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Theorem clwlkclwwlkf 28372

Description: 𝐹 is a function from the nonempty closed walks into the closed walks as word in a simple pseudograph. (Contributed by AV, 23-May-2022.) (Revised by AV, 29-Oct-2022.)

**Hypotheses**
Ref	Expression
clwlkclwwlkf.c	⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}
clwlkclwwlkf.f	⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)))

**Assertion**
Ref	Expression
clwlkclwwlkf	⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))

Distinct variable groups: 𝑤,𝐺,𝑐 𝐶,𝑐 Allowed substitution hints: 𝐶(𝑤) 𝐹(𝑤,𝑐)

Proof of Theorem clwlkclwwlkf Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwlkclwwlkf.c	. . . . 5 ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1^st ‘𝑤))}
2		eqid 2738	. . . . 5 ⊢ (1^st ‘𝑐) = (1^st ‘𝑐)
3		eqid 2738	. . . . 5 ⊢ (2^nd ‘𝑐) = (2^nd ‘𝑐)
4	1, 2, 3	clwlkclwwlkflem 28368	. . . 4 ⊢ (𝑐 ∈ 𝐶 → ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ ((2^nd ‘𝑐)‘0) = ((2^nd ‘𝑐)‘(♯‘(1^st ‘𝑐))) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ))
5		isclwlk 28141	. . . . . . . 8 ⊢ ((1^st ‘𝑐)(ClWalks‘𝐺)(2^nd ‘𝑐) ↔ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ ((2^nd ‘𝑐)‘0) = ((2^nd ‘𝑐)‘(♯‘(1^st ‘𝑐)))))
6		fvex 6787	. . . . . . . . 9 ⊢ (1^st ‘𝑐) ∈ V
7		breq1 5077	. . . . . . . . 9 ⊢ (𝑓 = (1^st ‘𝑐) → (𝑓(ClWalks‘𝐺)(2^nd ‘𝑐) ↔ (1^st ‘𝑐)(ClWalks‘𝐺)(2^nd ‘𝑐)))
8	6, 7	spcev 3545	. . . . . . . 8 ⊢ ((1^st ‘𝑐)(ClWalks‘𝐺)(2^nd ‘𝑐) → ∃𝑓 𝑓(ClWalks‘𝐺)(2^nd ‘𝑐))
9	5, 8	sylbir 234	. . . . . . 7 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ ((2^nd ‘𝑐)‘0) = ((2^nd ‘𝑐)‘(♯‘(1^st ‘𝑐)))) → ∃𝑓 𝑓(ClWalks‘𝐺)(2^nd ‘𝑐))
10	9	3adant3 1131	. . . . . 6 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ ((2^nd ‘𝑐)‘0) = ((2^nd ‘𝑐)‘(♯‘(1^st ‘𝑐))) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ) → ∃𝑓 𝑓(ClWalks‘𝐺)(2^nd ‘𝑐))
11	10	adantl 482	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ ((2^nd ‘𝑐)‘0) = ((2^nd ‘𝑐)‘(♯‘(1^st ‘𝑐))) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ)) → ∃𝑓 𝑓(ClWalks‘𝐺)(2^nd ‘𝑐))
12		simpl 483	. . . . . 6 ⊢ ((𝐺 ∈ USPGraph ∧ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ ((2^nd ‘𝑐)‘0) = ((2^nd ‘𝑐)‘(♯‘(1^st ‘𝑐))) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ)) → 𝐺 ∈ USPGraph)
13		eqid 2738	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
14	13	wlkpwrd 27984	. . . . . . . 8 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (2^nd ‘𝑐) ∈ Word (Vtx‘𝐺))
15	14	3ad2ant1 1132	. . . . . . 7 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ ((2^nd ‘𝑐)‘0) = ((2^nd ‘𝑐)‘(♯‘(1^st ‘𝑐))) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ) → (2^nd ‘𝑐) ∈ Word (Vtx‘𝐺))
16	15	adantl 482	. . . . . 6 ⊢ ((𝐺 ∈ USPGraph ∧ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ ((2^nd ‘𝑐)‘0) = ((2^nd ‘𝑐)‘(♯‘(1^st ‘𝑐))) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ)) → (2^nd ‘𝑐) ∈ Word (Vtx‘𝐺))
17		elnnnn0c 12278	. . . . . . . . . 10 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℕ ↔ ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ ∧ 1 ≤ (♯‘(1^st ‘𝑐))))
18		nn0re 12242	. . . . . . . . . . . . . 14 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ → (♯‘(1^st ‘𝑐)) ∈ ℝ)
19		1e2m1 12100	. . . . . . . . . . . . . . . . 17 ⊢ 1 = (2 − 1)
20	19	breq1i 5081	. . . . . . . . . . . . . . . 16 ⊢ (1 ≤ (♯‘(1^st ‘𝑐)) ↔ (2 − 1) ≤ (♯‘(1^st ‘𝑐)))
21	20	biimpi 215	. . . . . . . . . . . . . . 15 ⊢ (1 ≤ (♯‘(1^st ‘𝑐)) → (2 − 1) ≤ (♯‘(1^st ‘𝑐)))
22		2re 12047	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
23		1re 10975	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
24		lesubadd 11447	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℝ ∧ 1 ∈ ℝ ∧ (♯‘(1^st ‘𝑐)) ∈ ℝ) → ((2 − 1) ≤ (♯‘(1^st ‘𝑐)) ↔ 2 ≤ ((♯‘(1^st ‘𝑐)) + 1)))
25	22, 23, 24	mp3an12 1450	. . . . . . . . . . . . . . 15 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℝ → ((2 − 1) ≤ (♯‘(1^st ‘𝑐)) ↔ 2 ≤ ((♯‘(1^st ‘𝑐)) + 1)))
26	21, 25	syl5ib 243	. . . . . . . . . . . . . 14 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℝ → (1 ≤ (♯‘(1^st ‘𝑐)) → 2 ≤ ((♯‘(1^st ‘𝑐)) + 1)))
27	18, 26	syl 17	. . . . . . . . . . . . 13 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ → (1 ≤ (♯‘(1^st ‘𝑐)) → 2 ≤ ((♯‘(1^st ‘𝑐)) + 1)))
28	27	adantl 482	. . . . . . . . . . . 12 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ₀) → (1 ≤ (♯‘(1^st ‘𝑐)) → 2 ≤ ((♯‘(1^st ‘𝑐)) + 1)))
29		wlklenvp1 27985	. . . . . . . . . . . . . 14 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (♯‘(2^nd ‘𝑐)) = ((♯‘(1^st ‘𝑐)) + 1))
30	29	adantr 481	. . . . . . . . . . . . 13 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ₀) → (♯‘(2^nd ‘𝑐)) = ((♯‘(1^st ‘𝑐)) + 1))
31	30	breq2d 5086	. . . . . . . . . . . 12 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ₀) → (2 ≤ (♯‘(2^nd ‘𝑐)) ↔ 2 ≤ ((♯‘(1^st ‘𝑐)) + 1)))
32	28, 31	sylibrd 258	. . . . . . . . . . 11 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ₀) → (1 ≤ (♯‘(1^st ‘𝑐)) → 2 ≤ (♯‘(2^nd ‘𝑐))))
33	32	expimpd 454	. . . . . . . . . 10 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (((♯‘(1^st ‘𝑐)) ∈ ℕ₀ ∧ 1 ≤ (♯‘(1^st ‘𝑐))) → 2 ≤ (♯‘(2^nd ‘𝑐))))
34	17, 33	syl5bi 241	. . . . . . . . 9 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → ((♯‘(1^st ‘𝑐)) ∈ ℕ → 2 ≤ (♯‘(2^nd ‘𝑐))))
35	34	a1d 25	. . . . . . . 8 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (((2^nd ‘𝑐)‘0) = ((2^nd ‘𝑐)‘(♯‘(1^st ‘𝑐))) → ((♯‘(1^st ‘𝑐)) ∈ ℕ → 2 ≤ (♯‘(2^nd ‘𝑐)))))
36	35	3imp 1110	. . . . . . 7 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ ((2^nd ‘𝑐)‘0) = ((2^nd ‘𝑐)‘(♯‘(1^st ‘𝑐))) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ) → 2 ≤ (♯‘(2^nd ‘𝑐)))
37	36	adantl 482	. . . . . 6 ⊢ ((𝐺 ∈ USPGraph ∧ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ ((2^nd ‘𝑐)‘0) = ((2^nd ‘𝑐)‘(♯‘(1^st ‘𝑐))) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ)) → 2 ≤ (♯‘(2^nd ‘𝑐)))
38		eqid 2738	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
39	13, 38	clwlkclwwlk 28366	. . . . . 6 ⊢ ((𝐺 ∈ USPGraph ∧ (2^nd ‘𝑐) ∈ Word (Vtx‘𝐺) ∧ 2 ≤ (♯‘(2^nd ‘𝑐))) → (∃𝑓 𝑓(ClWalks‘𝐺)(2^nd ‘𝑐) ↔ ((lastS‘(2^nd ‘𝑐)) = ((2^nd ‘𝑐)‘0) ∧ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)) ∈ (ClWWalks‘𝐺))))
40	12, 16, 37, 39	syl3anc 1370	. . . . 5 ⊢ ((𝐺 ∈ USPGraph ∧ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ ((2^nd ‘𝑐)‘0) = ((2^nd ‘𝑐)‘(♯‘(1^st ‘𝑐))) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ)) → (∃𝑓 𝑓(ClWalks‘𝐺)(2^nd ‘𝑐) ↔ ((lastS‘(2^nd ‘𝑐)) = ((2^nd ‘𝑐)‘0) ∧ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)) ∈ (ClWWalks‘𝐺))))
41	11, 40	mpbid 231	. . . 4 ⊢ ((𝐺 ∈ USPGraph ∧ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ ((2^nd ‘𝑐)‘0) = ((2^nd ‘𝑐)‘(♯‘(1^st ‘𝑐))) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ)) → ((lastS‘(2^nd ‘𝑐)) = ((2^nd ‘𝑐)‘0) ∧ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)) ∈ (ClWWalks‘𝐺)))
42	4, 41	sylan2 593	. . 3 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑐 ∈ 𝐶) → ((lastS‘(2^nd ‘𝑐)) = ((2^nd ‘𝑐)‘0) ∧ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)) ∈ (ClWWalks‘𝐺)))
43	42	simprd 496	. 2 ⊢ ((𝐺 ∈ USPGraph ∧ 𝑐 ∈ 𝐶) → ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)) ∈ (ClWWalks‘𝐺))
44		clwlkclwwlkf.f	. 2 ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ ((2^nd ‘𝑐) prefix ((♯‘(2^nd ‘𝑐)) − 1)))
45	43, 44	fmptd 6988	1 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∃wex 1782 ∈ wcel 2106 {crab 3068 class class class wbr 5074 ↦ cmpt 5157 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 1^st c1st 7829 2^nd c2nd 7830 ℝcr 10870 0cc0 10871 1c1 10872 + caddc 10874 ≤ cle 11010 − cmin 11205 ℕcn 11973 2c2 12028 ℕ₀cn0 12233 ♯chash 14044 Word cword 14217 lastSclsw 14265 prefix cpfx 14383 Vtxcvtx 27366 iEdgciedg 27367 USPGraphcuspgr 27518 Walkscwlks 27963 ClWalkscclwlks 28138 ClWWalkscclwwlk 28345

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ifp 1061 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-xnn0 12306 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-hash 14045 df-word 14218 df-lsw 14266 df-substr 14354 df-pfx 14384 df-edg 27418 df-uhgr 27428 df-upgr 27452 df-uspgr 27520 df-wlks 27966 df-clwlks 28139 df-clwwlk 28346

This theorem is referenced by: clwlkclwwlkfo 28373 clwlkclwwlkf1 28374

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