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Theorem clwlkclwwlkf 29944

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 2730	. . . . 5 ⊢ (1^st ‘𝑐) = (1^st ‘𝑐)
3		eqid 2730	. . . . 5 ⊢ (2^nd ‘𝑐) = (2^nd ‘𝑐)
4	1, 2, 3	clwlkclwwlkflem 29940	. . . 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 29710	. . . . . . . 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 6874	. . . . . . . . 9 ⊢ (1^st ‘𝑐) ∈ V
7		breq1 5113	. . . . . . . . 9 ⊢ (𝑓 = (1^st ‘𝑐) → (𝑓(ClWalks‘𝐺)(2^nd ‘𝑐) ↔ (1^st ‘𝑐)(ClWalks‘𝐺)(2^nd ‘𝑐)))
8	6, 7	spcev 3575	. . . . . . . 8 ⊢ ((1^st ‘𝑐)(ClWalks‘𝐺)(2^nd ‘𝑐) → ∃𝑓 𝑓(ClWalks‘𝐺)(2^nd ‘𝑐))
9	5, 8	sylbir 235	. . . . . . 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 1132	. . . . . 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 481	. . . . 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 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)) → 𝐺 ∈ USPGraph)
13		eqid 2730	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
14	13	wlkpwrd 29552	. . . . . . . 8 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (2^nd ‘𝑐) ∈ Word (Vtx‘𝐺))
15	14	3ad2ant1 1133	. . . . . . 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 481	. . . . . 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 12494	. . . . . . . . . 10 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℕ ↔ ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ ∧ 1 ≤ (♯‘(1^st ‘𝑐))))
18		nn0re 12458	. . . . . . . . . . . . . 14 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ → (♯‘(1^st ‘𝑐)) ∈ ℝ)
19		1e2m1 12315	. . . . . . . . . . . . . . . . 17 ⊢ 1 = (2 − 1)
20	19	breq1i 5117	. . . . . . . . . . . . . . . 16 ⊢ (1 ≤ (♯‘(1^st ‘𝑐)) ↔ (2 − 1) ≤ (♯‘(1^st ‘𝑐)))
21	20	biimpi 216	. . . . . . . . . . . . . . 15 ⊢ (1 ≤ (♯‘(1^st ‘𝑐)) → (2 − 1) ≤ (♯‘(1^st ‘𝑐)))
22		2re 12267	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
23		1re 11181	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
24		lesubadd 11657	. . . . . . . . . . . . . . . 16 ⊢ ((2 ∈ ℝ ∧ 1 ∈ ℝ ∧ (♯‘(1^st ‘𝑐)) ∈ ℝ) → ((2 − 1) ≤ (♯‘(1^st ‘𝑐)) ↔ 2 ≤ ((♯‘(1^st ‘𝑐)) + 1)))
25	22, 23, 24	mp3an12 1453	. . . . . . . . . . . . . . 15 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℝ → ((2 − 1) ≤ (♯‘(1^st ‘𝑐)) ↔ 2 ≤ ((♯‘(1^st ‘𝑐)) + 1)))
26	21, 25	imbitrid 244	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ₀) → (1 ≤ (♯‘(1^st ‘𝑐)) → 2 ≤ ((♯‘(1^st ‘𝑐)) + 1)))
29		wlklenvp1 29553	. . . . . . . . . . . . . 14 ⊢ ((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) → (♯‘(2^nd ‘𝑐)) = ((♯‘(1^st ‘𝑐)) + 1))
30	29	adantr 480	. . . . . . . . . . . . 13 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ₀) → (♯‘(2^nd ‘𝑐)) = ((♯‘(1^st ‘𝑐)) + 1))
31	30	breq2d 5122	. . . . . . . . . . . 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 259	. . . . . . . . . . 11 ⊢ (((1^st ‘𝑐)(Walks‘𝐺)(2^nd ‘𝑐) ∧ (♯‘(1^st ‘𝑐)) ∈ ℕ₀) → (1 ≤ (♯‘(1^st ‘𝑐)) → 2 ≤ (♯‘(2^nd ‘𝑐))))
33	32	expimpd 453	. . . . . . . . . 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 242	. . . . . . . . 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 481	. . . . . 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 2730	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
39	13, 38	clwlkclwwlk 29938	. . . . . 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 1373	. . . . 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 232	. . . 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 495	. 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 7089	1 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∃wex 1779 ∈ wcel 2109 {crab 3408 class class class wbr 5110 ↦ cmpt 5191 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 1^st c1st 7969 2^nd c2nd 7970 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 ≤ cle 11216 − cmin 11412 ℕcn 12193 2c2 12248 ℕ₀cn0 12449 ♯chash 14302 Word cword 14485 lastSclsw 14534 prefix cpfx 14642 Vtxcvtx 28930 iEdgciedg 28931 USPGraphcuspgr 29082 Walkscwlks 29531 ClWalkscclwlks 29707 ClWWalkscclwwlk 29917

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-er 8674 df-map 8804 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-dju 9861 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-n0 12450 df-xnn0 12523 df-z 12537 df-uz 12801 df-fz 13476 df-fzo 13623 df-hash 14303 df-word 14486 df-lsw 14535 df-substr 14613 df-pfx 14643 df-edg 28982 df-uhgr 28992 df-upgr 29016 df-uspgr 29084 df-wlks 29534 df-clwlks 29708 df-clwwlk 29918

This theorem is referenced by: clwlkclwwlkfo 29945 clwlkclwwlkf1 29946

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