MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  crctprop Structured version   Visualization version   GIF version

Theorem crctprop 29824
Description: The properties of a circuit: A circuit is a closed trail. (Contributed by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
Assertion
Ref Expression
crctprop (𝐹(Circuits‘𝐺)𝑃 → (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))

Proof of Theorem crctprop
StepHypRef Expression
1 iscrct 29822 . 2 (𝐹(Circuits‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
21biimpi 216 1 (𝐹(Circuits‘𝐺)𝑃 → (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536   class class class wbr 5147  cfv 6562  0cc0 11152  chash 14365  Trailsctrls 29722  Circuitsccrcts 29816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fv 6570  df-trls 29724  df-crcts 29818
This theorem is referenced by:  crctisclwlk  29826  crctistrl  29827  usgr2trlncrct  29835  uspgrn2crct  29837  crctcshwlkn0  29850  eulercrct  30270  eucrct2eupth  30273
  Copyright terms: Public domain W3C validator