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Theorem clwlksf1clwwlklem0 26964

Description: Lemma 1 for clwlksf1clwwlklem 26968. (Contributed by AV, 3-May-2021.)

**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
clwlksf1clwwlklem0	⊢ (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))

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

Proof of Theorem clwlksf1clwwlklem0 Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwlksfclwwlk.1	. . . . . 6 ⊢ 𝐴 = (1^st ‘𝑐)
2		fveq2 6191	. . . . . 6 ⊢ (𝑐 = 𝑊 → (1^st ‘𝑐) = (1^st ‘𝑊))
3	1, 2	syl5eq 2668	. . . . 5 ⊢ (𝑐 = 𝑊 → 𝐴 = (1^st ‘𝑊))
4	3	fveq2d 6195	. . . 4 ⊢ (𝑐 = 𝑊 → (#‘𝐴) = (#‘(1^st ‘𝑊)))
5	4	eqeq1d 2624	. . 3 ⊢ (𝑐 = 𝑊 → ((#‘𝐴) = 𝑁 ↔ (#‘(1^st ‘𝑊)) = 𝑁))
6		clwlksfclwwlk.c	. . 3 ⊢ 𝐶 = {𝑐 ∈ (ClWalks‘𝐺) ∣ (#‘𝐴) = 𝑁}
7	5, 6	elrab2 3366	. 2 ⊢ (𝑊 ∈ 𝐶 ↔ (𝑊 ∈ (ClWalks‘𝐺) ∧ (#‘(1^st ‘𝑊)) = 𝑁))
8		eqid 2622	. . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
9		eqid 2622	. . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
10		eqid 2622	. . . . 5 ⊢ (1^st ‘𝑊) = (1^st ‘𝑊)
11		eqid 2622	. . . . 5 ⊢ (2^nd ‘𝑊) = (2^nd ‘𝑊)
12	8, 9, 10, 11	clwlkcompim 26676	. . . 4 ⊢ (𝑊 ∈ (ClWalks‘𝐺) → (((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(#‘(1^st ‘𝑊)))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘(1^st ‘𝑊)))if-(((2^nd ‘𝑊)‘𝑖) = ((2^nd ‘𝑊)‘(𝑖 + 1)), ((iEdg‘𝐺)‘((1^st ‘𝑊)‘𝑖)) = {((2^nd ‘𝑊)‘𝑖)}, {((2^nd ‘𝑊)‘𝑖), ((2^nd ‘𝑊)‘(𝑖 + 1))} ⊆ ((iEdg‘𝐺)‘((1^st ‘𝑊)‘𝑖))) ∧ ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘(#‘(1^st ‘𝑊))))))
13		simpr 477	. . . . . 6 ⊢ ((∀𝑖 ∈ (0..^(#‘(1^st ‘𝑊)))if-(((2^nd ‘𝑊)‘𝑖) = ((2^nd ‘𝑊)‘(𝑖 + 1)), ((iEdg‘𝐺)‘((1^st ‘𝑊)‘𝑖)) = {((2^nd ‘𝑊)‘𝑖)}, {((2^nd ‘𝑊)‘𝑖), ((2^nd ‘𝑊)‘(𝑖 + 1))} ⊆ ((iEdg‘𝐺)‘((1^st ‘𝑊)‘𝑖))) ∧ ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘(#‘(1^st ‘𝑊)))) → ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘(#‘(1^st ‘𝑊))))
14	13	anim2i 593	. . . . 5 ⊢ ((((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(#‘(1^st ‘𝑊)))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘(1^st ‘𝑊)))if-(((2^nd ‘𝑊)‘𝑖) = ((2^nd ‘𝑊)‘(𝑖 + 1)), ((iEdg‘𝐺)‘((1^st ‘𝑊)‘𝑖)) = {((2^nd ‘𝑊)‘𝑖)}, {((2^nd ‘𝑊)‘𝑖), ((2^nd ‘𝑊)‘(𝑖 + 1))} ⊆ ((iEdg‘𝐺)‘((1^st ‘𝑊)‘𝑖))) ∧ ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘(#‘(1^st ‘𝑊))))) → (((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)))))
15		df-3an 1039	. . . . 5 ⊢ (((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) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(#‘(1^st ‘𝑊)))⟶(Vtx‘𝐺)) ∧ ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘(#‘(1^st ‘𝑊)))))
16	14, 15	sylibr 224	. . . 4 ⊢ ((((1^st ‘𝑊) ∈ Word dom (iEdg‘𝐺) ∧ (2^nd ‘𝑊):(0...(#‘(1^st ‘𝑊)))⟶(Vtx‘𝐺)) ∧ (∀𝑖 ∈ (0..^(#‘(1^st ‘𝑊)))if-(((2^nd ‘𝑊)‘𝑖) = ((2^nd ‘𝑊)‘(𝑖 + 1)), ((iEdg‘𝐺)‘((1^st ‘𝑊)‘𝑖)) = {((2^nd ‘𝑊)‘𝑖)}, {((2^nd ‘𝑊)‘𝑖), ((2^nd ‘𝑊)‘(𝑖 + 1))} ⊆ ((iEdg‘𝐺)‘((1^st ‘𝑊)‘𝑖))) ∧ ((2^nd ‘𝑊)‘0) = ((2^nd ‘𝑊)‘(#‘(1^st ‘𝑊))))) → ((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)))))
17	12, 16	syl 17	. . 3 ⊢ (𝑊 ∈ (ClWalks‘𝐺) → ((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)))))
18	17	anim1i 592	. 2 ⊢ ((𝑊 ∈ (ClWalks‘𝐺) ∧ (#‘(1^st ‘𝑊)) = 𝑁) → (((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))
19	7, 18	sylbi 207	1 ⊢ (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))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 if-wif 1012 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 ⊆ wss 3574 {csn 4177 {cpr 4179 ⟨cop 4183 ↦ cmpt 4729 dom cdm 5114 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1^st c1st 7166 2^nd c2nd 7167 0cc0 9936 1c1 9937 + caddc 9939 ...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-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: clwlksf1clwwlklem1 26965 clwlksf1clwwlklem2 26966 clwlksf1clwwlklem3 26967

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