clwlkclwwlkf - Metamath Proof Explorer

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Theorem clwlkclwwlkf 30370

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 2762	. . . . 5 ⊢ (1^st ‘𝑐) = (1^st ‘𝑐)
3		eqid 2762	. . . . 5 ⊢ (2^nd ‘𝑐) = (2^nd ‘𝑐)
4	1, 2, 3	clwlkclwwlkflem 30366	. . . 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 30133	. . . . . . . 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 6894	. . . . . . . . 9 ⊢ (1^st ‘𝑐) ∈ V
7		breq1 5111	. . . . . . . . 9 ⊢ (𝑓 = (1^st ‘𝑐) → (𝑓(ClWalks‘𝐺)(2^nd ‘𝑐) ↔ (1^st ‘𝑐)(ClWalks‘𝐺)(2^nd ‘𝑐)))
8	6, 7	spcev 3564	. . . . . . . 8 ⊢ ((1^st ‘𝑐)(ClWalks‘𝐺)(2^nd ‘𝑐) → ∃𝑓 𝑓(ClWalks‘𝐺)(2^nd ‘𝑐))
9	5, 8	sylbir 238	. . . . . . 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 1149	. . . . . 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 486	. . . . 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 487	. . . . . 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 2762	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
14	13	wlkpwrd 29978	. . . . . . . 8 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (2^nd ‘𝑐) ∈ Word (Vtx‘𝐺))
15	14	3ad2ant1 1150	. . . . . . 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 486	. . . . . 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 12555	. . . . . . . . . 10 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℕ ↔ ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ ∧ 1 ≤ (♯‘(1^st ‘𝑐))))
18		nn0re 12519	. . . . . . . . . . . . . 14 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ → (♯‘(1^st ‘𝑐)) ∈ ℝ)
19		1e2m1 12373	. . . . . . . . . . . . . . . . 17 ⊢ 1 = (2 − 1)
20	19	breq1i 5115	. . . . . . . . . . . . . . . 16 ⊢ (1 ≤ (♯‘(1^st ‘𝑐)) ↔ (2 − 1) ≤ (♯‘(1^st ‘𝑐)))
21	20	biimpi 219	. . . . . . . . . . . . . . 15 ⊢ (1 ≤ (♯‘(1^st ‘𝑐)) → (2 − 1) ≤ (♯‘(1^st ‘𝑐)))
22		2re 12321	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
23		1re 11214	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
24		lesubadd 11692	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℝ ∧ 1 ∈ ℝ ∧ (♯‘(1^st ‘𝑐)) ∈ ℝ) → ((2 − 1) ≤ (♯‘(1^st ‘𝑐)) ↔ 2 ≤ ((♯‘(1^st ‘𝑐)) + 1)))
25	22, 23, 24	mp3an12 1479	. . . . . . . . . . . . . . 15 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℝ → ((2 − 1) ≤ (♯‘(1^st ‘𝑐)) ↔ 2 ≤ ((♯‘(1^st ‘𝑐)) + 1)))
26	21, 25	imbitrid 247	. . . . . . . . . . . . . 14 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℝ → (1 ≤ (♯‘(1^st ‘𝑐)) → 2 ≤ ((♯‘(1^st ‘𝑐)) + 1)))
27	18, 26	syl 18	. . . . . . . . . . . . 13 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ → (1 ≤ (♯‘(1^st ‘𝑐)) → 2 ≤ ((♯‘(1^st ‘𝑐)) + 1)))
28	27	adantl 486	. . . . . . . . . . . 12 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ₀) → (1 ≤ (♯‘(1^st ‘𝑐)) → 2 ≤ ((♯‘(1^st ‘𝑐)) + 1)))
29		wlklenvp1 29979	. . . . . . . . . . . . . 14 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (♯‘(2^nd ‘𝑐)) = ((♯‘(1^st ‘𝑐)) + 1))
30	29	adantr 485	. . . . . . . . . . . . 13 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ₀) → (♯‘(2^nd ‘𝑐)) = ((♯‘(1^st ‘𝑐)) + 1))
31	30	breq2d 5120	. . . . . . . . . . . 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 262	. . . . . . . . . . 11 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ₀) → (1 ≤ (♯‘(1^st ‘𝑐)) → 2 ≤ (♯‘(2^nd ‘𝑐))))
33	32	expimpd 458	. . . . . . . . . 10 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (((♯‘(1^st ‘𝑐)) ∈ ℕ₀ ∧ 1 ≤ (♯‘(1^st ‘𝑐))) → 2 ≤ (♯‘(2^nd ‘𝑐))))
34	17, 33	biimtrid 245	. . . . . . . . 9 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → ((♯‘(1^st ‘𝑐)) ∈ ℕ → 2 ≤ (♯‘(2^nd ‘𝑐))))
35	34	a1d 26	. . . . . . . 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 1127	. . . . . . 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 486	. . . . . 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 2762	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
39	13, 38	clwlkclwwlk 30364	. . . . . 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 1397	. . . . 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 235	. . . 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 604	. . 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 500	. 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 7109	1 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 ∧ w3a 1102 = wceq 1569 ∃wex 1808 ∈ wcel 2142 {crab 3415 class class class wbr 5108 ↦ cmpt 5191 ⟶wf 6532 ‘cfv 6536 (class class class)co 7412 1^st c1st 7982 2^nd c2nd 7983 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 ≤ cle 11250 − cmin 11447 ℕcn 12239 2c2 12301 ℕ₀cn0 12510 ♯chash 14373 Word cword 14557 lastSclsw 14606 prefix cpfx 14715 Vtxcvtx 29357 iEdgciedg 29358 USPGraphcuspgr 29509 Walkscwlks 29957 ClWalkscclwlks 30130 ClWWalkscclwwlk 30343

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 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 proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-ifp 1078 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3369 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5555 df-eprel 5560 df-po 5568 df-so 5569 df-fr 5613 df-we 5615 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7861 df-1st 7984 df-2nd 7985 df-frecs 8276 df-wrecs 8307 df-recs 8356 df-rdg 8395 df-1o 8451 df-2o 8452 df-oadd 8455 df-er 8692 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-dju 9894 df-card 9932 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11449 df-neg 11450 df-nn 12240 df-2 12309 df-n0 12511 df-xnn0 12584 df-z 12598 df-uz 12869 df-fz 13542 df-fzo 13690 df-hash 14374 df-word 14558 df-lsw 14607 df-substr 14686 df-pfx 14716 df-edg 29409 df-uhgr 29419 df-upgr 29443 df-uspgr 29511 df-wlks 29960 df-clwlks 30131 df-clwwlk 30344

This theorem is used by: clwlkclwwlkfo 30371 clwlkclwwlkf1 30372

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