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Theorem clwlkclwwlkf 29990

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 2733	. . . . 5 ⊢ (1^st ‘𝑐) = (1^st ‘𝑐)
3		eqid 2733	. . . . 5 ⊢ (2^nd ‘𝑐) = (2^nd ‘𝑐)
4	1, 2, 3	clwlkclwwlkflem 29986	. . . 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 29753	. . . . . . . 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 6841	. . . . . . . . 9 ⊢ (1^st ‘𝑐) ∈ V
7		breq1 5096	. . . . . . . . 9 ⊢ (𝑓 = (1^st ‘𝑐) → (𝑓(ClWalks‘𝐺)(2^nd ‘𝑐) ↔ (1^st ‘𝑐)(ClWalks‘𝐺)(2^nd ‘𝑐)))
8	6, 7	spcev 3557	. . . . . . . 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 2733	. . . . . . . . 9 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
14	13	wlkpwrd 29598	. . . . . . . 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 12433	. . . . . . . . . 10 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℕ ↔ ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ ∧ 1 ≤ (♯‘(1^st ‘𝑐))))
18		nn0re 12397	. . . . . . . . . . . . . 14 ⊢ ((♯‘(1^st ‘𝑐)) ∈ ℕ₀ → (♯‘(1^st ‘𝑐)) ∈ ℝ)
19		1e2m1 12254	. . . . . . . . . . . . . . . . 17 ⊢ 1 = (2 − 1)
20	19	breq1i 5100	. . . . . . . . . . . . . . . 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 12206	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℝ
23		1re 11119	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℝ
24		lesubadd 11596	. . . . . . . . . . . . . . . 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 29599	. . . . . . . . . . . . . 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 5105	. . . . . . . . . . . 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 2733	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
39	13, 38	clwlkclwwlk 29984	. . . . . 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 7053	1 ⊢ (𝐺 ∈ USPGraph → 𝐹:𝐶⟶(ClWWalks‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∃wex 1780 ∈ wcel 2113 {crab 3396 class class class wbr 5093 ↦ cmpt 5174 ⟶wf 6482 ‘cfv 6486 (class class class)co 7352 1^st c1st 7925 2^nd c2nd 7926 ℝcr 11012 0cc0 11013 1c1 11014 + caddc 11016 ≤ cle 11154 − cmin 11351 ℕcn 12132 2c2 12187 ℕ₀cn0 12388 ♯chash 14239 Word cword 14422 lastSclsw 14471 prefix cpfx 14580 Vtxcvtx 28976 iEdgciedg 28977 USPGraphcuspgr 29128 Walkscwlks 29577 ClWalkscclwlks 29750 ClWWalkscclwwlk 29963

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090

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 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-om 7803 df-1st 7927 df-2nd 7928 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-oadd 8395 df-er 8628 df-map 8758 df-pm 8759 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-dju 9801 df-card 9839 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-nn 12133 df-2 12195 df-n0 12389 df-xnn0 12462 df-z 12476 df-uz 12739 df-fz 13410 df-fzo 13557 df-hash 14240 df-word 14423 df-lsw 14472 df-substr 14551 df-pfx 14581 df-edg 29028 df-uhgr 29038 df-upgr 29062 df-uspgr 29130 df-wlks 29580 df-clwlks 29751 df-clwwlk 29964

This theorem is referenced by: clwlkclwwlkfo 29991 clwlkclwwlkf1 29992

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