Theorem clwlkclwwlkflem 27888

Description: Lemma for clwlkclwwlkf 27892. (Contributed by AV, 24-May-2022.)

Hypotheses

Ref	Expression
clwlkclwwlkf.c	⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}
clwlkclwwlkf.a	⊢ 𝐴 = (1st ‘𝑈)
clwlkclwwlkf.b	⊢ 𝐵 = (2nd ‘𝑈)

Assertion

Ref	Expression
clwlkclwwlkflem	⊢ (𝑈 ∈ 𝐶 → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))

Distinct variable groups:   𝑤,𝐺   𝑤,𝐴   𝑤,𝑈

Allowed substitution hints:   𝐵(𝑤)   𝐶(𝑤)

Proof of Theorem clwlkclwwlkflem

Step	Hyp	Ref	Expression
1		fveq2 6658	. . . . . 6 ⊢ (𝑤 = 𝑈 → (1st ‘𝑤) = (1st ‘𝑈))
2		clwlkclwwlkf.a	. . . . . 6 ⊢ 𝐴 = (1st ‘𝑈)
3	1, 2	eqtr4di 2811	. . . . 5 ⊢ (𝑤 = 𝑈 → (1st ‘𝑤) = 𝐴)
4	3	fveq2d 6662	. . . 4 ⊢ (𝑤 = 𝑈 → (♯‘(1st ‘𝑤)) = (♯‘𝐴))
5	4	breq2d 5044	. . 3 ⊢ (𝑤 = 𝑈 → (1 ≤ (♯‘(1st ‘𝑤)) ↔ 1 ≤ (♯‘𝐴)))
6		clwlkclwwlkf.c	. . 3 ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}
7	5, 6	elrab2 3605	. 2 ⊢ (𝑈 ∈ 𝐶 ↔ (𝑈 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘𝐴)))
8		clwlkwlk 27663	. . . 4 ⊢ (𝑈 ∈ (ClWalks‘𝐺) → 𝑈 ∈ (Walks‘𝐺))
9		wlkop 27516	. . . . 5 ⊢ (𝑈 ∈ (Walks‘𝐺) → 𝑈 = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩)
10		clwlkclwwlkf.b	. . . . . . . 8 ⊢ 𝐵 = (2nd ‘𝑈)
11	2, 10	opeq12i 4768	. . . . . . 7 ⊢ ⟨𝐴, 𝐵⟩ = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩
12	11	eqeq2i 2771	. . . . . 6 ⊢ (𝑈 = ⟨𝐴, 𝐵⟩ ↔ 𝑈 = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩)
13		eleq1 2839	. . . . . . 7 ⊢ (𝑈 = ⟨𝐴, 𝐵⟩ → (𝑈 ∈ (ClWalks‘𝐺) ↔ ⟨𝐴, 𝐵⟩ ∈ (ClWalks‘𝐺)))
14		df-br 5033	. . . . . . . 8 ⊢ (𝐴(ClWalks‘𝐺)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (ClWalks‘𝐺))
15		isclwlk 27661	. . . . . . . . 9 ⊢ (𝐴(ClWalks‘𝐺)𝐵 ↔ (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))))
16		wlkcl 27504	. . . . . . . . . . . . . . 15 ⊢ (𝐴(Walks‘𝐺)𝐵 → (♯‘𝐴) ∈ ℕ0)
17		elnnnn0c 11979	. . . . . . . . . . . . . . . 16 ⊢ ((♯‘𝐴) ∈ ℕ ↔ ((♯‘𝐴) ∈ ℕ0 ∧ 1 ≤ (♯‘𝐴)))
18	17	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝐴(Walks‘𝐺)𝐵 → ((♯‘𝐴) ∈ ℕ ↔ ((♯‘𝐴) ∈ ℕ0 ∧ 1 ≤ (♯‘𝐴))))
19	16, 18	mpbirand 706	. . . . . . . . . . . . . 14 ⊢ (𝐴(Walks‘𝐺)𝐵 → ((♯‘𝐴) ∈ ℕ ↔ 1 ≤ (♯‘𝐴)))
20	19	bicomd 226	. . . . . . . . . . . . 13 ⊢ (𝐴(Walks‘𝐺)𝐵 → (1 ≤ (♯‘𝐴) ↔ (♯‘𝐴) ∈ ℕ))
21	20	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) → (1 ≤ (♯‘𝐴) ↔ (♯‘𝐴) ∈ ℕ))
22	21	pm5.32i 578	. . . . . . . . . . 11 ⊢ (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) ∧ 1 ≤ (♯‘𝐴)) ↔ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) ∧ (♯‘𝐴) ∈ ℕ))
23		df-3an 1086	. . . . . . . . . . 11 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ↔ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) ∧ (♯‘𝐴) ∈ ℕ))
24	22, 23	sylbb2 241	. . . . . . . . . 10 ⊢ (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) ∧ 1 ≤ (♯‘𝐴)) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))
25	24	ex 416	. . . . . . . . 9 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ)))
26	15, 25	sylbi 220	. . . . . . . 8 ⊢ (𝐴(ClWalks‘𝐺)𝐵 → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ)))
27	14, 26	sylbir 238	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ)))
28	13, 27	syl6bi 256	. . . . . 6 ⊢ (𝑈 = ⟨𝐴, 𝐵⟩ → (𝑈 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))))
29	12, 28	sylbir 238	. . . . 5 ⊢ (𝑈 = ⟨(1st ‘𝑈), (2nd ‘𝑈)⟩ → (𝑈 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))))
30	9, 29	syl 17	. . . 4 ⊢ (𝑈 ∈ (Walks‘𝐺) → (𝑈 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))))
31	8, 30	mpcom 38	. . 3 ⊢ (𝑈 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ)))
32	31	imp 410	. 2 ⊢ ((𝑈 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘𝐴)) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))
33	7, 32	sylbi 220	1 ⊢ (𝑈 ∈ 𝐶 → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  {crab 3074  ⟨cop 4528   class class class wbr 5032  ‘cfv 6335  1^st c1st 7691  2^nd c2nd 7692  0cc0 10575  1c1 10576   ≤ cle 10714  ℕcn 11674  ℕ₀cn0 11934  ♯chash 13740  Walkscwlks 27485  ClWalkscclwlks 27658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ifp 1059  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-1o 8112  df-er 8299  df-map 8418  df-en 8528  df-dom 8529  df-sdom 8530  df-fin 8531  df-card 9401  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-n0 11935  df-z 12021  df-uz 12283  df-fz 12940  df-fzo 13083  df-hash 13741  df-word 13914  df-wlks 27488  df-clwlks 27659

This theorem is referenced by:  clwlkclwwlkf1lem2  27889  clwlkclwwlkf1lem3  27890  clwlkclwwlkf  27892  clwlkclwwlkf1  27894