Theorem clwlkcompbp 26913

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 26906	. . 3 ⊢ (𝑊 ∈ (ClWalks‘𝐺) → 𝑊 ∈ (Walks‘𝐺))
2		wlkop 26758	. . 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 2838	. . . 4 ⊢ (𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩ → (𝑊 ∈ (ClWalks‘𝐺) ↔ ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩ ∈ (ClWalks‘𝐺)))
5		df-br 4787	. . . 4 ⊢ ((1st ‘𝑊)(ClWalks‘𝐺)(2nd ‘𝑊) ↔ ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩ ∈ (ClWalks‘𝐺))
6	4, 5	syl6bbr 278	. . 3 ⊢ (𝑊 = ⟨(1st ‘𝑊), (2nd ‘𝑊)⟩ → (𝑊 ∈ (ClWalks‘𝐺) ↔ (1st ‘𝑊)(ClWalks‘𝐺)(2nd ‘𝑊)))
7		isclwlk 26904	. . . 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 4795	. . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 ↔ (1st ‘𝑊)(Walks‘𝐺)(2nd ‘𝑊))
11	9	fveq1i 6333	. . . . . 6 ⊢ (𝑃‘0) = ((2nd ‘𝑊)‘0)
12	8	fveq2i 6335	. . . . . . 7 ⊢ (♯‘𝐹) = (♯‘(1st ‘𝑊))
13	9, 12	fveq12i 6337	. . . . . 6 ⊢ (𝑃‘(♯‘𝐹)) = ((2nd ‘𝑊)‘(♯‘(1st ‘𝑊)))
14	11, 13	eqeq12i 2785	. . . . 5 ⊢ ((𝑃‘0) = (𝑃‘(♯‘𝐹)) ↔ ((2nd ‘𝑊)‘0) = ((2nd ‘𝑊)‘(♯‘(1st ‘𝑊))))
15	10, 14	anbi12i 612	. . . 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 228	. . 3 ⊢ ((1st ‘𝑊)(ClWalks‘𝐺)(2nd ‘𝑊) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
17	6, 16	syl6bi 243	. 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 382   = wceq 1631   ∈ wcel 2145  ⟨cop 4322   class class class wbr 4786  ‘cfv 6031  1^st c1st 7313  2^nd c2nd 7314  0cc0 10138  ♯chash 13321  Walkscwlks 26727  ClWalkscclwlks 26901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-ifp 1050  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-n0 11495  df-z 11580  df-uz 11889  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-wlks 26730  df-clwlks 26902

This theorem is referenced by: (None)