MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  clwlkclwwlkflem Structured version   Visualization version   GIF version

Theorem clwlkclwwlkflem 27147
Description: Lemma for clwlkclwwlkf 27151. (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 6408 . . . . . 6 (𝑤 = 𝑈 → (1st𝑤) = (1st𝑈))
2 clwlkclwwlkf.a . . . . . 6 𝐴 = (1st𝑈)
31, 2syl6eqr 2858 . . . . 5 (𝑤 = 𝑈 → (1st𝑤) = 𝐴)
43fveq2d 6412 . . . 4 (𝑤 = 𝑈 → (♯‘(1st𝑤)) = (♯‘𝐴))
54breq2d 4856 . . 3 (𝑤 = 𝑈 → (1 ≤ (♯‘(1st𝑤)) ↔ 1 ≤ (♯‘𝐴)))
6 clwlkclwwlkf.c . . 3 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st𝑤))}
75, 6elrab2 3562 . 2 (𝑈𝐶 ↔ (𝑈 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘𝐴)))
8 clwlkwlk 26899 . . . 4 (𝑈 ∈ (ClWalks‘𝐺) → 𝑈 ∈ (Walks‘𝐺))
9 wlkop 26751 . . . . 5 (𝑈 ∈ (Walks‘𝐺) → 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
10 clwlkclwwlkf.b . . . . . . . 8 𝐵 = (2nd𝑈)
112, 10opeq12i 4600 . . . . . . 7 𝐴, 𝐵⟩ = ⟨(1st𝑈), (2nd𝑈)⟩
1211eqeq2i 2818 . . . . . 6 (𝑈 = ⟨𝐴, 𝐵⟩ ↔ 𝑈 = ⟨(1st𝑈), (2nd𝑈)⟩)
13 eleq1 2873 . . . . . . 7 (𝑈 = ⟨𝐴, 𝐵⟩ → (𝑈 ∈ (ClWalks‘𝐺) ↔ ⟨𝐴, 𝐵⟩ ∈ (ClWalks‘𝐺)))
14 df-br 4845 . . . . . . . 8 (𝐴(ClWalks‘𝐺)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (ClWalks‘𝐺))
15 isclwlk 26897 . . . . . . . . 9 (𝐴(ClWalks‘𝐺)𝐵 ↔ (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))))
16 wlkcl 26739 . . . . . . . . . . . . . . 15 (𝐴(Walks‘𝐺)𝐵 → (♯‘𝐴) ∈ ℕ0)
17 elnnnn0c 11604 . . . . . . . . . . . . . . . 16 ((♯‘𝐴) ∈ ℕ ↔ ((♯‘𝐴) ∈ ℕ0 ∧ 1 ≤ (♯‘𝐴)))
1817a1i 11 . . . . . . . . . . . . . . 15 (𝐴(Walks‘𝐺)𝐵 → ((♯‘𝐴) ∈ ℕ ↔ ((♯‘𝐴) ∈ ℕ0 ∧ 1 ≤ (♯‘𝐴))))
1916, 18mpbirand 690 . . . . . . . . . . . . . 14 (𝐴(Walks‘𝐺)𝐵 → ((♯‘𝐴) ∈ ℕ ↔ 1 ≤ (♯‘𝐴)))
2019bicomd 214 . . . . . . . . . . . . 13 (𝐴(Walks‘𝐺)𝐵 → (1 ≤ (♯‘𝐴) ↔ (♯‘𝐴) ∈ ℕ))
2120adantr 468 . . . . . . . . . . . 12 ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) → (1 ≤ (♯‘𝐴) ↔ (♯‘𝐴) ∈ ℕ))
2221pm5.32i 566 . . . . . . . . . . 11 (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) ∧ 1 ≤ (♯‘𝐴)) ↔ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) ∧ (♯‘𝐴) ∈ ℕ))
23 df-3an 1102 . . . . . . . . . . 11 ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ↔ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) ∧ (♯‘𝐴) ∈ ℕ))
2422, 23sylbb2 229 . . . . . . . . . 10 (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) ∧ 1 ≤ (♯‘𝐴)) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))
2524ex 399 . . . . . . . . 9 ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴))) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ)))
2615, 25sylbi 208 . . . . . . . 8 (𝐴(ClWalks‘𝐺)𝐵 → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ)))
2714, 26sylbir 226 . . . . . . 7 (⟨𝐴, 𝐵⟩ ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ)))
2813, 27syl6bi 244 . . . . . 6 (𝑈 = ⟨𝐴, 𝐵⟩ → (𝑈 ∈ (ClWalks‘𝐺) → (1 ≤ (♯‘𝐴) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))))
2912, 28sylbir 226 . . . . 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 395 . 2 ((𝑈 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘𝐴)) → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))
337, 32sylbi 208 1 (𝑈𝐶 → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  {crab 3100  cop 4376   class class class wbr 4844  cfv 6101  1st c1st 7396  2nd c2nd 7397  0cc0 10221  1c1 10222  cle 10360  cn 11305  0cn0 11559  chash 13337  Walkscwlks 26720  ClWalkscclwlks 26894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179  ax-cnex 10277  ax-resscn 10278  ax-1cn 10279  ax-icn 10280  ax-addcl 10281  ax-addrcl 10282  ax-mulcl 10283  ax-mulrcl 10284  ax-mulcom 10285  ax-addass 10286  ax-mulass 10287  ax-distr 10288  ax-i2m1 10289  ax-1ne0 10290  ax-1rid 10291  ax-rnegex 10292  ax-rrecex 10293  ax-cnre 10294  ax-pre-lttri 10295  ax-pre-lttrn 10296  ax-pre-ltadd 10297  ax-pre-mulgt0 10298
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-ifp 1079  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6835  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-om 7296  df-1st 7398  df-2nd 7399  df-wrecs 7642  df-recs 7704  df-rdg 7742  df-1o 7796  df-oadd 7800  df-er 7979  df-map 8094  df-pm 8095  df-en 8193  df-dom 8194  df-sdom 8195  df-fin 8196  df-card 9048  df-pnf 10361  df-mnf 10362  df-xr 10363  df-ltxr 10364  df-le 10365  df-sub 10553  df-neg 10554  df-nn 11306  df-n0 11560  df-z 11644  df-uz 11905  df-fz 12550  df-fzo 12690  df-hash 13338  df-word 13510  df-wlks 26723  df-clwlks 26895
This theorem is referenced by:  clwlkclwwlkf1lem2  27148  clwlkclwwlkf1lem3  27149  clwlkclwwlkf  27151  clwlkclwwlkf1  27153
  Copyright terms: Public domain W3C validator