Theorem ccatopth 11490

Description: An opth 4377-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof shortened by AV, 12-Oct-2022.)

Assertion

Ref	Expression
ccatopth	⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Proof of Theorem ccatopth

Step	Hyp	Ref	Expression
1		oveq1 6092	. . . . 5 ⊢ ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → ((𝐴 ++ 𝐵) prefix (♯‘𝐴)) = ((𝐶 ++ 𝐷) prefix (♯‘𝐴)))
2		pfxccat1 11476	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) → ((𝐴 ++ 𝐵) prefix (♯‘𝐴)) = 𝐴)
3		oveq2 6093	. . . . . . 7 ⊢ ((♯‘𝐴) = (♯‘𝐶) → ((𝐶 ++ 𝐷) prefix (♯‘𝐴)) = ((𝐶 ++ 𝐷) prefix (♯‘𝐶)))
4		pfxccat1 11476	. . . . . . 7 ⊢ ((𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) → ((𝐶 ++ 𝐷) prefix (♯‘𝐶)) = 𝐶)
5	3, 4	sylan9eqr 2293	. . . . . 6 ⊢ (((𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐶 ++ 𝐷) prefix (♯‘𝐴)) = 𝐶)
6	2, 5	eqeqan12d 2254	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ ((𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶))) → (((𝐴 ++ 𝐵) prefix (♯‘𝐴)) = ((𝐶 ++ 𝐷) prefix (♯‘𝐴)) ↔ 𝐴 = 𝐶))
7	1, 6	imbitrid 154	. . . 4 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ ((𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶))) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → 𝐴 = 𝐶))
8	7	3impb 1230	. . 3 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → 𝐴 = 𝐶))
9		simpr 110	. . . . . 6 ⊢ ((((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷))
10		simpl3 1033	. . . . . . 7 ⊢ ((((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (♯‘𝐴) = (♯‘𝐶))
11	9	fveq2d 5699	. . . . . . . 8 ⊢ ((((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (♯‘(𝐴 ++ 𝐵)) = (♯‘(𝐶 ++ 𝐷)))
12		simpl1 1031	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋))
13		ccatlen 11365	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
14	12, 13	syl 14	. . . . . . . 8 ⊢ ((((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
15		simpl2 1032	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋))
16		ccatlen 11365	. . . . . . . . 9 ⊢ ((𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) → (♯‘(𝐶 ++ 𝐷)) = ((♯‘𝐶) + (♯‘𝐷)))
17	15, 16	syl 14	. . . . . . . 8 ⊢ ((((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (♯‘(𝐶 ++ 𝐷)) = ((♯‘𝐶) + (♯‘𝐷)))
18	11, 14, 17	3eqtr3d 2279	. . . . . . 7 ⊢ ((((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → ((♯‘𝐴) + (♯‘𝐵)) = ((♯‘𝐶) + (♯‘𝐷)))
19	10, 18	opeq12d 3912	. . . . . 6 ⊢ ((((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → ⟨(♯‘𝐴), ((♯‘𝐴) + (♯‘𝐵))⟩ = ⟨(♯‘𝐶), ((♯‘𝐶) + (♯‘𝐷))⟩)
20	9, 19	oveq12d 6103	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → ((𝐴 ++ 𝐵) substr ⟨(♯‘𝐴), ((♯‘𝐴) + (♯‘𝐵))⟩) = ((𝐶 ++ 𝐷) substr ⟨(♯‘𝐶), ((♯‘𝐶) + (♯‘𝐷))⟩))
21		swrdccat2 11445	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) → ((𝐴 ++ 𝐵) substr ⟨(♯‘𝐴), ((♯‘𝐴) + (♯‘𝐵))⟩) = 𝐵)
22	12, 21	syl 14	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → ((𝐴 ++ 𝐵) substr ⟨(♯‘𝐴), ((♯‘𝐴) + (♯‘𝐵))⟩) = 𝐵)
23		swrdccat2 11445	. . . . . 6 ⊢ ((𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) → ((𝐶 ++ 𝐷) substr ⟨(♯‘𝐶), ((♯‘𝐶) + (♯‘𝐷))⟩) = 𝐷)
24	15, 23	syl 14	. . . . 5 ⊢ ((((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → ((𝐶 ++ 𝐷) substr ⟨(♯‘𝐶), ((♯‘𝐶) + (♯‘𝐷))⟩) = 𝐷)
25	20, 22, 24	3eqtr3d 2279	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → 𝐵 = 𝐷)
26	25	ex 115	. . 3 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → 𝐵 = 𝐷))
27	8, 26	jcad 307	. 2 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
28		oveq12 6094	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷))
29	27, 28	impbid1 142	1 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ⟨cop 3712  ‘cfv 5377  (class class class)co 6085   + caddc 8182  ♯chash 11216  Word cword 11306   ++ cconcat 11360   substr csubstr 11419   prefix cpfx 11446

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9306  df-n0 9566  df-z 9647  df-uz 9924  df-fz 10414  df-fzo 10552  df-ihash 11217  df-word 11307  df-concat 11361  df-substr 11420  df-pfx 11447

This theorem is used by:  ccatopth2  11491  ccatlcan  11492