Theorem clwlkclwwlkf1lem2 30034

Description: Lemma 2 for clwlkclwwlkf1 30039. (Contributed by AV, 24-May-2022.) (Revised by AV, 30-Oct-2022.)

Hypotheses

Ref	Expression
clwlkclwwlkf.c	⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}
clwlkclwwlkf.a	⊢ 𝐴 = (1st ‘𝑈)
clwlkclwwlkf.b	⊢ 𝐵 = (2nd ‘𝑈)
clwlkclwwlkf.d	⊢ 𝐷 = (1st ‘𝑊)
clwlkclwwlkf.e	⊢ 𝐸 = (2nd ‘𝑊)

Assertion

Ref	Expression
clwlkclwwlkf1lem2	⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖)))

Distinct variable groups:   𝑖,𝐺   𝑤,𝐺   𝑤,𝐴   𝑤,𝑈   𝐴,𝑖   𝐵,𝑖   𝐷,𝑖   𝑤,𝐷   𝑖,𝐸   𝑤,𝑊

Allowed substitution hints:   𝐵(𝑤)   𝐶(𝑤,𝑖)   𝑈(𝑖)   𝐸(𝑤)   𝑊(𝑖)

Proof of Theorem clwlkclwwlkf1lem2

Step	Hyp	Ref	Expression
1		clwlkclwwlkf.c	. . . . 5 ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))}
2		clwlkclwwlkf.a	. . . . 5 ⊢ 𝐴 = (1st ‘𝑈)
3		clwlkclwwlkf.b	. . . . 5 ⊢ 𝐵 = (2nd ‘𝑈)
4	1, 2, 3	clwlkclwwlkflem 30033	. . . 4 ⊢ (𝑈 ∈ 𝐶 → (𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ))
5		clwlkclwwlkf.d	. . . . 5 ⊢ 𝐷 = (1st ‘𝑊)
6		clwlkclwwlkf.e	. . . . 5 ⊢ 𝐸 = (2nd ‘𝑊)
7	1, 5, 6	clwlkclwwlkflem 30033	. . . 4 ⊢ (𝑊 ∈ 𝐶 → (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ))
8	4, 7	anim12i 613	. . 3 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)))
9		eqid 2735	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
10	9	wlkpwrd 29650	. . . . . 6 ⊢ (𝐴(Walks‘𝐺)𝐵 → 𝐵 ∈ Word (Vtx‘𝐺))
11	10	3ad2ant1 1132	. . . . 5 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → 𝐵 ∈ Word (Vtx‘𝐺))
12	9	wlkpwrd 29650	. . . . . 6 ⊢ (𝐷(Walks‘𝐺)𝐸 → 𝐸 ∈ Word (Vtx‘𝐺))
13	12	3ad2ant1 1132	. . . . 5 ⊢ ((𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ) → 𝐸 ∈ Word (Vtx‘𝐺))
14	11, 13	anim12i 613	. . . 4 ⊢ (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) → (𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝐸 ∈ Word (Vtx‘𝐺)))
15		nnnn0 12531	. . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ ℕ0)
16	15	3ad2ant3 1134	. . . . 5 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → (♯‘𝐴) ∈ ℕ0)
17		nnnn0 12531	. . . . . 6 ⊢ ((♯‘𝐷) ∈ ℕ → (♯‘𝐷) ∈ ℕ0)
18	17	3ad2ant3 1134	. . . . 5 ⊢ ((𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ) → (♯‘𝐷) ∈ ℕ0)
19	16, 18	anim12i 613	. . . 4 ⊢ (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) → ((♯‘𝐴) ∈ ℕ0 ∧ (♯‘𝐷) ∈ ℕ0))
20		wlklenvp1 29651	. . . . . . . 8 ⊢ (𝐴(Walks‘𝐺)𝐵 → (♯‘𝐵) = ((♯‘𝐴) + 1))
21		nnre 12271	. . . . . . . . . 10 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ ℝ)
22	21	lep1d 12197	. . . . . . . . 9 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ≤ ((♯‘𝐴) + 1))
23		breq2 5152	. . . . . . . . 9 ⊢ ((♯‘𝐵) = ((♯‘𝐴) + 1) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ (♯‘𝐴) ≤ ((♯‘𝐴) + 1)))
24	22, 23	imbitrrid 246	. . . . . . . 8 ⊢ ((♯‘𝐵) = ((♯‘𝐴) + 1) → ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ≤ (♯‘𝐵)))
25	20, 24	syl 17	. . . . . . 7 ⊢ (𝐴(Walks‘𝐺)𝐵 → ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ≤ (♯‘𝐵)))
26	25	a1d 25	. . . . . 6 ⊢ (𝐴(Walks‘𝐺)𝐵 → ((𝐵‘0) = (𝐵‘(♯‘𝐴)) → ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ≤ (♯‘𝐵))))
27	26	3imp 1110	. . . . 5 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → (♯‘𝐴) ≤ (♯‘𝐵))
28		wlklenvp1 29651	. . . . . . . 8 ⊢ (𝐷(Walks‘𝐺)𝐸 → (♯‘𝐸) = ((♯‘𝐷) + 1))
29		nnre 12271	. . . . . . . . . 10 ⊢ ((♯‘𝐷) ∈ ℕ → (♯‘𝐷) ∈ ℝ)
30	29	lep1d 12197	. . . . . . . . 9 ⊢ ((♯‘𝐷) ∈ ℕ → (♯‘𝐷) ≤ ((♯‘𝐷) + 1))
31		breq2 5152	. . . . . . . . 9 ⊢ ((♯‘𝐸) = ((♯‘𝐷) + 1) → ((♯‘𝐷) ≤ (♯‘𝐸) ↔ (♯‘𝐷) ≤ ((♯‘𝐷) + 1)))
32	30, 31	imbitrrid 246	. . . . . . . 8 ⊢ ((♯‘𝐸) = ((♯‘𝐷) + 1) → ((♯‘𝐷) ∈ ℕ → (♯‘𝐷) ≤ (♯‘𝐸)))
33	28, 32	syl 17	. . . . . . 7 ⊢ (𝐷(Walks‘𝐺)𝐸 → ((♯‘𝐷) ∈ ℕ → (♯‘𝐷) ≤ (♯‘𝐸)))
34	33	a1d 25	. . . . . 6 ⊢ (𝐷(Walks‘𝐺)𝐸 → ((𝐸‘0) = (𝐸‘(♯‘𝐷)) → ((♯‘𝐷) ∈ ℕ → (♯‘𝐷) ≤ (♯‘𝐸))))
35	34	3imp 1110	. . . . 5 ⊢ ((𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ) → (♯‘𝐷) ≤ (♯‘𝐸))
36	27, 35	anim12i 613	. . . 4 ⊢ (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) → ((♯‘𝐴) ≤ (♯‘𝐵) ∧ (♯‘𝐷) ≤ (♯‘𝐸)))
37	14, 19, 36	3jca 1127	. . 3 ⊢ (((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) ∧ (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ)) → ((𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝐸 ∈ Word (Vtx‘𝐺)) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ (♯‘𝐷) ∈ ℕ0) ∧ ((♯‘𝐴) ≤ (♯‘𝐵) ∧ (♯‘𝐷) ≤ (♯‘𝐸))))
38		pfxeq 14731	. . 3 ⊢ (((𝐵 ∈ Word (Vtx‘𝐺) ∧ 𝐸 ∈ Word (Vtx‘𝐺)) ∧ ((♯‘𝐴) ∈ ℕ0 ∧ (♯‘𝐷) ∈ ℕ0) ∧ ((♯‘𝐴) ≤ (♯‘𝐵) ∧ (♯‘𝐷) ≤ (♯‘𝐸))) → ((𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷)) ↔ ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖))))
39	8, 37, 38	3syl 18	. 2 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → ((𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷)) ↔ ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖))))
40	39	biimp3a 1468	1 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ∀wral 3059  {crab 3433   class class class wbr 5148  ‘cfv 6563  (class class class)co 7431  1^st c1st 8011  2^nd c2nd 8012  0cc0 11153  1c1 11154   + caddc 11156   ≤ cle 11294  ℕcn 12264  ℕ₀cn0 12524  ..^cfzo 13691  ♯chash 14366  Word cword 14549   prefix cpfx 14705  Vtxcvtx 29028  Walkscwlks 29629  ClWalkscclwlks 29803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ifp 1063  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-substr 14676  df-pfx 14706  df-wlks 29632  df-clwlks 29804

This theorem is referenced by:  clwlkclwwlkf1lem3  30035  clwlkclwwlkf1  30039