Theorem clwlkcompbp 28051

Description: Basic properties of the components of a closed walk. (Contributed by AV, 23-May-2022.)

Hypotheses

Ref	Expression
clwlkcompbp.1	⊢ 𝐹 = (1st ‘𝑊)
clwlkcompbp.2	⊢ 𝑃 = (2nd ‘𝑊)

Assertion

Ref	Expression
clwlkcompbp	⊢ (𝑊 ∈ (ClWalks‘𝐺) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))

Proof of Theorem clwlkcompbp

Step	Hyp	Ref	Expression
1		clwlkwlk 28044	. . 3 ⊢ (𝑊 ∈ (ClWalks‘𝐺) → 𝑊 ∈ (Walks‘𝐺))
2		wlkop 27897	. . 3 ⊢ (𝑊 ∈ (Walks‘𝐺) → 𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩)
3	1, 2	syl 17	. 2 ⊢ (𝑊 ∈ (ClWalks‘𝐺) → 𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩)
4		eleq1 2826	. . . 4 ⊢ (𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩ → (𝑊 ∈ (ClWalks‘𝐺) ↔ ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩ ∈ (ClWalks‘𝐺)))
5		df-br 5071	. . . 4 ⊢ ((1st ‘𝑊)(ClWalks‘𝐺)(2nd ‘𝑊) ↔ ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩ ∈ (ClWalks‘𝐺))
6	4, 5	bitr4di 288	. . 3 ⊢ (𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩ → (𝑊 ∈ (ClWalks‘𝐺) ↔ (1st ‘𝑊)(ClWalks‘𝐺)(2nd ‘𝑊)))
7		isclwlk 28042	. . . 4 ⊢ ((1st ‘𝑊)(ClWalks‘𝐺)(2nd ‘𝑊) ↔ ((1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊) ∧ ((2nd ‘𝑊)‘0) = ((2nd ‘𝑊)‘(♯‘(1st ‘𝑊)))))
8		clwlkcompbp.1	. . . . . 6 ⊢ 𝐹 = (1st ‘𝑊)
9		clwlkcompbp.2	. . . . . 6 ⊢ 𝑃 = (2nd ‘𝑊)
10	8, 9	breq12i 5079	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 ↔ (1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊))
11	9	fveq1i 6757	. . . . . 6 ⊢ (𝑃‘0) = ((2nd ‘𝑊)‘0)
12	8	fveq2i 6759	. . . . . . 7 ⊢ (♯‘𝐹) = (♯‘(1st ‘𝑊))
13	9, 12	fveq12i 6762	. . . . . 6 ⊢ (𝑃‘(♯‘𝐹)) = ((2nd ‘𝑊)‘(♯‘(1st ‘𝑊)))
14	11, 13	eqeq12i 2756	. . . . 5 ⊢ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ↔ ((2nd ‘𝑊)‘0) = ((2nd ‘𝑊)‘(♯‘(1st ‘𝑊))))
15	10, 14	anbi12i 626	. . . 4 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))) ↔ ((1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊) ∧ ((2nd ‘𝑊)‘0) = ((2nd ‘𝑊)‘(♯‘(1st ‘𝑊)))))
16	7, 15	sylbb2 237	. . 3 ⊢ ((1st ‘𝑊)(ClWalks‘𝐺)(2nd ‘𝑊) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
17	6, 16	syl6bi 252	. 2 ⊢ (𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩ → (𝑊 ∈ (ClWalks‘𝐺) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))))
18	3, 17	mpcom 38	1 ⊢ (𝑊 ∈ (ClWalks‘𝐺) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ⟨cop 4564   class class class wbr 5070  ‘cfv 6418  1^st c1st 7802  2^nd c2nd 7803  0cc0 10802  ♯chash 13972  Walkscwlks 27866  ClWalkscclwlks 28039

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ifp 1060  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-fzo 13312  df-hash 13973  df-word 14146  df-wlks 27869  df-clwlks 28040

This theorem is referenced by: (None)