clwlksf1clwwlklem3OLD - Metamath Proof Explorer

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Theorem clwlksf1clwwlklem3OLD 27248

Description: Obsolete as of 24-May-2022. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
clwlksfclwwlk.1	⊢ 𝐴 = (1^st ‘𝑐)
clwlksfclwwlk.2	⊢ 𝐵 = (2^nd ‘𝑐)
clwlksfclwwlk.c	⊢ 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘𝐴) = 𝑁}
clwlksfclwwlk.f	⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))

**Assertion**
Ref	Expression
clwlksf1clwwlklem3OLD	⊢ (𝑊 ∈ 𝐶 → (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺))

Distinct variable groups: 𝐺,𝑐 𝑁,𝑐 𝑊,𝑐 𝐶,𝑐 𝐹,𝑐 Allowed substitution hints: 𝐴(𝑐) 𝐵(𝑐)

**Proof of Theorem clwlksf1clwwlklem3OLD**
Step	Hyp	Ref	Expression
1		clwlksfclwwlk.1	. . 3 ⊢ 𝐴 = (1^st ‘𝑐)
2		clwlksfclwwlk.2	. . 3 ⊢ 𝐵 = (2^nd ‘𝑐)
3		clwlksfclwwlk.c	. . 3 ⊢ 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘𝐴) = 𝑁}
4		clwlksfclwwlk.f	. . 3 ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))
5	1, 2, 3, 4	clwlksf1clwwlklem0OLD 27245	. 2 ⊢ (𝑊 ∈ 𝐶 → (((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺) ∧ ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘(♯‘(1^st ‘𝑊)))) ∧ (♯‘(1^st ‘𝑊)) = 𝑁))
6		lencl 13520	. . . . 5 ⊢ ((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) → (♯‘(1^st ‘𝑊)) ∈ ℕ₀)
7		nn0z 11602	. . . . . . . . 9 ⊢ ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → (♯‘(1^st ‘𝑊)) ∈ ℤ)
8		fzval3 12745	. . . . . . . . 9 ⊢ ((♯‘(1^st ‘𝑊)) ∈ ℤ → (0...(♯‘(1^st ‘𝑊))) = (0..^((♯‘(1^st ‘𝑊)) + 1)))
9	7, 8	syl 17	. . . . . . . 8 ⊢ ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → (0...(♯‘(1^st ‘𝑊))) = (0..^((♯‘(1^st ‘𝑊)) + 1)))
10	9	feq2d 6171	. . . . . . 7 ⊢ ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → ((2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺) ↔ (2^nd ‘𝑊):(0..^((♯‘(1^st ‘𝑊)) + 1))⟶(Vtx‘𝐺)))
11	10	biimpa 462	. . . . . 6 ⊢ (((♯‘(1^st ‘𝑊)) ∈ ℕ₀ ∧ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺)) → (2^nd ‘𝑊):(0..^((♯‘(1^st ‘𝑊)) + 1))⟶(Vtx‘𝐺))
12		iswrdi 13505	. . . . . 6 ⊢ ((2^nd ‘𝑊):(0..^((♯‘(1^st ‘𝑊)) + 1))⟶(Vtx‘𝐺) → (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺))
13	11, 12	syl 17	. . . . 5 ⊢ (((♯‘(1^st ‘𝑊)) ∈ ℕ₀ ∧ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺)) → (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺))
14	6, 13	sylan 569	. . . 4 ⊢ (((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺)) → (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺))
15	14	3adant3 1126	. . 3 ⊢ (((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺) ∧ ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘(♯‘(1^st ‘𝑊)))) → (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺))
16	15	adantr 466	. 2 ⊢ ((((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(♯‘(1^st ‘𝑊)))⟶(Vtx‘𝐺) ∧ ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘(♯‘(1^st ‘𝑊)))) ∧ (♯‘(1^st ‘𝑊)) = 𝑁) → (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺))
17	5, 16	syl 17	1 ⊢ (𝑊 ∈ 𝐶 → (2^nd ‘𝑊) ∈ Word (Vtx‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 {crab 3065 ⟨cop 4322 ↦ cmpt 4863 dom cdm 5249 ⟶wf 6027 ‘cfv 6031 (class class class)co 6793 1^st c1st 7313 2^nd c2nd 7314 0cc0 10138 1c1 10139 + caddc 10141 ℕ₀cn0 11494 ℤcz 11579 ...cfz 12533 ..^cfzo 12673 ♯chash 13321 Word cword 13487 substr csubstr 13491 Vtxcvtx 26095 iEdgciedg 26096 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-oadd 7717 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: clwlksf1clwwlklemOLD 27249

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