Theorem clwlkclwwlkf1lem2 30483

Description: Lemma 2 for clwlkclwwlkf1 30488. (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 30482	. . . 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 30482	. . . 4 ⊢ (𝑊 ∈ 𝐶 → (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ))
8	4, 7	anim12i 625	. . 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 2762	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
10	9	wlkpwrd 30085	. . . . . 6 ⊢ (𝐴(Walks‘𝐺)𝐵 → 𝐵 ∈ Word (Vtx‘𝐺))
11	10	3ad2ant1 1151	. . . . 5 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → 𝐵 ∈ Word (Vtx‘𝐺))
12	9	wlkpwrd 30085	. . . . . 6 ⊢ (𝐷(Walks‘𝐺)𝐸 → 𝐸 ∈ Word (Vtx‘𝐺))
13	12	3ad2ant1 1151	. . . . 5 ⊢ ((𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ) → 𝐸 ∈ Word (Vtx‘𝐺))
14	11, 13	anim12i 625	. . . 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 12539	. . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ ℕ0)
16	15	3ad2ant3 1153	. . . . 5 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → (♯‘𝐴) ∈ ℕ0)
17		nnnn0 12539	. . . . . 6 ⊢ ((♯‘𝐷) ∈ ℕ → (♯‘𝐷) ∈ ℕ0)
18	17	3ad2ant3 1153	. . . . 5 ⊢ ((𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ) → (♯‘𝐷) ∈ ℕ0)
19	16, 18	anim12i 625	. . . 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 30086	. . . . . . . 8 ⊢ (𝐴(Walks‘𝐺)𝐵 → (♯‘𝐵) = ((♯‘𝐴) + 1))
21		nnre 12268	. . . . . . . . . 10 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ ℝ)
22	21	lep1d 12174	. . . . . . . . 9 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ≤ ((♯‘𝐴) + 1))
23		breq2 5111	. . . . . . . . 9 ⊢ ((♯‘𝐵) = ((♯‘𝐴) + 1) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ (♯‘𝐴) ≤ ((♯‘𝐴) + 1)))
24	22, 23	imbitrrid 249	. . . . . . . 8 ⊢ ((♯‘𝐵) = ((♯‘𝐴) + 1) → ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ≤ (♯‘𝐵)))
25	20, 24	syl 18	. . . . . . 7 ⊢ (𝐴(Walks‘𝐺)𝐵 → ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ≤ (♯‘𝐵)))
26	25	a1d 26	. . . . . 6 ⊢ (𝐴(Walks‘𝐺)𝐵 → ((𝐵‘0) = (𝐵‘(♯‘𝐴)) → ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ≤ (♯‘𝐵))))
27	26	3imp 1128	. . . . 5 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → (♯‘𝐴) ≤ (♯‘𝐵))
28		wlklenvp1 30086	. . . . . . . 8 ⊢ (𝐷(Walks‘𝐺)𝐸 → (♯‘𝐸) = ((♯‘𝐷) + 1))
29		nnre 12268	. . . . . . . . . 10 ⊢ ((♯‘𝐷) ∈ ℕ → (♯‘𝐷) ∈ ℝ)
30	29	lep1d 12174	. . . . . . . . 9 ⊢ ((♯‘𝐷) ∈ ℕ → (♯‘𝐷) ≤ ((♯‘𝐷) + 1))
31		breq2 5111	. . . . . . . . 9 ⊢ ((♯‘𝐸) = ((♯‘𝐷) + 1) → ((♯‘𝐷) ≤ (♯‘𝐸) ↔ (♯‘𝐷) ≤ ((♯‘𝐷) + 1)))
32	30, 31	imbitrrid 249	. . . . . . . 8 ⊢ ((♯‘𝐸) = ((♯‘𝐷) + 1) → ((♯‘𝐷) ∈ ℕ → (♯‘𝐷) ≤ (♯‘𝐸)))
33	28, 32	syl 18	. . . . . . 7 ⊢ (𝐷(Walks‘𝐺)𝐸 → ((♯‘𝐷) ∈ ℕ → (♯‘𝐷) ≤ (♯‘𝐸)))
34	33	a1d 26	. . . . . 6 ⊢ (𝐷(Walks‘𝐺)𝐸 → ((𝐸‘0) = (𝐸‘(♯‘𝐷)) → ((♯‘𝐷) ∈ ℕ → (♯‘𝐷) ≤ (♯‘𝐸))))
35	34	3imp 1128	. . . . 5 ⊢ ((𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ) → (♯‘𝐷) ≤ (♯‘𝐸))
36	27, 35	anim12i 625	. . . 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 1146	. . 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 14769	. . 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 19	. 2 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶) → ((𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷)) ↔ ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖))))
40	39	biimp3a 1498	1 ⊢ ((𝑈 ∈ 𝐶 ∧ 𝑊 ∈ 𝐶 ∧ (𝐵 prefix (♯‘𝐴)) = (𝐸 prefix (♯‘𝐷))) → ((♯‘𝐴) = (♯‘𝐷) ∧ ∀𝑖 ∈ (0..^(♯‘𝐴))(𝐵‘𝑖) = (𝐸‘𝑖)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078  {crab 3414   class class class wbr 5107  ‘cfv 6537  (class class class)co 7417  1^st c1st 7988  2^nd c2nd 7989  0cc0 11128  1c1 11129   + caddc 11131   ≤ cle 11272  ℕcn 12261  ℕ₀cn0 12532  ..^cfzo 13713  ♯chash 14398  Word cword 14582   prefix cpfx 14744  Vtxcvtx 29461  Walkscwlks 30064  ClWalkscclwlks 30244

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740  ax-cnex 11184  ax-resscn 11185  ax-1cn 11186  ax-icn 11187  ax-addcl 11188  ax-addrcl 11189  ax-mulcl 11190  ax-mulrcl 11191  ax-mulcom 11192  ax-addass 11193  ax-mulass 11194  ax-distr 11195  ax-i2m1 11196  ax-1ne0 11197  ax-1rid 11198  ax-rnegex 11199  ax-rrecex 11200  ax-cnre 11201  ax-pre-lttri 11202  ax-pre-lttrn 11203  ax-pre-ltadd 11204  ax-pre-mulgt0 11205

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7374  df-ov 7420  df-oprab 7421  df-mpo 7422  df-om 7867  df-1st 7990  df-2nd 7991  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-er 8700  df-map 8832  df-en 8957  df-dom 8958  df-sdom 8959  df-fin 8960  df-card 9948  df-pnf 11273  df-mnf 11274  df-xr 11275  df-ltxr 11276  df-le 11277  df-sub 11471  df-neg 11472  df-nn 12262  df-n0 12533  df-z 12620  df-uz 12892  df-fz 13566  df-fzo 13714  df-hash 14399  df-word 14583  df-substr 14713  df-pfx 14745  df-wlks 30067  df-clwlks 30245

This theorem is used by:  clwlkclwwlkf1lem3  30484  clwlkclwwlkf1  30488