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Theorem clwlkclwwlkf 27771

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 2821	. . . . 5 ⊢ (1^st ‘𝑐) = (1^st ‘𝑐)
3		eqid 2821	. . . . 5 ⊢ (2^nd ‘𝑐) = (2^nd ‘𝑐)
4	1, 2, 3	clwlkclwwlkflem 27767	. . . 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 27540	. . . . . . . 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 6669	. . . . . . . . 9 ⊢ (1^st ‘𝑐) ∈ V
7		breq1 5055	. . . . . . . . 9 ⊢ (𝑓 = (1^st ‘𝑐) → (𝑓(ClWalks‘𝐺)(2^nd ‘𝑐) ↔ (1^st ‘𝑐)(ClWalks‘𝐺)(2^nd ‘𝑐)))
8	6, 7	spcev 3599	. . . . . . . 8 ⊢ ((1^st ‘𝑐)(ClWalks‘𝐺)(2^nd ‘𝑐) → ∃𝑓 𝑓(ClWalks‘𝐺)(2^nd ‘𝑐))
9	5, 8	sylbir 237	. . . . . . 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 1128	. . . . . 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 484	. . . . 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 485	. . . . . 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 2821	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
14	13	wlkpwrd 27385	. . . . . . . 8 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (2^nd ‘𝑐) ∈ Word (Vtx‘𝐺))
15	14	3ad2ant1 1129	. . . . . . 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 484	. . . . . 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 11929	. . . . . . . . . 10 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℕ ↔ ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ ∧ 1 ≤ (♯‘(1^st ‘𝑐))))
18		nn0re 11893	. . . . . . . . . . . . . 14 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ → (♯‘(1^st ‘𝑐)) ∈ ℝ)
19		1e2m1 11751	. . . . . . . . . . . . . . . . 17 ⊢ 1 = (2 − 1)
20	19	breq1i 5059	. . . . . . . . . . . . . . . 16 ⊢ (1 ≤ (♯‘(1^st ‘𝑐)) ↔ (2 − 1) ≤ (♯‘(1^st ‘𝑐)))
21	20	biimpi 218	. . . . . . . . . . . . . . 15 ⊢ (1 ≤ (♯‘(1^st ‘𝑐)) → (2 − 1) ≤ (♯‘(1^st ‘𝑐)))
22		2re 11698	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
23		1re 10627	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
24		lesubadd 11098	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℝ ∧ 1 ∈ ℝ ∧ (♯‘(1^st ‘𝑐)) ∈ ℝ) → ((2 − 1) ≤ (♯‘(1^st ‘𝑐)) ↔ 2 ≤ ((♯‘(1^st ‘𝑐)) + 1)))
25	22, 23, 24	mp3an12 1447	. . . . . . . . . . . . . . 15 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℝ → ((2 − 1) ≤ (♯‘(1^st ‘𝑐)) ↔ 2 ≤ ((♯‘(1^st ‘𝑐)) + 1)))
26	21, 25	syl5ib 246	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ₀) → (1 ≤ (♯‘(1^st ‘𝑐)) → 2 ≤ ((♯‘(1^st ‘𝑐)) + 1)))
29		wlklenvp1 27386	. . . . . . . . . . . . . 14 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (♯‘(2^nd ‘𝑐)) = ((♯‘(1^st ‘𝑐)) + 1))
30	29	adantr 483	. . . . . . . . . . . . 13 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ₀) → (♯‘(2^nd ‘𝑐)) = ((♯‘(1^st ‘𝑐)) + 1))
31	30	breq2d 5064	. . . . . . . . . . . 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 261	. . . . . . . . . . 11 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ₀) → (1 ≤ (♯‘(1^st ‘𝑐)) → 2 ≤ (♯‘(2^nd ‘𝑐))))
33	32	expimpd 456	. . . . . . . . . 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 244	. . . . . . . . 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 1107	. . . . . . 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 484	. . . . . 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 2821	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
39	13, 38	clwlkclwwlk 27765	. . . . . 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 1367	. . . . 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 234	. . . 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 594	. . 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 498	. 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 6864	1 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {crab 3142 class class class wbr 5052 ↦ cmpt 5132 ⟶wf 6337 ‘cfv 6341 (class class class)co 7142 1^st c1st 7673 2^nd c2nd 7674 ℝcr 10522 0cc0 10523 1c1 10524 + caddc 10526 ≤ cle 10662 − cmin 10856 ℕcn 11624 2c2 11679 ℕ₀cn0 11884 ♯chash 13680 Word cword 13851 lastSclsw 13899 prefix cpfx 14017 Vtxcvtx 26767 iEdgciedg 26768 USPGraphcuspgr 26919 Walkscwlks 27364 ClWalkscclwlks 27537 ClWWalkscclwwlk 27744

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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ifp 1058 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-2o 8089 df-oadd 8092 df-er 8275 df-map 8394 df-pm 8395 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-dju 9316 df-card 9354 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-n0 11885 df-xnn0 11955 df-z 11969 df-uz 12231 df-fz 12883 df-fzo 13024 df-hash 13681 df-word 13852 df-lsw 13900 df-substr 13988 df-pfx 14018 df-edg 26819 df-uhgr 26829 df-upgr 26853 df-uspgr 26921 df-wlks 27367 df-clwlks 27538 df-clwwlk 27745

This theorem is referenced by: clwlkclwwlkfo 27772 clwlkclwwlkf1 27773

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