Theorem usg2cwwk2dif 26348

Description: If a word represents a closed walk of length at least 2 in a undirected simple graph, the first two symbols of the word must be different. (Contributed by Alexander van der Vekens, 17-Jun-2018.)

Assertion

Ref	Expression
usg2cwwk2dif	⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ (ℤ≥‘2) ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑊‘1) ≠ (𝑊‘0))

Proof of Theorem usg2cwwk2dif

Dummy variable 𝑖 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwwlknimp 26304	. . . 4 ⊢ (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸))
2		simpr 476	. . . . . 6 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸) ∧ 𝑁 ∈ (ℤ≥‘2)) ∧ 𝑉 USGrph 𝐸) → 𝑉 USGrph 𝐸)
3		uz2m1nn 11639	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 − 1) ∈ ℕ)
4		lbfzo0 12375	. . . . . . . . . . . 12 ⊢ (0 ∈ (0..^(𝑁 − 1)) ↔ (𝑁 − 1) ∈ ℕ)
5	3, 4	sylibr 223	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ≥‘2) → 0 ∈ (0..^(𝑁 − 1)))
6		fveq2 6103	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 0 → (𝑊‘𝑖) = (𝑊‘0))
7	6	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑖 = 0) → (𝑊‘𝑖) = (𝑊‘0))
8		oveq1 6556	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 = 0 → (𝑖 + 1) = (0 + 1))
9	8	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑖 = 0) → (𝑖 + 1) = (0 + 1))
10		0p1e1 11009	. . . . . . . . . . . . . . 15 ⊢ (0 + 1) = 1
11	9, 10	syl6eq 2660	. . . . . . . . . . . . . 14 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑖 = 0) → (𝑖 + 1) = 1)
12	11	fveq2d 6107	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑖 = 0) → (𝑊‘(𝑖 + 1)) = (𝑊‘1))
13	7, 12	preq12d 4220	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑖 = 0) → {(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} = {(𝑊‘0), (𝑊‘1)})
14	13	eleq1d 2672	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑖 = 0) → ({(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ↔ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸))
15	5, 14	rspcdv 3285	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ≥‘2) → (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 → {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸))
16	15	com12 32	. . . . . . . . 9 ⊢ (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 → (𝑁 ∈ (ℤ≥‘2) → {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸))
17	16	3ad2ant2 1076	. . . . . . . 8 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸) → (𝑁 ∈ (ℤ≥‘2) → {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸))
18	17	imp 444	. . . . . . 7 ⊢ ((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸) ∧ 𝑁 ∈ (ℤ≥‘2)) → {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸)
19	18	adantr 480	. . . . . 6 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸) ∧ 𝑁 ∈ (ℤ≥‘2)) ∧ 𝑉 USGrph 𝐸) → {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸)
20		usgraedgrn 25910	. . . . . . 7 ⊢ ((𝑉 USGrph 𝐸 ∧ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸) → (𝑊‘0) ≠ (𝑊‘1))
21	20	necomd 2837	. . . . . 6 ⊢ ((𝑉 USGrph 𝐸 ∧ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸) → (𝑊‘1) ≠ (𝑊‘0))
22	2, 19, 21	syl2anc 691	. . . . 5 ⊢ (((((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸) ∧ 𝑁 ∈ (ℤ≥‘2)) ∧ 𝑉 USGrph 𝐸) → (𝑊‘1) ≠ (𝑊‘0))
23	22	exp31 628	. . . 4 ⊢ (((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸) → (𝑁 ∈ (ℤ≥‘2) → (𝑉 USGrph 𝐸 → (𝑊‘1) ≠ (𝑊‘0))))
24	1, 23	syl 17	. . 3 ⊢ (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑁 ∈ (ℤ≥‘2) → (𝑉 USGrph 𝐸 → (𝑊‘1) ≠ (𝑊‘0))))
25	24	com13 86	. 2 ⊢ (𝑉 USGrph 𝐸 → (𝑁 ∈ (ℤ≥‘2) → (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑊‘1) ≠ (𝑊‘0))))
26	25	3imp 1249	1 ⊢ ((𝑉 USGrph 𝐸 ∧ 𝑁 ∈ (ℤ≥‘2) ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑊‘1) ≠ (𝑊‘0))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {cpr 4127   class class class wbr 4583  ran crn 5039  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818   − cmin 10145  ℕcn 10897  2c2 10947  ℤ_≥cuz 11563  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   USGrph cusg 25859   ClWWalksN cclwwlkn 26277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-clwwlk 26279  df-clwwlkn 26280

This theorem is referenced by:  usg2cwwkdifex  26349