Theorem clwlkclwwlkf1lem2 30024

Description: Lemma 2 for clwlkclwwlkf1 30029. (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 30023	. . . 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 30023	. . . 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 2737	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
10	9	wlkpwrd 29635	. . . . . 6 ⊢ (𝐴(Walks‘𝐺)𝐵 → 𝐵 ∈ Word (Vtx‘𝐺))
11	10	3ad2ant1 1134	. . . . 5 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → 𝐵 ∈ Word (Vtx‘𝐺))
12	9	wlkpwrd 29635	. . . . . 6 ⊢ (𝐷(Walks‘𝐺)𝐸 → 𝐸 ∈ Word (Vtx‘𝐺))
13	12	3ad2ant1 1134	. . . . 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 12533	. . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ ℕ0)
16	15	3ad2ant3 1136	. . . . 5 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → (♯‘𝐴) ∈ ℕ0)
17		nnnn0 12533	. . . . . 6 ⊢ ((♯‘𝐷) ∈ ℕ → (♯‘𝐷) ∈ ℕ0)
18	17	3ad2ant3 1136	. . . . 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 29636	. . . . . . . 8 ⊢ (𝐴(Walks‘𝐺)𝐵 → (♯‘𝐵) = ((♯‘𝐴) + 1))
21		nnre 12273	. . . . . . . . . 10 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ ℝ)
22	21	lep1d 12199	. . . . . . . . 9 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ≤ ((♯‘𝐴) + 1))
23		breq2 5147	. . . . . . . . 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 1111	. . . . 5 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → (♯‘𝐴) ≤ (♯‘𝐵))
28		wlklenvp1 29636	. . . . . . . 8 ⊢ (𝐷(Walks‘𝐺)𝐸 → (♯‘𝐸) = ((♯‘𝐷) + 1))
29		nnre 12273	. . . . . . . . . 10 ⊢ ((♯‘𝐷) ∈ ℕ → (♯‘𝐷) ∈ ℝ)
30	29	lep1d 12199	. . . . . . . . 9 ⊢ ((♯‘𝐷) ∈ ℕ → (♯‘𝐷) ≤ ((♯‘𝐷) + 1))
31		breq2 5147	. . . . . . . . 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 1111	. . . . 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 1129	. . 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 14734	. . 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 1471	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 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  {crab 3436   class class class wbr 5143  ‘cfv 6561  (class class class)co 7431  1^st c1st 8012  2^nd c2nd 8013  0cc0 11155  1c1 11156   + caddc 11158   ≤ cle 11296  ℕcn 12266  ℕ₀cn0 12526  ..^cfzo 13694  ♯chash 14369  Word cword 14552   prefix cpfx 14708  Vtxcvtx 29013  Walkscwlks 29614  ClWalkscclwlks 29790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-substr 14679  df-pfx 14709  df-wlks 29617  df-clwlks 29791

This theorem is referenced by:  clwlkclwwlkf1lem3  30025  clwlkclwwlkf1  30029