Theorem clwlknf1oclwwlknlem1OLD 27449

Description: Obsolete version of clwlknf1oclwwlknlem1 27448 as of 12-Oct-2022. (Contributed by AV, 26-May-2022.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
clwlknf1oclwwlknlem1OLD	⊢ ((𝐶 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → (♯‘((2nd ‘𝐶) substr ⟨0, ((♯‘(2nd ‘𝐶)) − 1)⟩)) = (♯‘(1st ‘𝐶)))

Proof of Theorem clwlknf1oclwwlknlem1OLD

Step	Hyp	Ref	Expression
1		clwlkwlk 27078	. . 3 ⊢ (𝐶 ∈ (ClWalks‘𝐺) → 𝐶 ∈ (Walks‘𝐺))
2		wlkcpr 26927	. . . 4 ⊢ (𝐶 ∈ (Walks‘𝐺) ↔ (1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶))
3		eqid 2826	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	wlkpwrd 26916	. . . . . . 7 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (2nd ‘𝐶) ∈ Word (Vtx‘𝐺))
5		lencl 13594	. . . . . . . . 9 ⊢ ((2nd ‘𝐶) ∈ Word (Vtx‘𝐺) → (♯‘(2nd ‘𝐶)) ∈ ℕ0)
6	4, 5	syl 17	. . . . . . . 8 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (♯‘(2nd ‘𝐶)) ∈ ℕ0)
7		wlklenvm1 26920	. . . . . . . . . . 11 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (♯‘(1st ‘𝐶)) = ((♯‘(2nd ‘𝐶)) − 1))
8	7	breq2d 4886	. . . . . . . . . 10 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (1 ≤ (♯‘(1st ‘𝐶)) ↔ 1 ≤ ((♯‘(2nd ‘𝐶)) − 1)))
9		1red 10358	. . . . . . . . . . . . 13 ⊢ ((♯‘(2nd ‘𝐶)) ∈ ℕ0 → 1 ∈ ℝ)
10		nn0re 11629	. . . . . . . . . . . . 13 ⊢ ((♯‘(2nd ‘𝐶)) ∈ ℕ0 → (♯‘(2nd ‘𝐶)) ∈ ℝ)
11	9, 9, 10	leaddsub2d 10955	. . . . . . . . . . . 12 ⊢ ((♯‘(2nd ‘𝐶)) ∈ ℕ0 → ((1 + 1) ≤ (♯‘(2nd ‘𝐶)) ↔ 1 ≤ ((♯‘(2nd ‘𝐶)) − 1)))
12		1p1e2 11484	. . . . . . . . . . . . . 14 ⊢ (1 + 1) = 2
13	12	breq1i 4881	. . . . . . . . . . . . 13 ⊢ ((1 + 1) ≤ (♯‘(2nd ‘𝐶)) ↔ 2 ≤ (♯‘(2nd ‘𝐶)))
14	13	biimpi 208	. . . . . . . . . . . 12 ⊢ ((1 + 1) ≤ (♯‘(2nd ‘𝐶)) → 2 ≤ (♯‘(2nd ‘𝐶)))
15	11, 14	syl6bir 246	. . . . . . . . . . 11 ⊢ ((♯‘(2nd ‘𝐶)) ∈ ℕ0 → (1 ≤ ((♯‘(2nd ‘𝐶)) − 1) → 2 ≤ (♯‘(2nd ‘𝐶))))
16	4, 5, 15	3syl 18	. . . . . . . . . 10 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (1 ≤ ((♯‘(2nd ‘𝐶)) − 1) → 2 ≤ (♯‘(2nd ‘𝐶))))
17	8, 16	sylbid 232	. . . . . . . . 9 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (1 ≤ (♯‘(1st ‘𝐶)) → 2 ≤ (♯‘(2nd ‘𝐶))))
18	17	imp 397	. . . . . . . 8 ⊢ (((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → 2 ≤ (♯‘(2nd ‘𝐶)))
19		ige2m1fz 12725	. . . . . . . 8 ⊢ (((♯‘(2nd ‘𝐶)) ∈ ℕ0 ∧ 2 ≤ (♯‘(2nd ‘𝐶))) → ((♯‘(2nd ‘𝐶)) − 1) ∈ (0...(♯‘(2nd ‘𝐶))))
20	6, 18, 19	syl2an2r 677	. . . . . . 7 ⊢ (((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → ((♯‘(2nd ‘𝐶)) − 1) ∈ (0...(♯‘(2nd ‘𝐶))))
21		swrd0lenOLD 13709	. . . . . . 7 ⊢ (((2nd ‘𝐶) ∈ Word (Vtx‘𝐺) ∧ ((♯‘(2nd ‘𝐶)) − 1) ∈ (0...(♯‘(2nd ‘𝐶)))) → (♯‘((2nd ‘𝐶) substr ⟨0, ((♯‘(2nd ‘𝐶)) − 1)⟩)) = ((♯‘(2nd ‘𝐶)) − 1))
22	4, 20, 21	syl2an2r 677	. . . . . 6 ⊢ (((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → (♯‘((2nd ‘𝐶) substr ⟨0, ((♯‘(2nd ‘𝐶)) − 1)⟩)) = ((♯‘(2nd ‘𝐶)) − 1))
23	7	eqcomd 2832	. . . . . . 7 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → ((♯‘(2nd ‘𝐶)) − 1) = (♯‘(1st ‘𝐶)))
24	23	adantr 474	. . . . . 6 ⊢ (((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → ((♯‘(2nd ‘𝐶)) − 1) = (♯‘(1st ‘𝐶)))
25	22, 24	eqtrd 2862	. . . . 5 ⊢ (((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → (♯‘((2nd ‘𝐶) substr ⟨0, ((♯‘(2nd ‘𝐶)) − 1)⟩)) = (♯‘(1st ‘𝐶)))
26	25	ex 403	. . . 4 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (1 ≤ (♯‘(1st ‘𝐶)) → (♯‘((2nd ‘𝐶) substr ⟨0, ((♯‘(2nd ‘𝐶)) − 1)⟩)) = (♯‘(1st ‘𝐶))))
27	2, 26	sylbi 209	. . 3 ⊢ (𝐶 ∈ (Walks‘𝐺) → (1 ≤ (♯‘(1st ‘𝐶)) → (♯‘((2nd ‘𝐶) substr ⟨0, ((♯‘(2nd ‘𝐶)) − 1)⟩)) = (♯‘(1st ‘𝐶))))
28	1, 27	syl 17	. 2 ⊢ (𝐶 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘(1st ‘𝐶)) → (♯‘((2nd ‘𝐶) substr ⟨0, ((♯‘(2nd ‘𝐶)) − 1)⟩)) = (♯‘(1st ‘𝐶))))
29	28	imp 397	1 ⊢ ((𝐶 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → (♯‘((2nd ‘𝐶) substr ⟨0, ((♯‘(2nd ‘𝐶)) − 1)⟩)) = (♯‘(1st ‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ⟨cop 4404   class class class wbr 4874  ‘cfv 6124  (class class class)co 6906  1^st c1st 7427  2^nd c2nd 7428  0cc0 10253  1c1 10254   + caddc 10256   ≤ cle 10393   − cmin 10586  2c2 11407  ℕ₀cn0 11619  ...cfz 12620  ♯chash 13411  Word cword 13575   substr csubstr 13701  Vtxcvtx 26295  Walkscwlks 26895  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-2 11415  df-n0 11620  df-z 11706  df-uz 11970  df-fz 12621  df-fzo 12762  df-hash 13412  df-word 13576  df-substr 13702  df-wlks 26898  df-clwlks 27074

This theorem is referenced by:  clwlknf1oclwwlknOLD  27454