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Theorem eupai 26494
 Description: Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.)
Assertion
Ref Expression
eupai ((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) → (((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))–1-1-onto𝐴𝑃:(0...(#‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}))
Distinct variable groups:   𝐴,𝑘   𝑘,𝐸   𝑘,𝐹   𝑃,𝑘   𝑘,𝑉

Proof of Theorem eupai
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 fndm 5904 . . . . 5 (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
2 iseupa 26492 . . . . 5 (dom 𝐸 = 𝐴 → (𝐹(𝑉 EulPaths 𝐸)𝑃 ↔ (𝑉 UMGrph 𝐸 ∧ ∃𝑛 ∈ ℕ0 (𝐹:(1...𝑛)–1-1-onto𝐴𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}))))
31, 2syl 17 . . . 4 (𝐸 Fn 𝐴 → (𝐹(𝑉 EulPaths 𝐸)𝑃 ↔ (𝑉 UMGrph 𝐸 ∧ ∃𝑛 ∈ ℕ0 (𝐹:(1...𝑛)–1-1-onto𝐴𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}))))
43biimpac 502 . . 3 ((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) → (𝑉 UMGrph 𝐸 ∧ ∃𝑛 ∈ ℕ0 (𝐹:(1...𝑛)–1-1-onto𝐴𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)})))
54simprd 478 . 2 ((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) → ∃𝑛 ∈ ℕ0 (𝐹:(1...𝑛)–1-1-onto𝐴𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}))
6 f1ofn 6051 . . . . . . . . . . . . . 14 (𝐹:(1...𝑛)–1-1-onto𝐴𝐹 Fn (1...𝑛))
76ad2antll 761 . . . . . . . . . . . . 13 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → 𝐹 Fn (1...𝑛))
8 fzfid 12634 . . . . . . . . . . . . 13 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → (1...𝑛) ∈ Fin)
9 fndmeng 7919 . . . . . . . . . . . . 13 ((𝐹 Fn (1...𝑛) ∧ (1...𝑛) ∈ Fin) → (1...𝑛) ≈ 𝐹)
107, 8, 9syl2anc 691 . . . . . . . . . . . 12 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → (1...𝑛) ≈ 𝐹)
11 enfi 8061 . . . . . . . . . . . . . . 15 ((1...𝑛) ≈ 𝐹 → ((1...𝑛) ∈ Fin ↔ 𝐹 ∈ Fin))
1210, 11syl 17 . . . . . . . . . . . . . 14 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → ((1...𝑛) ∈ Fin ↔ 𝐹 ∈ Fin))
138, 12mpbid 221 . . . . . . . . . . . . 13 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → 𝐹 ∈ Fin)
14 hashen 12997 . . . . . . . . . . . . 13 (((1...𝑛) ∈ Fin ∧ 𝐹 ∈ Fin) → ((#‘(1...𝑛)) = (#‘𝐹) ↔ (1...𝑛) ≈ 𝐹))
158, 13, 14syl2anc 691 . . . . . . . . . . . 12 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → ((#‘(1...𝑛)) = (#‘𝐹) ↔ (1...𝑛) ≈ 𝐹))
1610, 15mpbird 246 . . . . . . . . . . 11 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → (#‘(1...𝑛)) = (#‘𝐹))
17 hashfz1 12996 . . . . . . . . . . . 12 (𝑛 ∈ ℕ0 → (#‘(1...𝑛)) = 𝑛)
1817ad2antrl 760 . . . . . . . . . . 11 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → (#‘(1...𝑛)) = 𝑛)
1916, 18eqtr3d 2646 . . . . . . . . . 10 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → (#‘𝐹) = 𝑛)
20 simprl 790 . . . . . . . . . 10 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → 𝑛 ∈ ℕ0)
2119, 20eqeltrd 2688 . . . . . . . . 9 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → (#‘𝐹) ∈ ℕ0)
2221a1d 25 . . . . . . . 8 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → (𝑃:(0...𝑛)⟶𝑉 → (#‘𝐹) ∈ ℕ0))
23 simprr 792 . . . . . . . . . 10 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → 𝐹:(1...𝑛)–1-1-onto𝐴)
2419oveq2d 6565 . . . . . . . . . . 11 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → (1...(#‘𝐹)) = (1...𝑛))
25 f1oeq2 6041 . . . . . . . . . . 11 ((1...(#‘𝐹)) = (1...𝑛) → (𝐹:(1...(#‘𝐹))–1-1-onto𝐴𝐹:(1...𝑛)–1-1-onto𝐴))
2624, 25syl 17 . . . . . . . . . 10 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → (𝐹:(1...(#‘𝐹))–1-1-onto𝐴𝐹:(1...𝑛)–1-1-onto𝐴))
2723, 26mpbird 246 . . . . . . . . 9 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → 𝐹:(1...(#‘𝐹))–1-1-onto𝐴)
2827a1d 25 . . . . . . . 8 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → (𝑃:(0...𝑛)⟶𝑉𝐹:(1...(#‘𝐹))–1-1-onto𝐴))
2919oveq2d 6565 . . . . . . . . . 10 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → (0...(#‘𝐹)) = (0...𝑛))
3029feq2d 5944 . . . . . . . . 9 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → (𝑃:(0...(#‘𝐹))⟶𝑉𝑃:(0...𝑛)⟶𝑉))
3130biimprd 237 . . . . . . . 8 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → (𝑃:(0...𝑛)⟶𝑉𝑃:(0...(#‘𝐹))⟶𝑉))
3222, 28, 313jcad 1236 . . . . . . 7 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → (𝑃:(0...𝑛)⟶𝑉 → ((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))–1-1-onto𝐴𝑃:(0...(#‘𝐹))⟶𝑉)))
3324raleqdv 3121 . . . . . . . 8 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → (∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} ↔ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}))
3433biimprd 237 . . . . . . 7 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → (∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}))
3532, 34anim12d 584 . . . . . 6 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → ((𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}) → (((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))–1-1-onto𝐴𝑃:(0...(#‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)})))
3635expd 451 . . . . 5 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ (𝑛 ∈ ℕ0𝐹:(1...𝑛)–1-1-onto𝐴)) → (𝑃:(0...𝑛)⟶𝑉 → (∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → (((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))–1-1-onto𝐴𝑃:(0...(#‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}))))
3736expr 641 . . . 4 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ 𝑛 ∈ ℕ0) → (𝐹:(1...𝑛)–1-1-onto𝐴 → (𝑃:(0...𝑛)⟶𝑉 → (∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)} → (((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))–1-1-onto𝐴𝑃:(0...(#‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)})))))
38373impd 1273 . . 3 (((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) ∧ 𝑛 ∈ ℕ0) → ((𝐹:(1...𝑛)–1-1-onto𝐴𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}) → (((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))–1-1-onto𝐴𝑃:(0...(#‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)})))
3938rexlimdva 3013 . 2 ((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) → (∃𝑛 ∈ ℕ0 (𝐹:(1...𝑛)–1-1-onto𝐴𝑃:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}) → (((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))–1-1-onto𝐴𝑃:(0...(#‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)})))
405, 39mpd 15 1 ((𝐹(𝑉 EulPaths 𝐸)𝑃𝐸 Fn 𝐴) → (((#‘𝐹) ∈ ℕ0𝐹:(1...(#‘𝐹))–1-1-onto𝐴𝑃:(0...(#‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (1...(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃‘(𝑘 − 1)), (𝑃𝑘)}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {cpr 4127   class class class wbr 4583  dom cdm 5038   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ≈ cen 7838  Fincfn 7841  0cc0 9815  1c1 9816   − cmin 10145  ℕ0cn0 11169  ...cfz 12197  #chash 12979   UMGrph cumg 25841   EulPaths ceup 26489 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-umgra 25842  df-eupa 26490 This theorem is referenced by:  eupacl  26496  eupaf1o  26497  eupapf  26499  eupaseg  26500
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