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Theorem wlksoneq1eq2 28921
Description: Two walks with identical sequences of vertices start and end at the same vertices. (Contributed by AV, 14-May-2021.)
Assertion
Ref Expression
wlksoneq1eq2 ((𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 ∧ 𝐻(𝐢(WalksOnβ€˜πΊ)𝐷)𝑃) β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷))
Proof of Theorem wlksoneq1eq2
StepHypRef Expression
1 eqid 2733 . . 3 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
21wlkonprop 28915 . 2 (𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 β†’ ((𝐺 ∈ V ∧ 𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)))
31wlkonprop 28915 . 2 (𝐻(𝐢(WalksOnβ€˜πΊ)𝐷)𝑃 β†’ ((𝐺 ∈ V ∧ 𝐢 ∈ (Vtxβ€˜πΊ) ∧ 𝐷 ∈ (Vtxβ€˜πΊ)) ∧ (𝐻 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐻(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐢 ∧ (π‘ƒβ€˜(β™―β€˜π»)) = 𝐷)))
4 simp2 1138 . . . . . . . . . 10 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ (π‘ƒβ€˜0) = 𝐴)
54eqcomd 2739 . . . . . . . . 9 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ 𝐴 = (π‘ƒβ€˜0))
6 simp2 1138 . . . . . . . . 9 ((𝐻(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐢 ∧ (π‘ƒβ€˜(β™―β€˜π»)) = 𝐷) β†’ (π‘ƒβ€˜0) = 𝐢)
75, 6sylan9eqr 2795 . . . . . . . 8 (((𝐻(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐢 ∧ (π‘ƒβ€˜(β™―β€˜π»)) = 𝐷) ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)) β†’ 𝐴 = 𝐢)
8 simp3 1139 . . . . . . . . . . 11 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)
98eqcomd 2739 . . . . . . . . . 10 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ 𝐡 = (π‘ƒβ€˜(β™―β€˜πΉ)))
109adantl 483 . . . . . . . . 9 (((𝐻(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐢 ∧ (π‘ƒβ€˜(β™―β€˜π»)) = 𝐷) ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)) β†’ 𝐡 = (π‘ƒβ€˜(β™―β€˜πΉ)))
11 wlklenvm1 28879 . . . . . . . . . . . 12 (𝐻(Walksβ€˜πΊ)𝑃 β†’ (β™―β€˜π») = ((β™―β€˜π‘ƒ) βˆ’ 1))
12 wlklenvm1 28879 . . . . . . . . . . . . . . 15 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1))
13 eqtr3 2759 . . . . . . . . . . . . . . . . 17 (((β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1) ∧ (β™―β€˜π») = ((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ (β™―β€˜πΉ) = (β™―β€˜π»))
1413fveq2d 6896 . . . . . . . . . . . . . . . 16 (((β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1) ∧ (β™―β€˜π») = ((β™―β€˜π‘ƒ) βˆ’ 1)) β†’ (π‘ƒβ€˜(β™―β€˜πΉ)) = (π‘ƒβ€˜(β™―β€˜π»)))
1514ex 414 . . . . . . . . . . . . . . 15 ((β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1) β†’ ((β™―β€˜π») = ((β™―β€˜π‘ƒ) βˆ’ 1) β†’ (π‘ƒβ€˜(β™―β€˜πΉ)) = (π‘ƒβ€˜(β™―β€˜π»))))
1612, 15syl 17 . . . . . . . . . . . . . 14 (𝐹(Walksβ€˜πΊ)𝑃 β†’ ((β™―β€˜π») = ((β™―β€˜π‘ƒ) βˆ’ 1) β†’ (π‘ƒβ€˜(β™―β€˜πΉ)) = (π‘ƒβ€˜(β™―β€˜π»))))
17163ad2ant1 1134 . . . . . . . . . . . . 13 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ ((β™―β€˜π») = ((β™―β€˜π‘ƒ) βˆ’ 1) β†’ (π‘ƒβ€˜(β™―β€˜πΉ)) = (π‘ƒβ€˜(β™―β€˜π»))))
1817com12 32 . . . . . . . . . . . 12 ((β™―β€˜π») = ((β™―β€˜π‘ƒ) βˆ’ 1) β†’ ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ (π‘ƒβ€˜(β™―β€˜πΉ)) = (π‘ƒβ€˜(β™―β€˜π»))))
1911, 18syl 17 . . . . . . . . . . 11 (𝐻(Walksβ€˜πΊ)𝑃 β†’ ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ (π‘ƒβ€˜(β™―β€˜πΉ)) = (π‘ƒβ€˜(β™―β€˜π»))))
20193ad2ant1 1134 . . . . . . . . . 10 ((𝐻(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐢 ∧ (π‘ƒβ€˜(β™―β€˜π»)) = 𝐷) β†’ ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ (π‘ƒβ€˜(β™―β€˜πΉ)) = (π‘ƒβ€˜(β™―β€˜π»))))
2120imp 408 . . . . . . . . 9 (((𝐻(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐢 ∧ (π‘ƒβ€˜(β™―β€˜π»)) = 𝐷) ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)) β†’ (π‘ƒβ€˜(β™―β€˜πΉ)) = (π‘ƒβ€˜(β™―β€˜π»)))
22 simpl3 1194 . . . . . . . . 9 (((𝐻(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐢 ∧ (π‘ƒβ€˜(β™―β€˜π»)) = 𝐷) ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)) β†’ (π‘ƒβ€˜(β™―β€˜π»)) = 𝐷)
2310, 21, 223eqtrd 2777 . . . . . . . 8 (((𝐻(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐢 ∧ (π‘ƒβ€˜(β™―β€˜π»)) = 𝐷) ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)) β†’ 𝐡 = 𝐷)
247, 23jca 513 . . . . . . 7 (((𝐻(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐢 ∧ (π‘ƒβ€˜(β™―β€˜π»)) = 𝐷) ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)) β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷))
2524ex 414 . . . . . 6 ((𝐻(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐢 ∧ (π‘ƒβ€˜(β™―β€˜π»)) = 𝐷) β†’ ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷)))
26253ad2ant3 1136 . . . . 5 (((𝐺 ∈ V ∧ 𝐢 ∈ (Vtxβ€˜πΊ) ∧ 𝐷 ∈ (Vtxβ€˜πΊ)) ∧ (𝐻 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐻(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐢 ∧ (π‘ƒβ€˜(β™―β€˜π»)) = 𝐷)) β†’ ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷)))
2726com12 32 . . . 4 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ (((𝐺 ∈ V ∧ 𝐢 ∈ (Vtxβ€˜πΊ) ∧ 𝐷 ∈ (Vtxβ€˜πΊ)) ∧ (𝐻 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐻(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐢 ∧ (π‘ƒβ€˜(β™―β€˜π»)) = 𝐷)) β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷)))
28273ad2ant3 1136 . . 3 (((𝐺 ∈ V ∧ 𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)) β†’ (((𝐺 ∈ V ∧ 𝐢 ∈ (Vtxβ€˜πΊ) ∧ 𝐷 ∈ (Vtxβ€˜πΊ)) ∧ (𝐻 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐻(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐢 ∧ (π‘ƒβ€˜(β™―β€˜π»)) = 𝐷)) β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷)))
2928imp 408 . 2 ((((𝐺 ∈ V ∧ 𝐴 ∈ (Vtxβ€˜πΊ) ∧ 𝐡 ∈ (Vtxβ€˜πΊ)) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡)) ∧ ((𝐺 ∈ V ∧ 𝐢 ∈ (Vtxβ€˜πΊ) ∧ 𝐷 ∈ (Vtxβ€˜πΊ)) ∧ (𝐻 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐻(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐢 ∧ (π‘ƒβ€˜(β™―β€˜π»)) = 𝐷))) β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷))
302, 3, 29syl2an 597 1 ((𝐹(𝐴(WalksOnβ€˜πΊ)𝐡)𝑃 ∧ 𝐻(𝐢(WalksOnβ€˜πΊ)𝐷)𝑃) β†’ (𝐴 = 𝐢 ∧ 𝐡 = 𝐷))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3475   class class class wbr 5149  β€˜cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111   βˆ’ cmin 11444  β™―chash 14290  Vtxcvtx 28256  Walkscwlks 28853  WalksOncwlkson 28854
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-wlkson 28857
This theorem is referenced by:  wspthneq1eq2  29114
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