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Theorem wlkiswwlks2lem4 29126
Description: Lemma 4 for wlkiswwlks2 29129. (Contributed by Alexander van der Vekens, 20-Jul-2018.) (Revised by AV, 10-Apr-2021.)
Hypotheses
Ref Expression
wlkiswwlks2lem.f 𝐹 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))
wlkiswwlks2lem.e 𝐸 = (iEdgβ€˜πΊ)
Assertion
Ref Expression
wlkiswwlks2lem4 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≀ (β™―β€˜π‘ƒ)) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
Distinct variable groups:   π‘₯,𝑃   π‘₯,𝐸   π‘₯,𝑉   𝑖,𝐹   𝑖,𝐺   𝑃,𝑖   𝑖,𝑉,π‘₯
Allowed substitution hints:   𝐸(𝑖)   𝐹(π‘₯)   𝐺(π‘₯)

Proof of Theorem wlkiswwlks2lem4
StepHypRef Expression
1 wlkiswwlks2lem.f . . . 4 𝐹 = (π‘₯ ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)) ↦ (β—‘πΈβ€˜{(π‘ƒβ€˜π‘₯), (π‘ƒβ€˜(π‘₯ + 1))}))
21wlkiswwlks2lem1 29123 . . 3 ((𝑃 ∈ Word 𝑉 ∧ 1 ≀ (β™―β€˜π‘ƒ)) β†’ (β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1))
323adant1 1131 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≀ (β™―β€˜π‘ƒ)) β†’ (β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1))
4 lencl 14483 . . . . . . . . . 10 (𝑃 ∈ Word 𝑉 β†’ (β™―β€˜π‘ƒ) ∈ β„•0)
543ad2ant2 1135 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≀ (β™―β€˜π‘ƒ)) β†’ (β™―β€˜π‘ƒ) ∈ β„•0)
61wlkiswwlks2lem2 29124 . . . . . . . . 9 (((β™―β€˜π‘ƒ) ∈ β„•0 ∧ 𝑖 ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))) β†’ (πΉβ€˜π‘–) = (β—‘πΈβ€˜{(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
75, 6sylan 581 . . . . . . . 8 (((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≀ (β™―β€˜π‘ƒ)) ∧ 𝑖 ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))) β†’ (πΉβ€˜π‘–) = (β—‘πΈβ€˜{(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
87adantr 482 . . . . . . 7 ((((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≀ (β™―β€˜π‘ƒ)) ∧ 𝑖 ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))) ∧ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸) β†’ (πΉβ€˜π‘–) = (β—‘πΈβ€˜{(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
98fveq2d 6896 . . . . . 6 ((((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≀ (β™―β€˜π‘ƒ)) ∧ 𝑖 ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))) ∧ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸) β†’ (πΈβ€˜(πΉβ€˜π‘–)) = (πΈβ€˜(β—‘πΈβ€˜{(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
10 wlkiswwlks2lem.e . . . . . . . . . . 11 𝐸 = (iEdgβ€˜πΊ)
1110uspgrf1oedg 28433 . . . . . . . . . 10 (𝐺 ∈ USPGraph β†’ 𝐸:dom 𝐸–1-1-ontoβ†’(Edgβ€˜πΊ))
1210rneqi 5937 . . . . . . . . . . . 12 ran 𝐸 = ran (iEdgβ€˜πΊ)
13 edgval 28309 . . . . . . . . . . . 12 (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ)
1412, 13eqtr4i 2764 . . . . . . . . . . 11 ran 𝐸 = (Edgβ€˜πΊ)
15 f1oeq3 6824 . . . . . . . . . . 11 (ran 𝐸 = (Edgβ€˜πΊ) β†’ (𝐸:dom 𝐸–1-1-ontoβ†’ran 𝐸 ↔ 𝐸:dom 𝐸–1-1-ontoβ†’(Edgβ€˜πΊ)))
1614, 15ax-mp 5 . . . . . . . . . 10 (𝐸:dom 𝐸–1-1-ontoβ†’ran 𝐸 ↔ 𝐸:dom 𝐸–1-1-ontoβ†’(Edgβ€˜πΊ))
1711, 16sylibr 233 . . . . . . . . 9 (𝐺 ∈ USPGraph β†’ 𝐸:dom 𝐸–1-1-ontoβ†’ran 𝐸)
18173ad2ant1 1134 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≀ (β™―β€˜π‘ƒ)) β†’ 𝐸:dom 𝐸–1-1-ontoβ†’ran 𝐸)
1918adantr 482 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≀ (β™―β€˜π‘ƒ)) ∧ 𝑖 ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))) β†’ 𝐸:dom 𝐸–1-1-ontoβ†’ran 𝐸)
20 f1ocnvfv2 7275 . . . . . . 7 ((𝐸:dom 𝐸–1-1-ontoβ†’ran 𝐸 ∧ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸) β†’ (πΈβ€˜(β—‘πΈβ€˜{(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})
2119, 20sylan 581 . . . . . 6 ((((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≀ (β™―β€˜π‘ƒ)) ∧ 𝑖 ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))) ∧ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸) β†’ (πΈβ€˜(β—‘πΈβ€˜{(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})
229, 21eqtrd 2773 . . . . 5 ((((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≀ (β™―β€˜π‘ƒ)) ∧ 𝑖 ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))) ∧ {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸) β†’ (πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})
2322ex 414 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≀ (β™―β€˜π‘ƒ)) ∧ 𝑖 ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))) β†’ ({(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 β†’ (πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
2423ralimdva 3168 . . 3 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≀ (β™―β€˜π‘ƒ)) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 β†’ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
25 oveq2 7417 . . . . 5 ((β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1) β†’ (0..^(β™―β€˜πΉ)) = (0..^((β™―β€˜π‘ƒ) βˆ’ 1)))
2625raleqdv 3326 . . . 4 ((β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1) β†’ (βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ↔ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
2726imbi2d 341 . . 3 ((β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1) β†’ ((βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}) ↔ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 β†’ βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
2824, 27imbitrrid 245 . 2 ((β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1) β†’ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≀ (β™―β€˜π‘ƒ)) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))})))
293, 28mpcom 38 1 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≀ (β™―β€˜π‘ƒ)) β†’ (βˆ€π‘– ∈ (0..^((β™―β€˜π‘ƒ) βˆ’ 1)){(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))} ∈ ran 𝐸 β†’ βˆ€π‘– ∈ (0..^(β™―β€˜πΉ))(πΈβ€˜(πΉβ€˜π‘–)) = {(π‘ƒβ€˜π‘–), (π‘ƒβ€˜(𝑖 + 1))}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {cpr 4631   class class class wbr 5149   ↦ cmpt 5232  β—‘ccnv 5676  dom cdm 5677  ran crn 5678  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111   + caddc 11113   ≀ cle 11249   βˆ’ cmin 11444  β„•0cn0 12472  ..^cfzo 13627  β™―chash 14290  Word cword 14464  iEdgciedg 28257  Edgcedg 28307  USPGraphcuspgr 28408
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-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-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-edg 28308  df-uspgr 28410
This theorem is referenced by:  wlkiswwlks2lem6  29128
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