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Theorem clwlkclwwlkflem 27888
 Description: Lemma for clwlkclwwlkf 27892. (Contributed by AV, 24-May-2022.)
Hypotheses
Ref Expression
clwlkclwwlkf.c 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}
clwlkclwwlkf.a 𝐴 = (1st𝑈)
clwlkclwwlkf.b 𝐵 = (2nd𝑈)
Assertion
Ref Expression
clwlkclwwlkflem (𝑈𝐶 → (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
StepHypRef Expression
1 fveq2 6658 . . . . . 6 (𝑤 = 𝑈 → (1st𝑤) = (1st𝑈))
2 clwlkclwwlkf.a . . . . . 6 𝐴 = (1st𝑈)
31, 2eqtr4di 2811 . . . . 5 (𝑤 = 𝑈 → (1st𝑤) = 𝐴)
43fveq2d 6662 . . . 4 (𝑤 = 𝑈 → (♯‘(1st𝑤)) = (♯‘𝐴))
54breq2d 5044 . . 3 (𝑤 = 𝑈 → (1 ≤ (♯‘(1st𝑤)) ↔ 1 ≤ (♯‘𝐴)))
6 clwlkclwwlkf.c . . 3 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}
75, 6elrab2 3605 . 2 (𝑈𝐶 ↔ (𝑈 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘𝐴)))
8 clwlkwlk 27663 . . . 4 (𝑈 ∈ (ClWalks‘𝐺) → 𝑈 ∈ (Walks‘𝐺))
9 wlkop 27516 . . . . 5 (𝑈 ∈ (Walks‘𝐺) → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
10 clwlkclwwlkf.b . . . . . . . 8 𝐵 = (2nd𝑈)
112, 10opeq12i 4768 . . . . . . 7 𝐴, 𝐵⟩ = ⟨(1st𝑈), (2nd𝑈)⟩
1211eqeq2i 2771 . . . . . 6 (𝑈 = ⟨𝐴, 𝐵⟩ ↔ 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
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)))
1817a1i 11 . . . . . . . . . . . . . . 15 (𝐴(Walks‘𝐺)𝐵 → ((♯‘𝐴) ∈ ℕ ↔ ((♯‘𝐴) ∈ ℕ0 ∧ 1 ≤ (♯‘𝐴))))
1916, 18mpbirand 706 . . . . . . . . . . . . . 14 (𝐴(Walks‘𝐺)𝐵 → ((♯‘𝐴) ∈ ℕ ↔ 1 ≤ (♯‘𝐴)))
2019bicomd 226 . . . . . . . . . . . . 13 (𝐴(Walks‘𝐺)𝐵 → (1 ≤ (♯‘𝐴) ↔ (♯‘𝐴) ∈ ℕ))
2120adantr 484 . . . . . . . . . . . 12 ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) → (1 ≤ (♯‘𝐴) ↔ (♯‘𝐴) ∈ ℕ))
2221pm5.32i 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))
2422, 23sylbb2 241 . . . . . . . . . 10 (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) ∧ 1 ≤ (♯‘𝐴)) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))
2524ex 416 . . . . . . . . 9 ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ)))
2615, 25sylbi 220 . . . . . . . 8 (𝐴(ClWalks‘𝐺)𝐵 → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ)))
2714, 26sylbir 238 . . . . . . 7 (⟨𝐴, 𝐵⟩ ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ)))
2813, 27syl6bi 256 . . . . . 6 (𝑈 = ⟨𝐴, 𝐵⟩ → (𝑈 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))))
2912, 28sylbir 238 . . . . 5 (𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩ → (𝑈 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))))
309, 29syl 17 . . . 4 (𝑈 ∈ (Walks‘𝐺) → (𝑈 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))))
318, 30mpcom 38 . . 3 (𝑈 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ)))
3231imp 410 . 2 ((𝑈 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘𝐴)) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))
337, 32sylbi 220 1 (𝑈𝐶 → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))
 Colors of variables: wff setvar class 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  1st c1st 7691  2nd 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
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