28770   WWalksN cwwlksn 28771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-wlks 28547  df-wwlks 28775  df-wwlksn 28776
This theorem is referenced by:  wlklnwwlkln2  28828  wlklnwwlklnupgr2  28830
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