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Theorem wlkiswwlks2lem4 29394
Description: Lemma 4 for wlkiswwlks2 29397. (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 29391 . . 3 ((𝑃 ∈ Word 𝑉 ∧ 1 ≀ (β™―β€˜π‘ƒ)) β†’ (β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1))
323adant1 1129 . 2 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≀ (β™―β€˜π‘ƒ)) β†’ (β™―β€˜πΉ) = ((β™―β€˜π‘ƒ) βˆ’ 1))
4 lencl 14488 . . . . . . . . . 10 (𝑃 ∈ Word 𝑉 β†’ (β™―β€˜π‘ƒ) ∈ β„•0)
543ad2ant2 1133 . . . . . . . . 9 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≀ (β™―β€˜π‘ƒ)) β†’ (β™―β€˜π‘ƒ) ∈ β„•0)
61wlkiswwlks2lem2 29392 . . . . . . . . 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 579 . . . . . . . 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 480 . . . . . . 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 6895 . . . . . 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 28701 . . . . . . . . . 10 (𝐺 ∈ USPGraph β†’ 𝐸:dom 𝐸–1-1-ontoβ†’(Edgβ€˜πΊ))
1210rneqi 5936 . . . . . . . . . . . 12 ran 𝐸 = ran (iEdgβ€˜πΊ)
13 edgval 28577 . . . . . . . . . . . 12 (Edgβ€˜πΊ) = ran (iEdgβ€˜πΊ)
1412, 13eqtr4i 2762 . . . . . . . . . . 11 ran 𝐸 = (Edgβ€˜πΊ)
15 f1oeq3 6823 . . . . . . . . . . 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 1132 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≀ (β™―β€˜π‘ƒ)) β†’ 𝐸:dom 𝐸–1-1-ontoβ†’ran 𝐸)
1918adantr 480 . . . . . . 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 7278 . . . . . . 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 579 . . . . . 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 2771 . . . . 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 412 . . . 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 3166 . . 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 7420 . . . . 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 3324 . . . 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 340 . . 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  {cpr 4630   class class class wbr 5148   ↦ cmpt 5231  β—‘ccnv 5675  dom cdm 5676  ran crn 5677  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7412  0cc0 11114  1c1 11115   + caddc 11117   ≀ cle 11254   βˆ’ cmin 11449  β„•0cn0 12477  ..^cfzo 13632  β™―chash 14295  Word cword 14469  iEdgciedg 28525  Edgcedg 28575  USPGraphcuspgr 28676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  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 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-card 9938  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-n0 12478  df-z 12564  df-uz 12828  df-fz 13490  df-fzo 13633  df-hash 14296  df-word 14470  df-edg 28576  df-uspgr 28678
This theorem is referenced by:  wlkiswwlks2lem6  29396
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