MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ccatopth2 Structured version   Visualization version   GIF version

Theorem ccatopth2 14752
Description: An opth 5487-like theorem for recovering the two halves of a concatenated word. (Contributed by Mario Carneiro, 1-Oct-2015.)
Assertion
Ref Expression
ccatopth2 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Proof of Theorem ccatopth2
StepHypRef Expression
1 fveq2 6907 . . . 4 ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → (♯‘(𝐴 ++ 𝐵)) = (♯‘(𝐶 ++ 𝐷)))
2 ccatlen 14610 . . . . . . . 8 ((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
323ad2ant1 1132 . . . . . . 7 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
4 simp3 1137 . . . . . . . 8 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (♯‘𝐵) = (♯‘𝐷))
54oveq2d 7447 . . . . . . 7 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → ((♯‘𝐴) + (♯‘𝐵)) = ((♯‘𝐴) + (♯‘𝐷)))
63, 5eqtrd 2775 . . . . . 6 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐷)))
7 ccatlen 14610 . . . . . . 7 ((𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) → (♯‘(𝐶 ++ 𝐷)) = ((♯‘𝐶) + (♯‘𝐷)))
873ad2ant2 1133 . . . . . 6 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (♯‘(𝐶 ++ 𝐷)) = ((♯‘𝐶) + (♯‘𝐷)))
96, 8eqeq12d 2751 . . . . 5 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → ((♯‘(𝐴 ++ 𝐵)) = (♯‘(𝐶 ++ 𝐷)) ↔ ((♯‘𝐴) + (♯‘𝐷)) = ((♯‘𝐶) + (♯‘𝐷))))
10 simp1l 1196 . . . . . . . 8 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → 𝐴 ∈ Word 𝑋)
11 lencl 14568 . . . . . . . 8 (𝐴 ∈ Word 𝑋 → (♯‘𝐴) ∈ ℕ0)
1210, 11syl 17 . . . . . . 7 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (♯‘𝐴) ∈ ℕ0)
1312nn0cnd 12587 . . . . . 6 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (♯‘𝐴) ∈ ℂ)
14 simp2l 1198 . . . . . . . 8 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → 𝐶 ∈ Word 𝑋)
15 lencl 14568 . . . . . . . 8 (𝐶 ∈ Word 𝑋 → (♯‘𝐶) ∈ ℕ0)
1614, 15syl 17 . . . . . . 7 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (♯‘𝐶) ∈ ℕ0)
1716nn0cnd 12587 . . . . . 6 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (♯‘𝐶) ∈ ℂ)
18 simp2r 1199 . . . . . . . 8 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → 𝐷 ∈ Word 𝑋)
19 lencl 14568 . . . . . . . 8 (𝐷 ∈ Word 𝑋 → (♯‘𝐷) ∈ ℕ0)
2018, 19syl 17 . . . . . . 7 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (♯‘𝐷) ∈ ℕ0)
2120nn0cnd 12587 . . . . . 6 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (♯‘𝐷) ∈ ℂ)
2213, 17, 21addcan2d 11463 . . . . 5 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → (((♯‘𝐴) + (♯‘𝐷)) = ((♯‘𝐶) + (♯‘𝐷)) ↔ (♯‘𝐴) = (♯‘𝐶)))
239, 22bitrd 279 . . . 4 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → ((♯‘(𝐴 ++ 𝐵)) = (♯‘(𝐶 ++ 𝐷)) ↔ (♯‘𝐴) = (♯‘𝐶)))
241, 23imbitrid 244 . . 3 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → (♯‘𝐴) = (♯‘𝐶)))
25 ccatopth 14751 . . . . . . 7 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
2625biimpd 229 . . . . . 6 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷)))
27263expia 1120 . . . . 5 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋)) → ((♯‘𝐴) = (♯‘𝐶) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷))))
2827com23 86 . . . 4 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → ((♯‘𝐴) = (♯‘𝐶) → (𝐴 = 𝐶𝐵 = 𝐷))))
29283adant3 1131 . . 3 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → ((♯‘𝐴) = (♯‘𝐶) → (𝐴 = 𝐶𝐵 = 𝐷))))
3024, 29mpdd 43 . 2 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷)))
31 oveq12 7440 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷))
3230, 31impbid1 225 1 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐵) = (♯‘𝐷)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431   + caddc 11156  0cn0 12524  chash 14366  Word cword 14549   ++ cconcat 14605
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:  ccatrcan  14754
  Copyright terms: Public domain W3C validator