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Theorem clwwlkextfrlem1 27208

Description: Lemma for numclwwlk2lem1 27235. (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Revised by AV, 27-May-2021.)

**Assertion**
Ref	Expression
clwwlkextfrlem1	⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑊) ≠ (𝑊‘0))) → (((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) ≠ (𝑊‘0)))

**Proof of Theorem clwwlkextfrlem1**
Step	Hyp	Ref	Expression
1		wwlknbp2 26752	. . . 4 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)))
2		simpll 790	. . . . . . 7 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (( lastS ‘𝑊) ≠ (𝑊‘0) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)))) → 𝑊 ∈ Word (Vtx‘𝐺))
3		s1cl 13382	. . . . . . . . . 10 ⊢ (𝑋 ∈ (Vtx‘𝐺) → ⟨“𝑋”⟩ ∈ Word (Vtx‘𝐺))
4	3	adantl 482	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) → ⟨“𝑋”⟩ ∈ Word (Vtx‘𝐺))
5	4	adantl 482	. . . . . . . 8 ⊢ ((( lastS ‘𝑊) ≠ (𝑊‘0) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺))) → ⟨“𝑋”⟩ ∈ Word (Vtx‘𝐺))
6	5	adantl 482	. . . . . . 7 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (( lastS ‘𝑊) ≠ (𝑊‘0) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)))) → ⟨“𝑋”⟩ ∈ Word (Vtx‘𝐺))
7		nn0p1gt0 11322	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ₀ → 0 < (𝑁 + 1))
8	7	adantr 481	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) → 0 < (𝑁 + 1))
9	8	adantl 482	. . . . . . . . 9 ⊢ ((( lastS ‘𝑊) ≠ (𝑊‘0) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺))) → 0 < (𝑁 + 1))
10	9	adantl 482	. . . . . . . 8 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (( lastS ‘𝑊) ≠ (𝑊‘0) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)))) → 0 < (𝑁 + 1))
11		breq2 4657	. . . . . . . . . 10 ⊢ ((#‘𝑊) = (𝑁 + 1) → (0 < (#‘𝑊) ↔ 0 < (𝑁 + 1)))
12	11	adantl 482	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → (0 < (#‘𝑊) ↔ 0 < (𝑁 + 1)))
13	12	adantr 481	. . . . . . . 8 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (( lastS ‘𝑊) ≠ (𝑊‘0) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)))) → (0 < (#‘𝑊) ↔ 0 < (𝑁 + 1)))
14	10, 13	mpbird 247	. . . . . . 7 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (( lastS ‘𝑊) ≠ (𝑊‘0) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)))) → 0 < (#‘𝑊))
15		ccatfv0 13367	. . . . . . 7 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ⟨“𝑋”⟩ ∈ Word (Vtx‘𝐺) ∧ 0 < (#‘𝑊)) → ((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0))
16	2, 6, 14, 15	syl3anc 1326	. . . . . 6 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (( lastS ‘𝑊) ≠ (𝑊‘0) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)))) → ((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0))
17		oveq1 6657	. . . . . . . . . . . . . . . 16 ⊢ ((#‘𝑊) = (𝑁 + 1) → ((#‘𝑊) − 1) = ((𝑁 + 1) − 1))
18	17	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → ((#‘𝑊) − 1) = ((𝑁 + 1) − 1))
19	18	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺))) → ((#‘𝑊) − 1) = ((𝑁 + 1) − 1))
20		nn0cn 11302	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
21		pncan1 10454	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ ℂ → ((𝑁 + 1) − 1) = 𝑁)
22	20, 21	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑁 ∈ ℕ₀ → ((𝑁 + 1) − 1) = 𝑁)
23	22	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) → ((𝑁 + 1) − 1) = 𝑁)
24	23	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺))) → ((𝑁 + 1) − 1) = 𝑁)
25	19, 24	eqtr2d 2657	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺))) → 𝑁 = ((#‘𝑊) − 1))
26	25	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺))) → ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) = ((𝑊 ++ ⟨“𝑋”⟩)‘((#‘𝑊) − 1)))
27		simpll 790	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺))) → 𝑊 ∈ Word (Vtx‘𝐺))
28	4	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺))) → ⟨“𝑋”⟩ ∈ Word (Vtx‘𝐺))
29	8	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺))) → 0 < (𝑁 + 1))
30	12	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺))) → (0 < (#‘𝑊) ↔ 0 < (𝑁 + 1)))
31	29, 30	mpbird 247	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺))) → 0 < (#‘𝑊))
32		hashneq0 13155	. . . . . . . . . . . . . . . . 17 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (0 < (#‘𝑊) ↔ 𝑊 ≠ ∅))
33	32	bicomd 213	. . . . . . . . . . . . . . . 16 ⊢ (𝑊 ∈ Word (Vtx‘𝐺) → (𝑊 ≠ ∅ ↔ 0 < (#‘𝑊)))
34	33	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → (𝑊 ≠ ∅ ↔ 0 < (#‘𝑊)))
35	34	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺))) → (𝑊 ≠ ∅ ↔ 0 < (#‘𝑊)))
36	31, 35	mpbird 247	. . . . . . . . . . . . 13 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺))) → 𝑊 ≠ ∅)
37		ccatval1lsw 13368	. . . . . . . . . . . . 13 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ ⟨“𝑋”⟩ ∈ Word (Vtx‘𝐺) ∧ 𝑊 ≠ ∅) → ((𝑊 ++ ⟨“𝑋”⟩)‘((#‘𝑊) − 1)) = ( lastS ‘𝑊))
38	27, 28, 36, 37	syl3anc 1326	. . . . . . . . . . . 12 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺))) → ((𝑊 ++ ⟨“𝑋”⟩)‘((#‘𝑊) − 1)) = ( lastS ‘𝑊))
39	26, 38	eqtr2d 2657	. . . . . . . . . . 11 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺))) → ( lastS ‘𝑊) = ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁))
40	39	neeq1d 2853	. . . . . . . . . 10 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺))) → (( lastS ‘𝑊) ≠ (𝑊‘0) ↔ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) ≠ (𝑊‘0)))
41	40	biimpd 219	. . . . . . . . 9 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺))) → (( lastS ‘𝑊) ≠ (𝑊‘0) → ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) ≠ (𝑊‘0)))
42	41	ex 450	. . . . . . . 8 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) → (( lastS ‘𝑊) ≠ (𝑊‘0) → ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) ≠ (𝑊‘0))))
43	42	com23 86	. . . . . . 7 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → (( lastS ‘𝑊) ≠ (𝑊‘0) → ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) → ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) ≠ (𝑊‘0))))
44	43	imp32 449	. . . . . 6 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (( lastS ‘𝑊) ≠ (𝑊‘0) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)))) → ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) ≠ (𝑊‘0))
45	16, 44	jca 554	. . . . 5 ⊢ (((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) ∧ (( lastS ‘𝑊) ≠ (𝑊‘0) ∧ (𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)))) → (((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) ≠ (𝑊‘0)))
46	45	exp32 631	. . . 4 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)) → (( lastS ‘𝑊) ≠ (𝑊‘0) → ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) → (((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) ≠ (𝑊‘0)))))
47	1, 46	syl 17	. . 3 ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (( lastS ‘𝑊) ≠ (𝑊‘0) → ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) → (((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) ≠ (𝑊‘0)))))
48	47	imp 445	. 2 ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑊) ≠ (𝑊‘0)) → ((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) → (((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) ≠ (𝑊‘0))))
49	48	impcom 446	1 ⊢ (((𝑁 ∈ ℕ₀ ∧ 𝑋 ∈ (Vtx‘𝐺)) ∧ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ( lastS ‘𝑊) ≠ (𝑊‘0))) → (((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) ≠ (𝑊‘0)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∅c0 3915 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 < clt 10074 − cmin 10266 ℕ₀cn0 11292 #chash 13117 Word cword 13291 lastS clsw 13292 ++ cconcat 13293 ⟨“cs1 13294 Vtxcvtx 25874 WWalksN cwwlksn 26718

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-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-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-lsw 13300 df-concat 13301 df-s1 13302 df-wwlks 26722 df-wwlksn 26723

This theorem is referenced by: numclwwlk2lem1 27235

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