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Theorem crctcshlem4 26581
 Description: Lemma for crctcsh 26585. (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 6612 . . . 4 (𝑆 = 0 → (𝐹 cyclShift 𝑆) = (𝐹 cyclShift 0))
3 crctcsh.d . . . . . 6 (𝜑𝐹(Circuits‘𝐺)𝑃)
4 crctiswlk 26560 . . . . . 6 (𝐹(Circuits‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)
5 crctcsh.i . . . . . . 7 𝐼 = (iEdg‘𝐺)
65wlkf 26380 . . . . . 6 (𝐹(Walks‘𝐺)𝑃𝐹 ∈ Word dom 𝐼)
73, 4, 63syl 18 . . . . 5 (𝜑𝐹 ∈ Word dom 𝐼)
8 cshw0 13477 . . . . 5 (𝐹 ∈ Word dom 𝐼 → (𝐹 cyclShift 0) = 𝐹)
97, 8syl 17 . . . 4 (𝜑 → (𝐹 cyclShift 0) = 𝐹)
102, 9sylan9eqr 2677 . . 3 ((𝜑𝑆 = 0) → (𝐹 cyclShift 𝑆) = 𝐹)
111, 10syl5eq 2667 . 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 6612 . . . . . . . . 9 (𝑆 = 0 → (𝑁𝑆) = (𝑁 − 0))
14 crctcsh.v . . . . . . . . . . . 12 𝑉 = (Vtx‘𝐺)
15 crctcsh.n . . . . . . . . . . . 12 𝑁 = (#‘𝐹)
1614, 5, 3, 15crctcshlem1 26578 . . . . . . . . . . 11 (𝜑𝑁 ∈ ℕ0)
1716nn0cnd 11297 . . . . . . . . . 10 (𝜑𝑁 ∈ ℂ)
1817subid1d 10325 . . . . . . . . 9 (𝜑 → (𝑁 − 0) = 𝑁)
1913, 18sylan9eqr 2677 . . . . . . . 8 ((𝜑𝑆 = 0) → (𝑁𝑆) = 𝑁)
2019breq2d 4625 . . . . . . 7 ((𝜑𝑆 = 0) → (𝑥 ≤ (𝑁𝑆) ↔ 𝑥𝑁))
2120adantr 481 . . . . . 6 (((𝜑𝑆 = 0) ∧ 𝑥 ∈ (0...𝑁)) → (𝑥 ≤ (𝑁𝑆) ↔ 𝑥𝑁))
22 oveq2 6612 . . . . . . . . 9 (𝑆 = 0 → (𝑥 + 𝑆) = (𝑥 + 0))
2322adantl 482 . . . . . . . 8 ((𝜑𝑆 = 0) → (𝑥 + 𝑆) = (𝑥 + 0))
24 elfzelz 12284 . . . . . . . . . 10 (𝑥 ∈ (0...𝑁) → 𝑥 ∈ ℤ)
2524zcnd 11427 . . . . . . . . 9 (𝑥 ∈ (0...𝑁) → 𝑥 ∈ ℂ)
2625addid1d 10180 . . . . . . . 8 (𝑥 ∈ (0...𝑁) → (𝑥 + 0) = 𝑥)
2723, 26sylan9eq 2675 . . . . . . 7 (((𝜑𝑆 = 0) ∧ 𝑥 ∈ (0...𝑁)) → (𝑥 + 𝑆) = 𝑥)
2827fveq2d 6152 . . . . . 6 (((𝜑𝑆 = 0) ∧ 𝑥 ∈ (0...𝑁)) → (𝑃‘(𝑥 + 𝑆)) = (𝑃𝑥))
2927oveq1d 6619 . . . . . . 7 (((𝜑𝑆 = 0) ∧ 𝑥 ∈ (0...𝑁)) → ((𝑥 + 𝑆) − 𝑁) = (𝑥𝑁))
3029fveq2d 6152 . . . . . 6 (((𝜑𝑆 = 0) ∧ 𝑥 ∈ (0...𝑁)) → (𝑃‘((𝑥 + 𝑆) − 𝑁)) = (𝑃‘(𝑥𝑁)))
3121, 28, 30ifbieq12d 4085 . . . . 5 (((𝜑𝑆 = 0) ∧ 𝑥 ∈ (0...𝑁)) → if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))) = if(𝑥𝑁, (𝑃𝑥), (𝑃‘(𝑥𝑁))))
3231mpteq2dva 4704 . . . 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 12287 . . . . . . . . 9 (𝑥 ∈ (0...𝑁) → 𝑥𝑁)
3433adantl 482 . . . . . . . 8 ((𝜑𝑥 ∈ (0...𝑁)) → 𝑥𝑁)
3534iftrued 4066 . . . . . . 7 ((𝜑𝑥 ∈ (0...𝑁)) → if(𝑥𝑁, (𝑃𝑥), (𝑃‘(𝑥𝑁))) = (𝑃𝑥))
3635mpteq2dva 4704 . . . . . 6 (𝜑 → (𝑥 ∈ (0...𝑁) ↦ if(𝑥𝑁, (𝑃𝑥), (𝑃‘(𝑥𝑁)))) = (𝑥 ∈ (0...𝑁) ↦ (𝑃𝑥)))
3714wlkp 26382 . . . . . . . 8 (𝐹(Walks‘𝐺)𝑃𝑃:(0...(#‘𝐹))⟶𝑉)
383, 4, 373syl 18 . . . . . . 7 (𝜑𝑃:(0...(#‘𝐹))⟶𝑉)
39 ffn 6002 . . . . . . . . . . 11 (𝑃:(0...(#‘𝐹))⟶𝑉𝑃 Fn (0...(#‘𝐹)))
4015eqcomi 2630 . . . . . . . . . . . . 13 (#‘𝐹) = 𝑁
4140oveq2i 6615 . . . . . . . . . . . 12 (0...(#‘𝐹)) = (0...𝑁)
4241fneq2i 5944 . . . . . . . . . . 11 (𝑃 Fn (0...(#‘𝐹)) ↔ 𝑃 Fn (0...𝑁))
4339, 42sylib 208 . . . . . . . . . 10 (𝑃:(0...(#‘𝐹))⟶𝑉𝑃 Fn (0...𝑁))
4443adantl 482 . . . . . . . . 9 ((𝜑𝑃:(0...(#‘𝐹))⟶𝑉) → 𝑃 Fn (0...𝑁))
45 dffn5 6198 . . . . . . . . 9 (𝑃 Fn (0...𝑁) ↔ 𝑃 = (𝑥 ∈ (0...𝑁) ↦ (𝑃𝑥)))
4644, 45sylib 208 . . . . . . . 8 ((𝜑𝑃:(0...(#‘𝐹))⟶𝑉) → 𝑃 = (𝑥 ∈ (0...𝑁) ↦ (𝑃𝑥)))
4746eqcomd 2627 . . . . . . 7 ((𝜑𝑃:(0...(#‘𝐹))⟶𝑉) → (𝑥 ∈ (0...𝑁) ↦ (𝑃𝑥)) = 𝑃)
4838, 47mpdan 701 . . . . . 6 (𝜑 → (𝑥 ∈ (0...𝑁) ↦ (𝑃𝑥)) = 𝑃)
4936, 48eqtrd 2655 . . . . 5 (𝜑 → (𝑥 ∈ (0...𝑁) ↦ if(𝑥𝑁, (𝑃𝑥), (𝑃‘(𝑥𝑁)))) = 𝑃)
5049adantr 481 . . . 4 ((𝜑𝑆 = 0) → (𝑥 ∈ (0...𝑁) ↦ if(𝑥𝑁, (𝑃𝑥), (𝑃‘(𝑥𝑁)))) = 𝑃)
5132, 50eqtrd 2655 . . 3 ((𝜑𝑆 = 0) → (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁)))) = 𝑃)
5212, 51syl5eq 2667 . 2 ((𝜑𝑆 = 0) → 𝑄 = 𝑃)
5311, 52jca 554 1 ((𝜑𝑆 = 0) → (𝐻 = 𝐹𝑄 = 𝑃))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ifcif 4058   class class class wbr 4613   ↦ cmpt 4673  dom cdm 5074   Fn wfn 5842  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604  0cc0 9880   + caddc 9883   ≤ cle 10019   − cmin 10210  ...cfz 12268  ..^cfzo 12406  #chash 13057  Word cword 13230   cyclShift ccsh 13471  Vtxcvtx 25774  iEdgciedg 25775  Walkscwlks 26362  Circuitsccrcts 26548 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-sup 8292  df-inf 8293  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-rp 11777  df-fz 12269  df-fzo 12407  df-fl 12533  df-mod 12609  df-hash 13058  df-word 13238  df-concat 13240  df-substr 13242  df-csh 13472  df-wlks 26365  df-trls 26458  df-crcts 26550 This theorem is referenced by:  crctcshwlk  26583  crctcsh  26585
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