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Theorem clwlknf1oclwwlknlem1 29323

Description: Lemma 1 for clwlknf1oclwwlkn 29326. (Contributed by AV, 26-May-2022.) (Revised by AV, 1-Nov-2022.)

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

**Proof of Theorem clwlknf1oclwwlknlem1**
Step	Hyp	Ref	Expression
1		clwlkwlk 29021	. . 3 ⊢ (𝐶 ∈ (ClWalks‘𝐺) → 𝐶 ∈ (Walks‘𝐺))
2		wlkcpr 28875	. . . 4 ⊢ (𝐶 ∈ (Walks‘𝐺) ↔ (1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶))
3		eqid 2732	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	wlkpwrd 28863	. . . . . . 7 ⊢ ((1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶) → (2^nd ‘𝐶) ∈ Word (Vtx‘𝐺))
5		lencl 14479	. . . . . . . . 9 ⊢ ((2^nd ‘𝐶) ∈ Word (Vtx‘𝐺) → (♯‘(2^nd ‘𝐶)) ∈ ℕ₀)
6	4, 5	syl 17	. . . . . . . 8 ⊢ ((1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶) → (♯‘(2^nd ‘𝐶)) ∈ ℕ₀)
7		wlklenvm1 28868	. . . . . . . . . . 11 ⊢ ((1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶) → (♯‘(1^st ‘𝐶)) = ((♯‘(2^nd ‘𝐶)) − 1))
8	7	breq2d 5159	. . . . . . . . . 10 ⊢ ((1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶) → (1 ≤ (♯‘(1^st ‘𝐶)) ↔ 1 ≤ ((♯‘(2^nd ‘𝐶)) − 1)))
9		1red 11211	. . . . . . . . . . . . 13 ⊢ ((♯‘(2^nd ‘𝐶)) ∈ ℕ₀ → 1 ∈ ℝ)
10		nn0re 12477	. . . . . . . . . . . . 13 ⊢ ((♯‘(2^nd ‘𝐶)) ∈ ℕ₀ → (♯‘(2^nd ‘𝐶)) ∈ ℝ)
11	9, 9, 10	leaddsub2d 11812	. . . . . . . . . . . 12 ⊢ ((♯‘(2^nd ‘𝐶)) ∈ ℕ₀ → ((1 + 1) ≤ (♯‘(2^nd ‘𝐶)) ↔ 1 ≤ ((♯‘(2^nd ‘𝐶)) − 1)))
12		1p1e2 12333	. . . . . . . . . . . . . 14 ⊢ (1 + 1) = 2
13	12	breq1i 5154	. . . . . . . . . . . . 13 ⊢ ((1 + 1) ≤ (♯‘(2^nd ‘𝐶)) ↔ 2 ≤ (♯‘(2^nd ‘𝐶)))
14	13	biimpi 215	. . . . . . . . . . . 12 ⊢ ((1 + 1) ≤ (♯‘(2^nd ‘𝐶)) → 2 ≤ (♯‘(2^nd ‘𝐶)))
15	11, 14	syl6bir 253	. . . . . . . . . . 11 ⊢ ((♯‘(2^nd ‘𝐶)) ∈ ℕ₀ → (1 ≤ ((♯‘(2^nd ‘𝐶)) − 1) → 2 ≤ (♯‘(2^nd ‘𝐶))))
16	4, 5, 15	3syl 18	. . . . . . . . . 10 ⊢ ((1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶) → (1 ≤ ((♯‘(2^nd ‘𝐶)) − 1) → 2 ≤ (♯‘(2^nd ‘𝐶))))
17	8, 16	sylbid 239	. . . . . . . . 9 ⊢ ((1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶) → (1 ≤ (♯‘(1^st ‘𝐶)) → 2 ≤ (♯‘(2^nd ‘𝐶))))
18	17	imp 407	. . . . . . . 8 ⊢ (((1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶) ∧ 1 ≤ (♯‘(1^st ‘𝐶))) → 2 ≤ (♯‘(2^nd ‘𝐶)))
19		ige2m1fz 13587	. . . . . . . 8 ⊢ (((♯‘(2^nd ‘𝐶)) ∈ ℕ₀ ∧ 2 ≤ (♯‘(2^nd ‘𝐶))) → ((♯‘(2^nd ‘𝐶)) − 1) ∈ (0...(♯‘(2^nd ‘𝐶))))
20	6, 18, 19	syl2an2r 683	. . . . . . 7 ⊢ (((1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶) ∧ 1 ≤ (♯‘(1^st ‘𝐶))) → ((♯‘(2^nd ‘𝐶)) − 1) ∈ (0...(♯‘(2^nd ‘𝐶))))
21		pfxlen 14629	. . . . . . 7 ⊢ (((2^nd ‘𝐶) ∈ Word (Vtx‘𝐺) ∧ ((♯‘(2^nd ‘𝐶)) − 1) ∈ (0...(♯‘(2^nd ‘𝐶)))) → (♯‘((2^nd ‘𝐶) prefix ((♯‘(2^nd ‘𝐶)) − 1))) = ((♯‘(2^nd ‘𝐶)) − 1))
22	4, 20, 21	syl2an2r 683	. . . . . 6 ⊢ (((1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶) ∧ 1 ≤ (♯‘(1^st ‘𝐶))) → (♯‘((2^nd ‘𝐶) prefix ((♯‘(2^nd ‘𝐶)) − 1))) = ((♯‘(2^nd ‘𝐶)) − 1))
23	7	eqcomd 2738	. . . . . . 7 ⊢ ((1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶) → ((♯‘(2^nd ‘𝐶)) − 1) = (♯‘(1^st ‘𝐶)))
24	23	adantr 481	. . . . . 6 ⊢ (((1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶) ∧ 1 ≤ (♯‘(1^st ‘𝐶))) → ((♯‘(2^nd ‘𝐶)) − 1) = (♯‘(1^st ‘𝐶)))
25	22, 24	eqtrd 2772	. . . . 5 ⊢ (((1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶) ∧ 1 ≤ (♯‘(1^st ‘𝐶))) → (♯‘((2^nd ‘𝐶) prefix ((♯‘(2^nd ‘𝐶)) − 1))) = (♯‘(1^st ‘𝐶)))
26	25	ex 413	. . . 4 ⊢ ((1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶) → (1 ≤ (♯‘(1^st ‘𝐶)) → (♯‘((2^nd ‘𝐶) prefix ((♯‘(2^nd ‘𝐶)) − 1))) = (♯‘(1^st ‘𝐶))))
27	2, 26	sylbi 216	. . 3 ⊢ (𝐶 ∈ (Walks‘𝐺) → (1 ≤ (♯‘(1^st ‘𝐶)) → (♯‘((2^nd ‘𝐶) prefix ((♯‘(2^nd ‘𝐶)) − 1))) = (♯‘(1^st ‘𝐶))))
28	1, 27	syl 17	. 2 ⊢ (𝐶 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘(1^st ‘𝐶)) → (♯‘((2^nd ‘𝐶) prefix ((♯‘(2^nd ‘𝐶)) − 1))) = (♯‘(1^st ‘𝐶))))
29	28	imp 407	1 ⊢ ((𝐶 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1^st ‘𝐶))) → (♯‘((2^nd ‘𝐶) prefix ((♯‘(2^nd ‘𝐶)) − 1))) = (♯‘(1^st ‘𝐶)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 1^st c1st 7969 2^nd c2nd 7970 0cc0 11106 1c1 11107 + caddc 11109 ≤ cle 11245 − cmin 11440 2c2 12263 ℕ₀cn0 12468 ...cfz 13480 ♯chash 14286 Word cword 14460 prefix cpfx 14616 Vtxcvtx 28245 Walkscwlks 28842 ClWalkscclwlks 29016

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ifp 1062 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-fzo 13624 df-hash 14287 df-word 14461 df-substr 14587 df-pfx 14617 df-wlks 28845 df-clwlks 29017

This theorem is referenced by: clwlknf1oclwwlkn 29326

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