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Theorem upgrclwlkcompim 29651
Description: Implications for the properties of the components of a closed walk in a pseudograph. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 2-May-2021.)
Hypotheses
Ref Expression
isclwlke.v 𝑉 = (Vtxβ€˜πΊ)
isclwlke.i 𝐼 = (iEdgβ€˜πΊ)
clwlkcomp.1 𝐹 = (1st β€˜π‘Š)
clwlkcomp.2 𝑃 = (2nd β€˜π‘Š)
Assertion
Ref Expression
upgrclwlkcompim ((𝐺 ∈ UPGraph ∧ π‘Š ∈ (ClWalksβ€˜πΊ)) β†’ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))
Distinct variable groups:   π‘˜,𝐹   π‘˜,𝐺   𝑃,π‘˜   π‘˜,𝐼   π‘˜,𝑉
Allowed substitution hint:   π‘Š(π‘˜)
Proof of Theorem upgrclwlkcompim
StepHypRef Expression
1 isclwlke.v . . . 4 𝑉 = (Vtxβ€˜πΊ)
2 isclwlke.i . . . 4 𝐼 = (iEdgβ€˜πΊ)
3 clwlkcomp.1 . . . 4 𝐹 = (1st β€˜π‘Š)
4 clwlkcomp.2 . . . 4 𝑃 = (2nd β€˜π‘Š)
51, 2, 3, 4clwlkcompim 29650 . . 3 (π‘Š ∈ (ClWalksβ€˜πΊ) β†’ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰) ∧ (βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), (πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘˜))) ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)))))
65adantl 480 . 2 ((𝐺 ∈ UPGraph ∧ π‘Š ∈ (ClWalksβ€˜πΊ)) β†’ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰) ∧ (βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), (πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘˜))) ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)))))
7 simprl 769 . . 3 (((𝐺 ∈ UPGraph ∧ π‘Š ∈ (ClWalksβ€˜πΊ)) ∧ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰) ∧ (βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), (πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘˜))) ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))) β†’ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰))
8 clwlkwlk 29645 . . . . 5 (π‘Š ∈ (ClWalksβ€˜πΊ) β†’ π‘Š ∈ (Walksβ€˜πΊ))
91, 2, 3, 4upgrwlkcompim 29513 . . . . . 6 ((𝐺 ∈ UPGraph ∧ π‘Š ∈ (Walksβ€˜πΊ)) β†’ (𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰ ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))}))
109simp3d 1141 . . . . 5 ((𝐺 ∈ UPGraph ∧ π‘Š ∈ (Walksβ€˜πΊ)) β†’ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))})
118, 10sylan2 591 . . . 4 ((𝐺 ∈ UPGraph ∧ π‘Š ∈ (ClWalksβ€˜πΊ)) β†’ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))})
1211adantr 479 . . 3 (((𝐺 ∈ UPGraph ∧ π‘Š ∈ (ClWalksβ€˜πΊ)) ∧ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰) ∧ (βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), (πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘˜))) ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))) β†’ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))})
13 simprrr 780 . . 3 (((𝐺 ∈ UPGraph ∧ π‘Š ∈ (ClWalksβ€˜πΊ)) ∧ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰) ∧ (βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), (πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘˜))) ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))) β†’ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ)))
147, 12, 133jca 1125 . 2 (((𝐺 ∈ UPGraph ∧ π‘Š ∈ (ClWalksβ€˜πΊ)) ∧ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰) ∧ (βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))if-((π‘ƒβ€˜π‘˜) = (π‘ƒβ€˜(π‘˜ + 1)), (πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜)}, {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} βŠ† (πΌβ€˜(πΉβ€˜π‘˜))) ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))) β†’ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))
156, 14mpdan 685 1 ((𝐺 ∈ UPGraph ∧ π‘Š ∈ (ClWalksβ€˜πΊ)) β†’ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(β™―β€˜πΉ))βŸΆπ‘‰) ∧ βˆ€π‘˜ ∈ (0..^(β™―β€˜πΉ))(πΌβ€˜(πΉβ€˜π‘˜)) = {(π‘ƒβ€˜π‘˜), (π‘ƒβ€˜(π‘˜ + 1))} ∧ (π‘ƒβ€˜0) = (π‘ƒβ€˜(β™―β€˜πΉ))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394  if-wif 1060   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3051   βŠ† wss 3945  {csn 4629  {cpr 4631  dom cdm 5677  βŸΆwf 6543  β€˜cfv 6547  (class class class)co 7417  1st c1st 7990  2nd c2nd 7991  0cc0 11138  1c1 11139   + caddc 11141  ...cfz 13516  ..^cfzo 13659  β™―chash 14321  Word cword 14496  Vtxcvtx 28865  iEdgciedg 28866  UPGraphcupgr 28949  Walkscwlks 29466  ClWalkscclwlks 29640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5364  ax-pr 5428  ax-un 7739  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3775  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3965  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  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 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6499  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7373  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7870  df-1st 7992  df-2nd 7993  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-oadd 8489  df-er 8723  df-map 8845  df-pm 8846  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-dju 9924  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-2 12305  df-n0 12503  df-xnn0 12575  df-z 12589  df-uz 12853  df-fz 13517  df-fzo 13660  df-hash 14322  df-word 14497  df-edg 28917  df-uhgr 28927  df-upgr 28951  df-wlks 29469  df-clwlks 29641
This theorem is referenced by: (None)
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