clwlknf1oclwwlknlem1 - Metamath Proof Explorer

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Theorem clwlknf1oclwwlknlem1 29334

Description: Lemma 1 for clwlknf1oclwwlkn 29337. (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 29032	. . 3 ⊢ (𝐶 ∈ (ClWalks‘𝐺) → 𝐶 ∈ (Walks‘𝐺))
2		wlkcpr 28886	. . . 4 ⊢ (𝐶 ∈ (Walks‘𝐺) ↔ (1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶))
3		eqid 2733	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
4	3	wlkpwrd 28874	. . . . . . 7 ⊢ ((1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶) → (2^nd ‘𝐶) ∈ Word (Vtx‘𝐺))
5		lencl 14483	. . . . . . . . 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 28879	. . . . . . . . . . 11 ⊢ ((1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶) → (♯‘(1^st ‘𝐶)) = ((♯‘(2^nd ‘𝐶)) − 1))
8	7	breq2d 5161	. . . . . . . . . 10 ⊢ ((1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶) → (1 ≤ (♯‘(1^st ‘𝐶)) ↔ 1 ≤ ((♯‘(2^nd ‘𝐶)) − 1)))
9		1red 11215	. . . . . . . . . . . . 13 ⊢ ((♯‘(2^nd ‘𝐶)) ∈ ℕ₀ → 1 ∈ ℝ)
10		nn0re 12481	. . . . . . . . . . . . 13 ⊢ ((♯‘(2^nd ‘𝐶)) ∈ ℕ₀ → (♯‘(2^nd ‘𝐶)) ∈ ℝ)
11	9, 9, 10	leaddsub2d 11816	. . . . . . . . . . . 12 ⊢ ((♯‘(2^nd ‘𝐶)) ∈ ℕ₀ → ((1 + 1) ≤ (♯‘(2^nd ‘𝐶)) ↔ 1 ≤ ((♯‘(2^nd ‘𝐶)) − 1)))
12		1p1e2 12337	. . . . . . . . . . . . . 14 ⊢ (1 + 1) = 2
13	12	breq1i 5156	. . . . . . . . . . . . 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 254	. . . . . . . . . . 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 408	. . . . . . . 8 ⊢ (((1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶) ∧ 1 ≤ (♯‘(1^st ‘𝐶))) → 2 ≤ (♯‘(2^nd ‘𝐶)))
19		ige2m1fz 13591	. . . . . . . 8 ⊢ (((♯‘(2^nd ‘𝐶)) ∈ ℕ₀ ∧ 2 ≤ (♯‘(2^nd ‘𝐶))) → ((♯‘(2^nd ‘𝐶)) − 1) ∈ (0...(♯‘(2^nd ‘𝐶))))
20	6, 18, 19	syl2an2r 684	. . . . . . 7 ⊢ (((1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶) ∧ 1 ≤ (♯‘(1^st ‘𝐶))) → ((♯‘(2^nd ‘𝐶)) − 1) ∈ (0...(♯‘(2^nd ‘𝐶))))
21		pfxlen 14633	. . . . . . 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 684	. . . . . 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 2739	. . . . . . 7 ⊢ ((1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶) → ((♯‘(2^nd ‘𝐶)) − 1) = (♯‘(1^st ‘𝐶)))
24	23	adantr 482	. . . . . 6 ⊢ (((1^st ‘𝐶)(Walks‘𝐺)(2^nd ‘𝐶) ∧ 1 ≤ (♯‘(1^st ‘𝐶))) → ((♯‘(2^nd ‘𝐶)) − 1) = (♯‘(1^st ‘𝐶)))
25	22, 24	eqtrd 2773	. . . . 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 414	. . . 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 408	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 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5149 ‘cfv 6544 (class class class)co 7409 1^st c1st 7973 2^nd c2nd 7974 0cc0 11110 1c1 11111 + caddc 11113 ≤ cle 11249 − cmin 11444 2c2 12267 ℕ₀cn0 12472 ...cfz 13484 ♯chash 14290 Word cword 14464 prefix cpfx 14620 Vtxcvtx 28256 Walkscwlks 28853 ClWalkscclwlks 29027

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-n0 12473 df-z 12559 df-uz 12823 df-fz 13485 df-fzo 13628 df-hash 14291 df-word 14465 df-substr 14591 df-pfx 14621 df-wlks 28856 df-clwlks 29028

This theorem is referenced by: clwlknf1oclwwlkn 29337

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