Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  crctcshlem4 Structured version   Visualization version   GIF version

Theorem crctcshlem4 41023
 Description: Lemma for crctcsh 41027. (Contributed by AV, 10-Mar-2021.)
Hypotheses
Ref Expression
crctcsh.v 𝑉 = (Vtx‘𝐺)
crctcsh.i 𝐼 = (iEdg‘𝐺)
crctcsh.d (𝜑𝐹(CircuitS‘𝐺)𝑃)
crctcsh.n 𝑁 = (#‘𝐹)
crctcsh.s (𝜑𝑆 ∈ (0..^𝑁))
crctcsh.h 𝐻 = (𝐹 cyclShift 𝑆)
crctcsh.q 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))
Assertion
Ref Expression
crctcshlem4 ((𝜑𝑆 = 0) → (𝐻 = 𝐹𝑄 = 𝑃))
Distinct variable groups:   𝑥,𝑁   𝑥,𝑃   𝑥,𝑆   𝜑,𝑥
Allowed substitution hints:   𝑄(𝑥)   𝐹(𝑥)   𝐺(𝑥)   𝐻(𝑥)   𝐼(𝑥)   𝑉(𝑥)

Proof of Theorem crctcshlem4
StepHypRef Expression
1 crctcsh.h . . 3 𝐻 = (𝐹 cyclShift 𝑆)
2 oveq2 6557 . . . 4 (𝑆 = 0 → (𝐹 cyclShift 𝑆) = (𝐹 cyclShift 0))
3 crctcsh.d . . . . . 6 (𝜑𝐹(CircuitS‘𝐺)𝑃)
4 crctis1wlk 41002 . . . . . 6 (𝐹(CircuitS‘𝐺)𝑃𝐹(1Walks‘𝐺)𝑃)
5 crctcsh.i . . . . . . 7 𝐼 = (iEdg‘𝐺)
651wlkf 40819 . . . . . 6 (𝐹(1Walks‘𝐺)𝑃𝐹 ∈ Word dom 𝐼)
73, 4, 63syl 18 . . . . 5 (𝜑𝐹 ∈ Word dom 𝐼)
8 cshw0 13391 . . . . 5 (𝐹 ∈ Word dom 𝐼 → (𝐹 cyclShift 0) = 𝐹)
97, 8syl 17 . . . 4 (𝜑 → (𝐹 cyclShift 0) = 𝐹)
102, 9sylan9eqr 2666 . . 3 ((𝜑𝑆 = 0) → (𝐹 cyclShift 𝑆) = 𝐹)
111, 10syl5eq 2656 . 2 ((𝜑𝑆 = 0) → 𝐻 = 𝐹)
12 crctcsh.q . . 3 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))
13 oveq2 6557 . . . . . . . . 9 (𝑆 = 0 → (𝑁𝑆) = (𝑁 − 0))
14 crctcsh.v . . . . . . . . . . . 12 𝑉 = (Vtx‘𝐺)
15 crctcsh.n . . . . . . . . . . . 12 𝑁 = (#‘𝐹)
1614, 5, 3, 15crctcshlem1 41020 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℕ0)
1716nn0cnd 11230 . . . . . . . . . 10 (𝜑𝑁 ∈ ℂ)
1817subid1d 10260 . . . . . . . . 9 (𝜑 → (𝑁 − 0) = 𝑁)
1913, 18sylan9eqr 2666 . . . . . . . 8 ((𝜑𝑆 = 0) → (𝑁𝑆) = 𝑁)
2019breq2d 4595 . . . . . . 7 ((𝜑𝑆 = 0) → (𝑥 ≤ (𝑁𝑆) ↔ 𝑥𝑁))
2120adantr 480 . . . . . 6 (((𝜑𝑆 = 0) ∧ 𝑥 ∈ (0...𝑁)) → (𝑥 ≤ (𝑁𝑆) ↔ 𝑥𝑁))
22 oveq2 6557 . . . . . . . . 9 (𝑆 = 0 → (𝑥 + 𝑆) = (𝑥 + 0))
2322adantl 481 . . . . . . . 8 ((𝜑𝑆 = 0) → (𝑥 + 𝑆) = (𝑥 + 0))
24 elfzelz 12213 . . . . . . . . . 10 (𝑥 ∈ (0...𝑁) → 𝑥 ∈ ℤ)
2524zcnd 11359 . . . . . . . . 9 (𝑥 ∈ (0...𝑁) → 𝑥 ∈ ℂ)
2625addid1d 10115 . . . . . . . 8 (𝑥 ∈ (0...𝑁) → (𝑥 + 0) = 𝑥)
2723, 26sylan9eq 2664 . . . . . . 7 (((𝜑𝑆 = 0) ∧ 𝑥 ∈ (0...𝑁)) → (𝑥 + 𝑆) = 𝑥)
2827fveq2d 6107 . . . . . 6 (((𝜑𝑆 = 0) ∧ 𝑥 ∈ (0...𝑁)) → (𝑃‘(𝑥 + 𝑆)) = (𝑃𝑥))
2927oveq1d 6564 . . . . . . 7 (((𝜑𝑆 = 0) ∧ 𝑥 ∈ (0...𝑁)) → ((𝑥 + 𝑆) − 𝑁) = (𝑥𝑁))
3029fveq2d 6107 . . . . . 6 (((𝜑𝑆 = 0) ∧ 𝑥 ∈ (0...𝑁)) → (𝑃‘((𝑥 + 𝑆) − 𝑁)) = (𝑃‘(𝑥𝑁)))
3121, 28, 30ifbieq12d 4063 . . . . 5 (((𝜑𝑆 = 0) ∧ 𝑥 ∈ (0...𝑁)) → if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) = if(𝑥𝑁, (𝑃𝑥), (𝑃‘(𝑥𝑁))))
3231mpteq2dva 4672 . . . 4 ((𝜑𝑆 = 0) → (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁)))) = (𝑥 ∈ (0...𝑁) ↦ if(𝑥𝑁, (𝑃𝑥), (𝑃‘(𝑥𝑁)))))
33 elfzle2 12216 . . . . . . . . 9 (𝑥 ∈ (0...𝑁) → 𝑥𝑁)
3433adantl 481 . . . . . . . 8 ((𝜑𝑥 ∈ (0...𝑁)) → 𝑥𝑁)
3534iftrued 4044 . . . . . . 7 ((𝜑𝑥 ∈ (0...𝑁)) → if(𝑥𝑁, (𝑃𝑥), (𝑃‘(𝑥𝑁))) = (𝑃𝑥))
3635mpteq2dva 4672 . . . . . 6 (𝜑 → (𝑥 ∈ (0...𝑁) ↦ if(𝑥𝑁, (𝑃𝑥), (𝑃‘(𝑥𝑁)))) = (𝑥 ∈ (0...𝑁) ↦ (𝑃𝑥)))
37141wlkp 40821 . . . . . . . 8 (𝐹(1Walks‘𝐺)𝑃𝑃:(0...(#‘𝐹))⟶𝑉)
383, 4, 373syl 18 . . . . . . 7 (𝜑𝑃:(0...(#‘𝐹))⟶𝑉)
39 ffn 5958 . . . . . . . . . . 11 (𝑃:(0...(#‘𝐹))⟶𝑉𝑃 Fn (0...(#‘𝐹)))
4015eqcomi 2619 . . . . . . . . . . . . 13 (#‘𝐹) = 𝑁
4140oveq2i 6560 . . . . . . . . . . . 12 (0...(#‘𝐹)) = (0...𝑁)
4241fneq2i 5900 . . . . . . . . . . 11 (𝑃 Fn (0...(#‘𝐹)) ↔ 𝑃 Fn (0...𝑁))
4339, 42sylib 207 . . . . . . . . . 10 (𝑃:(0...(#‘𝐹))⟶𝑉𝑃 Fn (0...𝑁))
4443adantl 481 . . . . . . . . 9 ((𝜑𝑃:(0...(#‘𝐹))⟶𝑉) → 𝑃 Fn (0...𝑁))
45 dffn5 6151 . . . . . . . . 9 (𝑃 Fn (0...𝑁) ↔ 𝑃 = (𝑥 ∈ (0...𝑁) ↦ (𝑃𝑥)))
4644, 45sylib 207 . . . . . . . 8 ((𝜑𝑃:(0...(#‘𝐹))⟶𝑉) → 𝑃 = (𝑥 ∈ (0...𝑁) ↦ (𝑃𝑥)))
4746eqcomd 2616 . . . . . . 7 ((𝜑𝑃:(0...(#‘𝐹))⟶𝑉) → (𝑥 ∈ (0...𝑁) ↦ (𝑃𝑥)) = 𝑃)
4838, 47mpdan 699 . . . . . 6 (𝜑 → (𝑥 ∈ (0...𝑁) ↦ (𝑃𝑥)) = 𝑃)
4936, 48eqtrd 2644 . . . . 5 (𝜑 → (𝑥 ∈ (0...𝑁) ↦ if(𝑥𝑁, (𝑃𝑥), (𝑃‘(𝑥𝑁)))) = 𝑃)
5049adantr 480 . . . 4 ((𝜑𝑆 = 0) → (𝑥 ∈ (0...𝑁) ↦ if(𝑥𝑁, (𝑃𝑥), (𝑃‘(𝑥𝑁)))) = 𝑃)
5132, 50eqtrd 2644 . . 3 ((𝜑𝑆 = 0) → (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁)))) = 𝑃)
5212, 51syl5eq 2656 . 2 ((𝜑𝑆 = 0) → 𝑄 = 𝑃)
5311, 52jca 553 1 ((𝜑𝑆 = 0) → (𝐻 = 𝐹𝑄 = 𝑃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ifcif 4036   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  0cc0 9815   + caddc 9818   ≤ cle 9954   − cmin 10145  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   cyclShift ccsh 13385  Vtxcvtx 25673  iEdgciedg 25674  1Walksc1wlks 40796  CircuitSccrcts 40990 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  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  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-rmo 2904  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-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  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-div 10564  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-hash 12980  df-word 13154  df-concat 13156  df-substr 13158  df-csh 13386  df-1wlks 40800  df-trls 40901  df-crcts 40992 This theorem is referenced by:  crctcsh1wlk  41025  crctcsh  41027
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