Theorem ccatopth 14751

Description: An opth 5487-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 7438	. . . . 5 ⊢ ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → ((𝐴 ++ 𝐵) prefix (♯‘𝐴)) = ((𝐶 ++ 𝐷) prefix (♯‘𝐴)))
2		pfxccat1 14737	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) → ((𝐴 ++ 𝐵) prefix (♯‘𝐴)) = 𝐴)
3		oveq2 7439	. . . . . . 7 ⊢ ((♯‘𝐴) = (♯‘𝐶) → ((𝐶 ++ 𝐷) prefix (♯‘𝐴)) = ((𝐶 ++ 𝐷) prefix (♯‘𝐶)))
4		pfxccat1 14737	. . . . . . 7 ⊢ ((𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) → ((𝐶 ++ 𝐷) prefix (♯‘𝐶)) = 𝐶)
5	3, 4	sylan9eqr 2797	. . . . . 6 ⊢ (((𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐶 ++ 𝐷) prefix (♯‘𝐴)) = 𝐶)
6	2, 5	eqeqan12d 2749	. . . . 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 244	. . . 4 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ ((𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶))) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → 𝐴 = 𝐶))
8	7	3impb 1114	. . 3 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → 𝐴 = 𝐶))
9		simpr 484	. . . . . 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 1192	. . . . . . 7 ⊢ ((((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (♯‘𝐴) = (♯‘𝐶))
11	9	fveq2d 6911	. . . . . . . 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 1190	. . . . . . . . 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 14610	. . . . . . . . 9 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
14	12, 13	syl 17	. . . . . . . 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 1191	. . . . . . . . 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 14610	. . . . . . . . 9 ⊢ ((𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) → (♯‘(𝐶 ++ 𝐷)) = ((♯‘𝐶) + (♯‘𝐷)))
17	15, 16	syl 17	. . . . . . . 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 2783	. . . . . . 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 4886	. . . . . 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 7449	. . . . 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 14704	. . . . . 6 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) → ((𝐴 ++ 𝐵) substr ⟨(♯‘𝐴), ((♯‘𝐴) + (♯‘𝐵))⟩) = 𝐵)
22	12, 21	syl 17	. . . . 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 14704	. . . . . 6 ⊢ ((𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) → ((𝐶 ++ 𝐷) substr ⟨(♯‘𝐶), ((♯‘𝐶) + (♯‘𝐷))⟩) = 𝐷)
24	15, 23	syl 17	. . . . 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 2783	. . . 4 ⊢ ((((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → 𝐵 = 𝐷)
26	25	ex 412	. . 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 512	. 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 7440	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷))
29	27, 28	impbid1 225	1 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106  ⟨cop 4637  ‘cfv 6563  (class class class)co 7431   + caddc 11156  ♯chash 14366  Word cword 14549   ++ cconcat 14605   substr csubstr 14675   prefix cpfx 14705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-concat 14606  df-substr 14676  df-pfx 14706

This theorem is referenced by:  ccatopth2  14752  ccatlcan  14753  splval2  14792  s2eq2s1eq  14972  s3eqs2s1eq  14974  efgredleme  19776  efgredlemc  19778