clwlksf1clwwlklem3OLD - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > clwlksf1clwwlklem3OLD		Structured version Visualization version GIF version

Theorem clwlksf1clwwlklem3OLD 27408

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
clwlksfclwwlkOLD.1	⊢ 𝐴 = (1^st ‘𝑐)
clwlksfclwwlkOLD.2	⊢ 𝐵 = (2^nd ‘𝑐)
clwlksfclwwlkOLD.c	⊢ 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘𝐴) = 𝑁}
clwlksfclwwlkOLD.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		clwlksfclwwlkOLD.1	. . 3 ⊢ 𝐴 = (1^st ‘𝑐)
2		clwlksfclwwlkOLD.2	. . 3 ⊢ 𝐵 = (2^nd ‘𝑐)
3		clwlksfclwwlkOLD.c	. . 3 ⊢ 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘𝐴) = 𝑁}
4		clwlksfclwwlkOLD.f	. . 3 ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr ⟨0, (♯‘𝐴)⟩))
5	1, 2, 3, 4	clwlksf1clwwlklem0OLD 27405	. 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 13553	. . . . 5 ⊢ ((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) → (♯‘(1^st ‘𝑊)) ∈ ℕ₀)
7		nn0z 11690	. . . . . . . . 9 ⊢ ((♯‘(1^st ‘𝑊)) ∈ ℕ₀ → (♯‘(1^st ‘𝑊)) ∈ ℤ)
8		fzval3 12792	. . . . . . . . 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 6242	. . . . . . 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 469	. . . . . 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 13538	. . . . . 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 576	. . . 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 1163	. . 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 473	. 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 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 {crab 3093 ⟨cop 4374 ↦ cmpt 4922 dom cdm 5312 ⟶wf 6097 ‘cfv 6101 (class class class)co 6878 1^st c1st 7399 2^nd c2nd 7400 0cc0 10224 1c1 10225 + caddc 10227 ℕ₀cn0 11580 ℤcz 11666 ...cfz 12580 ..^cfzo 12720 ♯chash 13370 Word cword 13534 substr csubstr 13664 Vtxcvtx 26231 iEdgciedg 26232 ClWalkscclwlks 27024

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-ifp 1087 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 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 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-map 8097 df-pm 8098 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-card 9051 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-n0 11581 df-z 11667 df-uz 11931 df-fz 12581 df-fzo 12721 df-hash 13371 df-word 13535 df-wlks 26849 df-clwlks 27025

This theorem is referenced by: clwlksf1clwwlklemOLD 27409

W3C validator