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Theorem wlksoneq1eq2 28959
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 2732 . . 3 (Vtxβ€˜πΊ) = (Vtxβ€˜πΊ)
21wlkonprop 28953 . 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 28953 . 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 1137 . . . . . . . . . 10 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ (π‘ƒβ€˜0) = 𝐴)
54eqcomd 2738 . . . . . . . . 9 ((𝐹(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐴 ∧ (π‘ƒβ€˜(β™―β€˜πΉ)) = 𝐡) β†’ 𝐴 = (π‘ƒβ€˜0))
6 simp2 1137 . . . . . . . . 9 ((𝐻(Walksβ€˜πΊ)𝑃 ∧ (π‘ƒβ€˜0) = 𝐢 ∧ (π‘ƒβ€˜(β™―β€˜π»)) = 𝐷) β†’ (π‘ƒβ€˜0) = 𝐢)
75, 6sylan9eqr 2794 . . . . . . . 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 1138 . . . . . . . . . . 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 2738 . . . . . . . . . 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 482 . . . . . . . . 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 28917 . . . . . . . . . . . 12 (𝐻(Walksβ€˜πΊ)𝑃 β†’ (β™―β€˜π») = ((β™―β€˜π‘ƒ) βˆ’ 1))
12 wlklenvm1 28917 . . . . . . . . . . . . . . 15 (𝐹(Walksβ€˜πΊ)𝑃 β†’ (β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1))
13 eqtr3 2758 . . . . . . . . . . . . . . . . 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 6895 . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . 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 1133 . . . . . . . . . . . . 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 1133 . . . . . . . . . 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 407 . . . . . . . . 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 1193 . . . . . . . . 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 2776 . . . . . . . 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 512 . . . . . . 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 413 . . . . . 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 1135 . . . . 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 1135 . . 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 407 . 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 596 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3474   class class class wbr 5148  β€˜cfv 6543  (class class class)co 7411  0cc0 11112  1c1 11113   βˆ’ cmin 11446  β™―chash 14292  Vtxcvtx 28294  Walkscwlks 28891  WalksOncwlkson 28892
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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11252  df-mnf 11253  df-xr 11254  df-ltxr 11255  df-le 11256  df-sub 11448  df-neg 11449  df-nn 12215  df-n0 12475  df-z 12561  df-uz 12825  df-fz 13487  df-fzo 13630  df-hash 14293  df-word 14467  df-wlks 28894  df-wlkson 28895
This theorem is referenced by:  wspthneq1eq2  29152
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