Theorem clwlknf1oclwwlknlem1 29768

Description: Lemma 1 for clwlknf1oclwwlkn 29771. (Contributed by AV, 26-May-2022.) (Revised by AV, 1-Nov-2022.)

Assertion

Ref	Expression
clwlknf1oclwwlknlem1	⊢ ((𝐶 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → (♯‘((2nd ‘𝐶) prefix ((♯‘(2nd ‘𝐶)) − 1))) = (♯‘(1st ‘𝐶)))

Proof of Theorem clwlknf1oclwwlknlem1

Step	Hyp	Ref	Expression
1		clwlkwlk 29466	. . 3 ⊢ (𝐶 ∈ (ClWalks‘𝐺) → 𝐶 ∈ (Walks‘𝐺))
2		wlkcpr 29320	. . . 4 ⊢ (𝐶 ∈ (Walks‘𝐺) ↔ (1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶))
3		eqid 2731	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	wlkpwrd 29308	. . . . . . 7 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (2nd ‘𝐶) ∈ Word (Vtx‘𝐺))
5		lencl 14490	. . . . . . . . 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 29313	. . . . . . . . . . 11 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (♯‘(1st ‘𝐶)) = ((♯‘(2nd ‘𝐶)) − 1))
8	7	breq2d 5160	. . . . . . . . . 10 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (1 ≤ (♯‘(1st ‘𝐶)) ↔ 1 ≤ ((♯‘(2nd ‘𝐶)) − 1)))
9		1red 11222	. . . . . . . . . . . . 13 ⊢ ((♯‘(2nd ‘𝐶)) ∈ ℕ0 → 1 ∈ ℝ)
10		nn0re 12488	. . . . . . . . . . . . 13 ⊢ ((♯‘(2nd ‘𝐶)) ∈ ℕ0 → (♯‘(2nd ‘𝐶)) ∈ ℝ)
11	9, 9, 10	leaddsub2d 11823	. . . . . . . . . . . 12 ⊢ ((♯‘(2nd ‘𝐶)) ∈ ℕ0 → ((1 + 1) ≤ (♯‘(2nd ‘𝐶)) ↔ 1 ≤ ((♯‘(2nd ‘𝐶)) − 1)))
12		1p1e2 12344	. . . . . . . . . . . . . 14 ⊢ (1 + 1) = 2
13	12	breq1i 5155	. . . . . . . . . . . . 13 ⊢ ((1 + 1) ≤ (♯‘(2nd ‘𝐶)) ↔ 2 ≤ (♯‘(2nd ‘𝐶)))
14	13	biimpi 215	. . . . . . . . . . . 12 ⊢ ((1 + 1) ≤ (♯‘(2nd ‘𝐶)) → 2 ≤ (♯‘(2nd ‘𝐶)))
15	11, 14	syl6bir 254	. . . . . . . . . . 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 239	. . . . . . . . 9 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (1 ≤ (♯‘(1st ‘𝐶)) → 2 ≤ (♯‘(2nd ‘𝐶))))
18	17	imp 406	. . . . . . . 8 ⊢ (((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → 2 ≤ (♯‘(2nd ‘𝐶)))
19		ige2m1fz 13598	. . . . . . . 8 ⊢ (((♯‘(2nd ‘𝐶)) ∈ ℕ0 ∧ 2 ≤ (♯‘(2nd ‘𝐶))) → ((♯‘(2nd ‘𝐶)) − 1) ∈ (0...(♯‘(2nd ‘𝐶))))
20	6, 18, 19	syl2an2r 682	. . . . . . 7 ⊢ (((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → ((♯‘(2nd ‘𝐶)) − 1) ∈ (0...(♯‘(2nd ‘𝐶))))
21		pfxlen 14640	. . . . . . 7 ⊢ (((2nd ‘𝐶) ∈ Word (Vtx‘𝐺) ∧ ((♯‘(2nd ‘𝐶)) − 1) ∈ (0...(♯‘(2nd ‘𝐶)))) → (♯‘((2nd ‘𝐶) prefix ((♯‘(2nd ‘𝐶)) − 1))) = ((♯‘(2nd ‘𝐶)) − 1))
22	4, 20, 21	syl2an2r 682	. . . . . 6 ⊢ (((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → (♯‘((2nd ‘𝐶) prefix ((♯‘(2nd ‘𝐶)) − 1))) = ((♯‘(2nd ‘𝐶)) − 1))
23	7	eqcomd 2737	. . . . . . 7 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → ((♯‘(2nd ‘𝐶)) − 1) = (♯‘(1st ‘𝐶)))
24	23	adantr 480	. . . . . 6 ⊢ (((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → ((♯‘(2nd ‘𝐶)) − 1) = (♯‘(1st ‘𝐶)))
25	22, 24	eqtrd 2771	. . . . 5 ⊢ (((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → (♯‘((2nd ‘𝐶) prefix ((♯‘(2nd ‘𝐶)) − 1))) = (♯‘(1st ‘𝐶)))
26	25	ex 412	. . . 4 ⊢ ((1st ‘𝐶)(Walks‘𝐺)(2nd ‘𝐶) → (1 ≤ (♯‘(1st ‘𝐶)) → (♯‘((2nd ‘𝐶) prefix ((♯‘(2nd ‘𝐶)) − 1))) = (♯‘(1st ‘𝐶))))
27	2, 26	sylbi 216	. . 3 ⊢ (𝐶 ∈ (Walks‘𝐺) → (1 ≤ (♯‘(1st ‘𝐶)) → (♯‘((2nd ‘𝐶) prefix ((♯‘(2nd ‘𝐶)) − 1))) = (♯‘(1st ‘𝐶))))
28	1, 27	syl 17	. 2 ⊢ (𝐶 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘(1st ‘𝐶)) → (♯‘((2nd ‘𝐶) prefix ((♯‘(2nd ‘𝐶)) − 1))) = (♯‘(1st ‘𝐶))))
29	28	imp 406	1 ⊢ ((𝐶 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → (♯‘((2nd ‘𝐶) prefix ((♯‘(2nd ‘𝐶)) − 1))) = (♯‘(1st ‘𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105   class class class wbr 5148  ‘cfv 6543  (class class class)co 7412  1^st c1st 7977  2^nd c2nd 7978  0cc0 11116  1c1 11117   + caddc 11119   ≤ cle 11256   − cmin 11451  2c2 12274  ℕ₀cn0 12479  ...cfz 13491  ♯chash 14297  Word cword 14471   prefix cpfx 14627  Vtxcvtx 28690  Walkscwlks 29287  ClWalkscclwlks 29461

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11172  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ifp 1061  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-1o 8472  df-er 8709  df-map 8828  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-card 9940  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454  df-nn 12220  df-2 12282  df-n0 12480  df-z 12566  df-uz 12830  df-fz 13492  df-fzo 13635  df-hash 14298  df-word 14472  df-substr 14598  df-pfx 14628  df-wlks 29290  df-clwlks 29462

This theorem is referenced by:  clwlknf1oclwwlkn  29771