Theorem swrdccatidOLD 13806

Description: Obsolete proof of pfxccatid 13805 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
swrdccatidOLD	⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝐴)) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = 𝐴)

Proof of Theorem swrdccatidOLD

Step	Hyp	Ref	Expression
1		3simpa 1179	. . 3 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝐴)) → (𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉))
2		lencl 13550	. . . . 5 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℕ0)
3		lencl 13550	. . . . . 6 ⊢ (𝐵 ∈ Word 𝑉 → (♯‘𝐵) ∈ ℕ0)
4		simplr 786	. . . . . . . . 9 ⊢ ((((♯‘𝐵) ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 = (♯‘𝐴)) → (♯‘𝐴) ∈ ℕ0)
5		eleq1 2865	. . . . . . . . . 10 ⊢ (𝑁 = (♯‘𝐴) → (𝑁 ∈ ℕ0 ↔ (♯‘𝐴) ∈ ℕ0))
6	5	adantl 474	. . . . . . . . 9 ⊢ ((((♯‘𝐵) ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 = (♯‘𝐴)) → (𝑁 ∈ ℕ0 ↔ (♯‘𝐴) ∈ ℕ0))
7	4, 6	mpbird 249	. . . . . . . 8 ⊢ ((((♯‘𝐵) ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 = (♯‘𝐴)) → 𝑁 ∈ ℕ0)
8		nn0addcl 11614	. . . . . . . . . 10 ⊢ (((♯‘𝐴) ∈ ℕ0 ∧ (♯‘𝐵) ∈ ℕ0) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℕ0)
9	8	ancoms 451	. . . . . . . . 9 ⊢ (((♯‘𝐵) ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℕ0)
10	9	adantr 473	. . . . . . . 8 ⊢ ((((♯‘𝐵) ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 = (♯‘𝐴)) → ((♯‘𝐴) + (♯‘𝐵)) ∈ ℕ0)
11		nn0re 11587	. . . . . . . . . . . . 13 ⊢ ((♯‘𝐴) ∈ ℕ0 → (♯‘𝐴) ∈ ℝ)
12	11	anim1i 609	. . . . . . . . . . . 12 ⊢ (((♯‘𝐴) ∈ ℕ0 ∧ (♯‘𝐵) ∈ ℕ0) → ((♯‘𝐴) ∈ ℝ ∧ (♯‘𝐵) ∈ ℕ0))
13	12	ancoms 451	. . . . . . . . . . 11 ⊢ (((♯‘𝐵) ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) → ((♯‘𝐴) ∈ ℝ ∧ (♯‘𝐵) ∈ ℕ0))
14		nn0addge1 11625	. . . . . . . . . . 11 ⊢ (((♯‘𝐴) ∈ ℝ ∧ (♯‘𝐵) ∈ ℕ0) → (♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝐵)))
15	13, 14	syl 17	. . . . . . . . . 10 ⊢ (((♯‘𝐵) ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) → (♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝐵)))
16	15	adantr 473	. . . . . . . . 9 ⊢ ((((♯‘𝐵) ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 = (♯‘𝐴)) → (♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝐵)))
17		breq1 4845	. . . . . . . . . 10 ⊢ (𝑁 = (♯‘𝐴) → (𝑁 ≤ ((♯‘𝐴) + (♯‘𝐵)) ↔ (♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝐵))))
18	17	adantl 474	. . . . . . . . 9 ⊢ ((((♯‘𝐵) ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 = (♯‘𝐴)) → (𝑁 ≤ ((♯‘𝐴) + (♯‘𝐵)) ↔ (♯‘𝐴) ≤ ((♯‘𝐴) + (♯‘𝐵))))
19	16, 18	mpbird 249	. . . . . . . 8 ⊢ ((((♯‘𝐵) ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 = (♯‘𝐴)) → 𝑁 ≤ ((♯‘𝐴) + (♯‘𝐵)))
20		elfz2nn0 12682	. . . . . . . 8 ⊢ (𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))) ↔ (𝑁 ∈ ℕ0 ∧ ((♯‘𝐴) + (♯‘𝐵)) ∈ ℕ0 ∧ 𝑁 ≤ ((♯‘𝐴) + (♯‘𝐵))))
21	7, 10, 19, 20	syl3anbrc 1444	. . . . . . 7 ⊢ ((((♯‘𝐵) ∈ ℕ0 ∧ (♯‘𝐴) ∈ ℕ0) ∧ 𝑁 = (♯‘𝐴)) → 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))
22	21	exp31 411	. . . . . 6 ⊢ ((♯‘𝐵) ∈ ℕ0 → ((♯‘𝐴) ∈ ℕ0 → (𝑁 = (♯‘𝐴) → 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))))
23	3, 22	syl 17	. . . . 5 ⊢ (𝐵 ∈ Word 𝑉 → ((♯‘𝐴) ∈ ℕ0 → (𝑁 = (♯‘𝐴) → 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))))
24	2, 23	syl5com 31	. . . 4 ⊢ (𝐴 ∈ Word 𝑉 → (𝐵 ∈ Word 𝑉 → (𝑁 = (♯‘𝐴) → 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))))
25	24	3imp 1138	. . 3 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝐴)) → 𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))))
26		eqid 2798	. . . 4 ⊢ (♯‘𝐴) = (♯‘𝐴)
27	26	swrdccat3aOLD 13801	. . 3 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉) → (𝑁 ∈ (0...((♯‘𝐴) + (♯‘𝐵))) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = if(𝑁 ≤ (♯‘𝐴), (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − (♯‘𝐴))⟩)))))
28	1, 25, 27	sylc 65	. 2 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝐴)) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = if(𝑁 ≤ (♯‘𝐴), (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − (♯‘𝐴))⟩))))
29	2, 11	syl 17	. . . . . 6 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ∈ ℝ)
30	29	leidd 10885	. . . . 5 ⊢ (𝐴 ∈ Word 𝑉 → (♯‘𝐴) ≤ (♯‘𝐴))
31	30	3ad2ant1 1164	. . . 4 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝐴)) → (♯‘𝐴) ≤ (♯‘𝐴))
32		breq1 4845	. . . . 5 ⊢ (𝑁 = (♯‘𝐴) → (𝑁 ≤ (♯‘𝐴) ↔ (♯‘𝐴) ≤ (♯‘𝐴)))
33	32	3ad2ant3 1166	. . . 4 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝐴)) → (𝑁 ≤ (♯‘𝐴) ↔ (♯‘𝐴) ≤ (♯‘𝐴)))
34	31, 33	mpbird 249	. . 3 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝐴)) → 𝑁 ≤ (♯‘𝐴))
35	34	iftrued 4284	. 2 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝐴)) → if(𝑁 ≤ (♯‘𝐴), (𝐴 substr ⟨0, 𝑁⟩), (𝐴 ++ (𝐵 substr ⟨0, (𝑁 − (♯‘𝐴))⟩))) = (𝐴 substr ⟨0, 𝑁⟩))
36		swrdidOLD 13676	. . . 4 ⊢ (𝐴 ∈ Word 𝑉 → (𝐴 substr ⟨0, (♯‘𝐴)⟩) = 𝐴)
37	36	3ad2ant1 1164	. . 3 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝐴)) → (𝐴 substr ⟨0, (♯‘𝐴)⟩) = 𝐴)
38		opeq2 4593	. . . . . 6 ⊢ (𝑁 = (♯‘𝐴) → ⟨0, 𝑁⟩ = ⟨0, (♯‘𝐴)⟩)
39	38	oveq2d 6893	. . . . 5 ⊢ (𝑁 = (♯‘𝐴) → (𝐴 substr ⟨0, 𝑁⟩) = (𝐴 substr ⟨0, (♯‘𝐴)⟩))
40	39	eqeq1d 2800	. . . 4 ⊢ (𝑁 = (♯‘𝐴) → ((𝐴 substr ⟨0, 𝑁⟩) = 𝐴 ↔ (𝐴 substr ⟨0, (♯‘𝐴)⟩) = 𝐴))
41	40	3ad2ant3 1166	. . 3 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝐴)) → ((𝐴 substr ⟨0, 𝑁⟩) = 𝐴 ↔ (𝐴 substr ⟨0, (♯‘𝐴)⟩) = 𝐴))
42	37, 41	mpbird 249	. 2 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝐴)) → (𝐴 substr ⟨0, 𝑁⟩) = 𝐴)
43	28, 35, 42	3eqtrd 2836	1 ⊢ ((𝐴 ∈ Word 𝑉 ∧ 𝐵 ∈ Word 𝑉 ∧ 𝑁 = (♯‘𝐴)) → ((𝐴 ++ 𝐵) substr ⟨0, 𝑁⟩) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   ∧ w3a 1108   = wceq 1653   ∈ wcel 2157  ifcif 4276  ⟨cop 4373   class class class wbr 4842  ‘cfv 6100  (class class class)co 6877  ℝcr 10222  0cc0 10223   + caddc 10226   ≤ cle 10363   − cmin 10555  ℕ₀cn0 11577  ...cfz 12577  ♯chash 13367  Word cword 13531   ++ cconcat 13587   substr csubstr 13661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-rep 4963  ax-sep 4974  ax-nul 4982  ax-pow 5034  ax-pr 5096  ax-un 7182  ax-cnex 10279  ax-resscn 10280  ax-1cn 10281  ax-icn 10282  ax-addcl 10283  ax-addrcl 10284  ax-mulcl 10285  ax-mulrcl 10286  ax-mulcom 10287  ax-addass 10288  ax-mulass 10289  ax-distr 10290  ax-i2m1 10291  ax-1ne0 10292  ax-1rid 10293  ax-rnegex 10294  ax-rrecex 10295  ax-cnre 10296  ax-pre-lttri 10297  ax-pre-lttrn 10298  ax-pre-ltadd 10299  ax-pre-mulgt0 10300

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-nel 3074  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3386  df-sbc 3633  df-csb 3728  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-pss 3784  df-nul 4115  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-tp 4372  df-op 4374  df-uni 4628  df-int 4667  df-iun 4711  df-br 4843  df-opab 4905  df-mpt 4922  df-tr 4945  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5897  df-ord 5943  df-on 5944  df-lim 5945  df-suc 5946  df-iota 6063  df-fun 6102  df-fn 6103  df-f 6104  df-f1 6105  df-fo 6106  df-f1o 6107  df-fv 6108  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-om 7299  df-1st 7400  df-2nd 7401  df-wrecs 7644  df-recs 7706  df-rdg 7744  df-1o 7798  df-oadd 7802  df-er 7981  df-en 8195  df-dom 8196  df-sdom 8197  df-fin 8198  df-card 9050  df-pnf 10364  df-mnf 10365  df-xr 10366  df-ltxr 10367  df-le 10368  df-sub 10557  df-neg 10558  df-nn 11312  df-n0 11578  df-z 11664  df-uz 11928  df-fz 12578  df-fzo 12718  df-hash 13368  df-word 13532  df-concat 13588  df-substr 13662

This theorem is referenced by:  ccats1swrdeqbiOLD  13808  clwlkclwwlk2OLD  27291  clwlkclwwlkfoOLD  27299  clwlksfoclwwlkOLD  27397  numclwwlk1lem2foalemOLD  27702  numclwwlk1lem2foOLD  27717