Theorem ccatopth2 14722

Description: An opth 5448-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.)

Assertion

Ref	Expression
ccatopth2	⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Proof of Theorem ccatopth2

Step	Hyp	Ref	Expression
1		fveq2 6872	. . . 4 ⊢ ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → (♯‘(𝐴 ++ 𝐵)) = (♯‘(𝐶 ++ 𝐷)))
2		ccatlen 14580	. . . . . . . 8 ⊢ ((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
3	2	3ad2ant1 1133	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
4		simp3 1138	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (♯‘𝐵) = (♯‘𝐷))
5	4	oveq2d 7415	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → ((♯‘𝐴) + (♯‘𝐵)) = ((♯‘𝐴) + (♯‘𝐷)))
6	3, 5	eqtrd 2769	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐷)))
7		ccatlen 14580	. . . . . . 7 ⊢ ((𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) → (♯‘(𝐶 ++ 𝐷)) = ((♯‘𝐶) + (♯‘𝐷)))
8	7	3ad2ant2 1134	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (♯‘(𝐶 ++ 𝐷)) = ((♯‘𝐶) + (♯‘𝐷)))
9	6, 8	eqeq12d 2750	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → ((♯‘(𝐴 ++ 𝐵)) = (♯‘(𝐶 ++ 𝐷)) ↔ ((♯‘𝐴) + (♯‘𝐷)) = ((♯‘𝐶) + (♯‘𝐷))))
10		simp1l 1197	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → 𝐴 ∈ Word 𝑋)
11		lencl 14538	. . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑋 → (♯‘𝐴) ∈ ℕ0)
12	10, 11	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (♯‘𝐴) ∈ ℕ0)
13	12	nn0cnd 12556	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (♯‘𝐴) ∈ ℂ)
14		simp2l 1199	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → 𝐶 ∈ Word 𝑋)
15		lencl 14538	. . . . . . . 8 ⊢ (𝐶 ∈ Word 𝑋 → (♯‘𝐶) ∈ ℕ0)
16	14, 15	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (♯‘𝐶) ∈ ℕ0)
17	16	nn0cnd 12556	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (♯‘𝐶) ∈ ℂ)
18		simp2r 1200	. . . . . . . 8 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → 𝐷 ∈ Word 𝑋)
19		lencl 14538	. . . . . . . 8 ⊢ (𝐷 ∈ Word 𝑋 → (♯‘𝐷) ∈ ℕ0)
20	18, 19	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (♯‘𝐷) ∈ ℕ0)
21	20	nn0cnd 12556	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (♯‘𝐷) ∈ ℂ)
22	13, 17, 21	addcan2d 11431	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (((♯‘𝐴) + (♯‘𝐷)) = ((♯‘𝐶) + (♯‘𝐷)) ↔ (♯‘𝐴) = (♯‘𝐶)))
23	9, 22	bitrd 279	. . . 4 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → ((♯‘(𝐴 ++ 𝐵)) = (♯‘(𝐶 ++ 𝐷)) ↔ (♯‘𝐴) = (♯‘𝐶)))
24	1, 23	imbitrid 244	. . 3 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → (♯‘𝐴) = (♯‘𝐶)))
25		ccatopth 14721	. . . . . . 7 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
26	25	biimpd 229	. . . . . 6 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
27	26	3expia 1121	. . . . 5 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋)) → ((♯‘𝐴) = (♯‘𝐶) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))))
28	27	com23 86	. . . 4 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → ((♯‘𝐴) = (♯‘𝐶) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))))
29	28	3adant3 1132	. . 3 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → ((♯‘𝐴) = (♯‘𝐶) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))))
30	24, 29	mpdd 43	. 2 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
31		oveq12 7408	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷))
32	30, 31	impbid1 225	1 ⊢ (((𝐴 ∈ Word 𝑋 ∧ 𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋 ∧ 𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ‘cfv 6527  (class class class)co 7399   + caddc 11124  ℕ₀cn0 12493  ♯chash 14336  Word cword 14519   ++ cconcat 14575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-rep 5246  ax-sep 5263  ax-nul 5273  ax-pow 5332  ax-pr 5399  ax-un 7723  ax-cnex 11177  ax-resscn 11178  ax-1cn 11179  ax-icn 11180  ax-addcl 11181  ax-addrcl 11182  ax-mulcl 11183  ax-mulrcl 11184  ax-mulcom 11185  ax-addass 11186  ax-mulass 11187  ax-distr 11188  ax-i2m1 11189  ax-1ne0 11190  ax-1rid 11191  ax-rnegex 11192  ax-rrecex 11193  ax-cnre 11194  ax-pre-lttri 11195  ax-pre-lttrn 11196  ax-pre-ltadd 11197  ax-pre-mulgt0 11198

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-reu 3358  df-rab 3414  df-v 3459  df-sbc 3764  df-csb 3873  df-dif 3927  df-un 3929  df-in 3931  df-ss 3941  df-pss 3944  df-nul 4307  df-if 4499  df-pw 4575  df-sn 4600  df-pr 4602  df-op 4606  df-uni 4881  df-int 4920  df-iun 4966  df-br 5117  df-opab 5179  df-mpt 5199  df-tr 5227  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6287  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6480  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7356  df-ov 7402  df-oprab 7403  df-mpo 7404  df-om 7856  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8379  df-rdg 8418  df-1o 8474  df-er 8713  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-card 9945  df-pnf 11263  df-mnf 11264  df-xr 11265  df-ltxr 11266  df-le 11267  df-sub 11460  df-neg 11461  df-nn 12233  df-n0 12494  df-z 12581  df-uz 12845  df-fz 13514  df-fzo 13661  df-hash 14337  df-word 14520  df-concat 14576  df-substr 14646  df-pfx 14676

This theorem is referenced by:  ccatrcan  14724