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Mirrors > Home > MPE Home > Th. List > clwlksf1clwwlklem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for clwlksf1clwwlklem 26968. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) |
Ref | Expression |
---|---|
clwlksfclwwlk.1 | ⊢ 𝐴 = (1st ‘𝑐) |
clwlksfclwwlk.2 | ⊢ 𝐵 = (2nd ‘𝑐) |
clwlksfclwwlk.c | ⊢ 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁} |
clwlksfclwwlk.f | ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 substr 〈0, (#‘𝐴)〉)) |
Ref | Expression |
---|---|
clwlksf1clwwlklem3 | ⊢ (𝑊 ∈ 𝐶 → (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clwlksfclwwlk.1 | . . 3 ⊢ 𝐴 = (1st ‘𝑐) | |
2 | clwlksfclwwlk.2 | . . 3 ⊢ 𝐵 = (2nd ‘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 | clwlksf1clwwlklem0 26964 | . 2 ⊢ (𝑊 ∈ 𝐶 → (((1st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝑊):(0...(#‘(1st ‘𝑊)))⟶(Vtx‘𝐺) ∧ ((2nd ‘𝑊)‘0) = ((2nd ‘𝑊)‘(#‘(1st ‘𝑊)))) ∧ (#‘(1st ‘𝑊)) = 𝑁)) |
6 | lencl 13324 | . . . . 5 ⊢ ((1st ‘𝑊) ∈ Word dom (iEdg‘𝐺) → (#‘(1st ‘𝑊)) ∈ ℕ0) | |
7 | nn0z 11400 | . . . . . . . . 9 ⊢ ((#‘(1st ‘𝑊)) ∈ ℕ0 → (#‘(1st ‘𝑊)) ∈ ℤ) | |
8 | fzval3 12536 | . . . . . . . . 9 ⊢ ((#‘(1st ‘𝑊)) ∈ ℤ → (0...(#‘(1st ‘𝑊))) = (0..^((#‘(1st ‘𝑊)) + 1))) | |
9 | 7, 8 | syl 17 | . . . . . . . 8 ⊢ ((#‘(1st ‘𝑊)) ∈ ℕ0 → (0...(#‘(1st ‘𝑊))) = (0..^((#‘(1st ‘𝑊)) + 1))) |
10 | 9 | feq2d 6031 | . . . . . . 7 ⊢ ((#‘(1st ‘𝑊)) ∈ ℕ0 → ((2nd ‘𝑊):(0...(#‘(1st ‘𝑊)))⟶(Vtx‘𝐺) ↔ (2nd ‘𝑊):(0..^((#‘(1st ‘𝑊)) + 1))⟶(Vtx‘𝐺))) |
11 | 10 | biimpa 501 | . . . . . 6 ⊢ (((#‘(1st ‘𝑊)) ∈ ℕ0 ∧ (2nd ‘𝑊):(0...(#‘(1st ‘𝑊)))⟶(Vtx‘𝐺)) → (2nd ‘𝑊):(0..^((#‘(1st ‘𝑊)) + 1))⟶(Vtx‘𝐺)) |
12 | iswrdi 13309 | . . . . . 6 ⊢ ((2nd ‘𝑊):(0..^((#‘(1st ‘𝑊)) + 1))⟶(Vtx‘𝐺) → (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ (((#‘(1st ‘𝑊)) ∈ ℕ0 ∧ (2nd ‘𝑊):(0...(#‘(1st ‘𝑊)))⟶(Vtx‘𝐺)) → (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)) |
14 | 6, 13 | sylan 488 | . . . 4 ⊢ (((1st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝑊):(0...(#‘(1st ‘𝑊)))⟶(Vtx‘𝐺)) → (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)) |
15 | 14 | 3adant3 1081 | . . 3 ⊢ (((1st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝑊):(0...(#‘(1st ‘𝑊)))⟶(Vtx‘𝐺) ∧ ((2nd ‘𝑊)‘0) = ((2nd ‘𝑊)‘(#‘(1st ‘𝑊)))) → (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)) |
16 | 15 | adantr 481 | . 2 ⊢ ((((1st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2nd ‘𝑊):(0...(#‘(1st ‘𝑊)))⟶(Vtx‘𝐺) ∧ ((2nd ‘𝑊)‘0) = ((2nd ‘𝑊)‘(#‘(1st ‘𝑊)))) ∧ (#‘(1st ‘𝑊)) = 𝑁) → (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)) |
17 | 5, 16 | syl 17 | 1 ⊢ (𝑊 ∈ 𝐶 → (2nd ‘𝑊) ∈ Word (Vtx‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 {crab 2916 〈cop 4183 ↦ cmpt 4729 dom cdm 5114 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1st c1st 7166 2nd c2nd 7167 0cc0 9936 1c1 9937 + caddc 9939 ℕ0cn0 11292 ℤcz 11377 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 substr csubstr 13295 Vtxcvtx 25874 iEdgciedg 25875 ClWalkscclwlks 26666 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ifp 1013 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-wlks 26495 df-clwlks 26667 |
This theorem is referenced by: clwlksf1clwwlklem 26968 |
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