Theorem clwlksfclwwlk1hashOLD 27437

Description: Obsolete as of 23-May-2022. (Contributed by Alexander van der Vekens, 25-Jun-2018.) (Revised by AV, 2-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
clwlksfclwwlkOLD.1	⊢ 𝐴 = (1st ‘𝑐)
clwlksfclwwlkOLD.2	⊢ 𝐵 = (2nd ‘𝑐)
clwlksfclwwlkOLD.c	⊢ 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘𝐴) = 𝑁}
clwlksfclwwlkOLD.f	⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))

Assertion

Ref	Expression
clwlksfclwwlk1hashOLD	⊢ (𝑐 ∈ 𝐶 → (♯‘𝐴) ∈ (0...(♯‘𝐵)))

Distinct variable groups:   𝐺,𝑐   𝑁,𝑐

Allowed substitution hints:   𝐴(𝑐)   𝐵(𝑐)   𝐶(𝑐)   𝐹(𝑐)

Proof of Theorem clwlksfclwwlk1hashOLD

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwlksfclwwlkOLD.c	. . 3 ⊢ 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘𝐴) = 𝑁}
2	1	rabeq2i 3411	. 2 ⊢ (𝑐 ∈ 𝐶 ↔ (𝑐 ∈ (ClWalks‘𝐺) ∧ (♯‘𝐴) = 𝑁))
3		eqid 2826	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4		eqid 2826	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
5		clwlksfclwwlkOLD.1	. . . . 5 ⊢ 𝐴 = (1st ‘𝑐)
6		clwlksfclwwlkOLD.2	. . . . 5 ⊢ 𝐵 = (2nd ‘𝑐)
7	3, 4, 5, 6	clwlkcompim 27083	. . . 4 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → ((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(♯‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(♯‘𝐴))if-((𝐵‘𝑖) = (𝐵‘(𝑖 + 1)), ((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖)}, {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐴‘𝑖))) ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)))))
8		lencl 13594	. . . . . 6 ⊢ (𝐴 ∈ Word dom (iEdg‘𝐺) → (♯‘𝐴) ∈ ℕ0)
9		ffn 6279	. . . . . 6 ⊢ (𝐵:(0...(♯‘𝐴))⟶(Vtx‘𝐺) → 𝐵 Fn (0...(♯‘𝐴)))
10		fnfz0hash 13520	. . . . . . 7 ⊢ (((♯‘𝐴) ∈ ℕ0 ∧ 𝐵 Fn (0...(♯‘𝐴))) → (♯‘𝐵) = ((♯‘𝐴) + 1))
11		nn0fz0 12733	. . . . . . . . . 10 ⊢ ((♯‘𝐴) ∈ ℕ0 ↔ (♯‘𝐴) ∈ (0...(♯‘𝐴)))
12		fzelp1 12687	. . . . . . . . . 10 ⊢ ((♯‘𝐴) ∈ (0...(♯‘𝐴)) → (♯‘𝐴) ∈ (0...((♯‘𝐴) + 1)))
13	11, 12	sylbi 209	. . . . . . . . 9 ⊢ ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ (0...((♯‘𝐴) + 1)))
14		oveq2 6914	. . . . . . . . . 10 ⊢ ((♯‘𝐵) = ((♯‘𝐴) + 1) → (0...(♯‘𝐵)) = (0...((♯‘𝐴) + 1)))
15	14	eleq2d 2893	. . . . . . . . 9 ⊢ ((♯‘𝐵) = ((♯‘𝐴) + 1) → ((♯‘𝐴) ∈ (0...(♯‘𝐵)) ↔ (♯‘𝐴) ∈ (0...((♯‘𝐴) + 1))))
16	13, 15	syl5ibrcom 239	. . . . . . . 8 ⊢ ((♯‘𝐴) ∈ ℕ0 → ((♯‘𝐵) = ((♯‘𝐴) + 1) → (♯‘𝐴) ∈ (0...(♯‘𝐵))))
17	16	adantr 474	. . . . . . 7 ⊢ (((♯‘𝐴) ∈ ℕ0 ∧ 𝐵 Fn (0...(♯‘𝐴))) → ((♯‘𝐵) = ((♯‘𝐴) + 1) → (♯‘𝐴) ∈ (0...(♯‘𝐵))))
18	10, 17	mpd 15	. . . . . 6 ⊢ (((♯‘𝐴) ∈ ℕ0 ∧ 𝐵 Fn (0...(♯‘𝐴))) → (♯‘𝐴) ∈ (0...(♯‘𝐵)))
19	8, 9, 18	syl2an 591	. . . . 5 ⊢ ((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(♯‘𝐴))⟶(Vtx‘𝐺)) → (♯‘𝐴) ∈ (0...(♯‘𝐵)))
20	19	adantr 474	. . . 4 ⊢ (((𝐴 ∈ Word dom (iEdg‘𝐺) ∧ 𝐵:(0...(♯‘𝐴))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(♯‘𝐴))if-((𝐵‘𝑖) = (𝐵‘(𝑖 + 1)), ((iEdg‘𝐺)‘(𝐴‘𝑖)) = {(𝐵‘𝑖)}, {(𝐵‘𝑖), (𝐵‘(𝑖 + 1))} ⊆ ((iEdg‘𝐺)‘(𝐴‘𝑖))) ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)))) → (♯‘𝐴) ∈ (0...(♯‘𝐵)))
21	7, 20	syl 17	. . 3 ⊢ (𝑐 ∈ (ClWalks‘𝐺) → (♯‘𝐴) ∈ (0...(♯‘𝐵)))
22	21	adantr 474	. 2 ⊢ ((𝑐 ∈ (ClWalks‘𝐺) ∧ (♯‘𝐴) = 𝑁) → (♯‘𝐴) ∈ (0...(♯‘𝐵)))
23	2, 22	sylbi 209	1 ⊢ (𝑐 ∈ 𝐶 → (♯‘𝐴) ∈ (0...(♯‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 386  if-wif 1091   = wceq 1658   ∈ wcel 2166  ∀wral 3118  {crab 3122   ⊆ wss 3799  {csn 4398  {cpr 4400  ⟨cop 4404   ↦ cmpt 4953  dom cdm 5343   Fn wfn 6119  ⟶wf 6120  ‘cfv 6124  (class class class)co 6906  1^st c1st 7427  2^nd c2nd 7428  0cc0 10253  1c1 10254   + caddc 10256  ℕ₀cn0 11619  ...cfz 12620  ..^cfzo 12761  ♯chash 13411  Word cword 13575   substr csubstr 13701  Vtxcvtx 26295  iEdgciedg 26296  ClWalkscclwlks 27073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210  ax-cnex 10309  ax-resscn 10310  ax-1cn 10311  ax-icn 10312  ax-addcl 10313  ax-addrcl 10314  ax-mulcl 10315  ax-mulrcl 10316  ax-mulcom 10317  ax-addass 10318  ax-mulass 10319  ax-distr 10320  ax-i2m1 10321  ax-1ne0 10322  ax-1rid 10323  ax-rnegex 10324  ax-rrecex 10325  ax-cnre 10326  ax-pre-lttri 10327  ax-pre-lttrn 10328  ax-pre-ltadd 10329  ax-pre-mulgt0 10330

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ifp 1092  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-nel 3104  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4660  df-int 4699  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-tr 4977  df-id 5251  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-pred 5921  df-ord 5967  df-on 5968  df-lim 5969  df-suc 5970  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-riota 6867  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-om 7328  df-1st 7429  df-2nd 7430  df-wrecs 7673  df-recs 7735  df-rdg 7773  df-1o 7827  df-oadd 7831  df-er 8010  df-map 8125  df-pm 8126  df-en 8224  df-dom 8225  df-sdom 8226  df-fin 8227  df-card 9079  df-pnf 10394  df-mnf 10395  df-xr 10396  df-ltxr 10397  df-le 10398  df-sub 10588  df-neg 10589  df-nn 11352  df-n0 11620  df-z 11706  df-uz 11970  df-fz 12621  df-fzo 12762  df-hash 13412  df-word 13576  df-wlks 26898  df-clwlks 27074

This theorem is referenced by:  clwlksfclwwlk2sswdOLD  27438  clwlksfclwwlkOLD  27439