Theorem clwlkclwwlkf1lem2 30153

Description: Lemma 2 for clwlkclwwlkf1 30158. (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 30152	. . . 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 30152	. . . 4 ⊢ (𝑊 ∈ 𝐶 → (𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ))
8	4, 7	anim12i 622	. . 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 2761	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
10	9	wlkpwrd 29764	. . . . . 6 ⊢ (𝐴(Walks‘𝐺)𝐵 → 𝐵 ∈ Word (Vtx‘𝐺))
11	10	3ad2ant1 1145	. . . . 5 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → 𝐵 ∈ Word (Vtx‘𝐺))
12	9	wlkpwrd 29764	. . . . . 6 ⊢ (𝐷(Walks‘𝐺)𝐸 → 𝐸 ∈ Word (Vtx‘𝐺))
13	12	3ad2ant1 1145	. . . . 5 ⊢ ((𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ) → 𝐸 ∈ Word (Vtx‘𝐺))
14	11, 13	anim12i 622	. . . 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 12485	. . . . . 6 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ ℕ0)
16	15	3ad2ant3 1147	. . . . 5 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → (♯‘𝐴) ∈ ℕ0)
17		nnnn0 12485	. . . . . 6 ⊢ ((♯‘𝐷) ∈ ℕ → (♯‘𝐷) ∈ ℕ0)
18	17	3ad2ant3 1147	. . . . 5 ⊢ ((𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ) → (♯‘𝐷) ∈ ℕ0)
19	16, 18	anim12i 622	. . . 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 29765	. . . . . . . 8 ⊢ (𝐴(Walks‘𝐺)𝐵 → (♯‘𝐵) = ((♯‘𝐴) + 1))
21		nnre 12214	. . . . . . . . . 10 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ∈ ℝ)
22	21	lep1d 12120	. . . . . . . . 9 ⊢ ((♯‘𝐴) ∈ ℕ → (♯‘𝐴) ≤ ((♯‘𝐴) + 1))
23		breq2 5103	. . . . . . . . 9 ⊢ ((♯‘𝐵) = ((♯‘𝐴) + 1) → ((♯‘𝐴) ≤ (♯‘𝐵) ↔ (♯‘𝐴) ≤ ((♯‘𝐴) + 1)))
24	22, 23	imbitrrid 248	. . . . . . . 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 1122	. . . . 5 ⊢ ((𝐴(Walks‘𝐺)𝐵 ∧ (𝐵‘0) = (𝐵‘(♯‘𝐴)) ∧ (♯‘𝐴) ∈ ℕ) → (♯‘𝐴) ≤ (♯‘𝐵))
28		wlklenvp1 29765	. . . . . . . 8 ⊢ (𝐷(Walks‘𝐺)𝐸 → (♯‘𝐸) = ((♯‘𝐷) + 1))
29		nnre 12214	. . . . . . . . . 10 ⊢ ((♯‘𝐷) ∈ ℕ → (♯‘𝐷) ∈ ℝ)
30	29	lep1d 12120	. . . . . . . . 9 ⊢ ((♯‘𝐷) ∈ ℕ → (♯‘𝐷) ≤ ((♯‘𝐷) + 1))
31		breq2 5103	. . . . . . . . 9 ⊢ ((♯‘𝐸) = ((♯‘𝐷) + 1) → ((♯‘𝐷) ≤ (♯‘𝐸) ↔ (♯‘𝐷) ≤ ((♯‘𝐷) + 1)))
32	30, 31	imbitrrid 248	. . . . . . . 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 1122	. . . . 5 ⊢ ((𝐷(Walks‘𝐺)𝐸 ∧ (𝐸‘0) = (𝐸‘(♯‘𝐷)) ∧ (♯‘𝐷) ∈ ℕ) → (♯‘𝐷) ≤ (♯‘𝐸))
36	27, 35	anim12i 622	. . . 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 1140	. . 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 14706	. . 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 1489	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 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  {crab 3413   class class class wbr 5099  ‘cfv 6517  (class class class)co 7392  1^st c1st 7964  2^nd c2nd 7965  0cc0 11070  1c1 11071   + caddc 11073   ≤ cle 11214  ℕcn 12207  ℕ₀cn0 12478  ..^cfzo 13656  ♯chash 14340  Word cword 14523   prefix cpfx 14681  Vtxcvtx 29143  Walkscwlks 29743  ClWalkscclwlks 29916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714  ax-cnex 11126  ax-resscn 11127  ax-1cn 11128  ax-icn 11129  ax-addcl 11130  ax-addrcl 11131  ax-mulcl 11132  ax-mulrcl 11133  ax-mulcom 11134  ax-addass 11135  ax-mulass 11136  ax-distr 11137  ax-i2m1 11138  ax-1ne0 11139  ax-1rid 11140  ax-rnegex 11141  ax-rrecex 11142  ax-cnre 11143  ax-pre-lttri 11144  ax-pre-lttrn 11145  ax-pre-ltadd 11146  ax-pre-mulgt0 11147

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-ifp 1074  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-lim 6347  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-om 7843  df-1st 7966  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-rdg 8376  df-1o 8432  df-er 8673  df-map 8805  df-en 8924  df-dom 8925  df-sdom 8926  df-fin 8927  df-card 9894  df-pnf 11215  df-mnf 11216  df-xr 11217  df-ltxr 11218  df-le 11219  df-sub 11413  df-neg 11414  df-nn 12208  df-n0 12479  df-z 12566  df-uz 12837  df-fz 13510  df-fzo 13657  df-hash 14341  df-word 14524  df-substr 14652  df-pfx 14682  df-wlks 29746  df-clwlks 29917

This theorem is referenced by:  clwlkclwwlkf1lem3  30154  clwlkclwwlkf1  30158