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Theorem clwlkclwwlkfolem 30607

Description: Lemma for clwlkclwwlkfo 30609. (Contributed by AV, 25-May-2022.)

**Hypothesis**
Ref	Expression
clwlkclwwlkf.c	⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}

**Assertion**
Ref	Expression
clwlkclwwlkfolem	⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ 𝐶)

Distinct variable group: 𝑤,𝐺,𝑓 Allowed substitution hints: 𝐶(𝑤, 𝑓) 𝑊(𝑤, 𝑓)

Proof of Theorem clwlkclwwlkfolem Dummy variable 𝑐 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp3 1156	. 2 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
2		wrdlenccats1lenm1 14778	. . . . . . 7 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1) = (♯‘𝑊))
3	2	eqcomd 2766	. . . . . 6 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (♯‘𝑊) = ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1))
4	3	breq2d 5114	. . . . 5 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (1 ≤ (♯‘𝑊) ↔ 1 ≤ ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1)))
5	4	biimpa 482	. . . 4 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊)) → 1 ≤ ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1))
6	5	3adant3 1150	. . 3 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → 1 ≤ ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1))
7		df-br 5103	. . . . 5 ⊢ (𝑓(ClWalks‘𝐺)(𝑊 ++ ⟨“(𝑊‘0)”⟩) ↔ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺))
8		clwlkiswlk 30370	. . . . . 6 ⊢ (𝑓(ClWalks‘𝐺)(𝑊 ++ ⟨“(𝑊‘0)”⟩) → 𝑓(Walks‘𝐺)(𝑊 ++ ⟨“(𝑊‘0)”⟩))
9		wlklenvm1 30211	. . . . . 6 ⊢ (𝑓(Walks‘𝐺)(𝑊 ++ ⟨“(𝑊‘0)”⟩) → (♯‘𝑓) = ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1))
10	8, 9	syl 18	. . . . 5 ⊢ (𝑓(ClWalks‘𝐺)(𝑊 ++ ⟨“(𝑊‘0)”⟩) → (♯‘𝑓) = ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1))
11	7, 10	sylbir 238	. . . 4 ⊢ (⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) → (♯‘𝑓) = ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1))
12	11	3ad2ant3 1153	. . 3 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → (♯‘𝑓) = ((♯‘(𝑊 ++ ⟨“(𝑊‘0)”⟩)) − 1))
13	6, 12	breqtrrd 5132	. 2 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → 1 ≤ (♯‘𝑓))
14		vex 3454	. . . . . 6 ⊢ 𝑓 ∈ V
15		ovex 7457	. . . . . 6 ⊢ (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ V
16	14, 15	op1std 8015	. . . . 5 ⊢ (𝑐 = ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ → (1^st ‘𝑐) = 𝑓)
17	16	fveq2d 6894	. . . 4 ⊢ (𝑐 = ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ → (♯‘(1^st ‘𝑐)) = (♯‘𝑓))
18	17	breq2d 5114	. . 3 ⊢ (𝑐 = ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ → (1 ≤ (♯‘(1^st ‘𝑐)) ↔ 1 ≤ (♯‘𝑓)))
19		clwlkclwwlkf.c	. . . 4 ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}
20		2fveq3 6895	. . . . . 6 ⊢ (𝑤 = 𝑐 → (♯‘(1^st ‘𝑤)) = (♯‘(1^st ‘𝑐)))
21	20	breq2d 5114	. . . . 5 ⊢ (𝑤 = 𝑐 → (1 ≤ (♯‘(1^st ‘𝑤)) ↔ 1 ≤ (♯‘(1^st ‘𝑐))))
22	21	cbvrabv 3422	. . . 4 ⊢ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))} = {𝑐 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑐))}
23	19, 22	eqtri 2783	. . 3 ⊢ 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑐))}
24	18, 23	elrab2 3648	. 2 ⊢ (⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ 𝐶 ↔ (⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘𝑓)))
25	1, 13, 24	sylanbrc 595	1 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (♯‘𝑊) ∧ ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (ClWalks‘𝐺)) → ⟨𝑓, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ 𝐶)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 {crab 3412 ⟨cop 4834 class class class wbr 5102 ‘cfv 6543 (class class class)co 7424 1^st c1st 8003 0cc0 11207 1c1 11208 ≤ cle 11351 − cmin 11548 ♯chash 14481 Word cword 14665 ++ cconcat 14722 ⟨“cs1 14749 Vtxcvtx 29583 Walkscwlks 30186 ClWalkscclwlks 30366

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-ifp 1079 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7882 df-1st 8005 df-2nd 8006 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-er 8725 df-map 8857 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-nn 12343 df-n0 12614 df-z 12701 df-uz 12973 df-fz 13647 df-fzo 13796 df-hash 14482 df-word 14666 df-concat 14723 df-s1 14750 df-wlks 30189 df-clwlks 30367

This theorem is used by: clwlkclwwlkfo 30609

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