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

Theorem ccatopthOLD 13769
Description: Obsolete proof of ccatopth 13768 as of 12-Oct-2022. (Contributed by Mario Carneiro, 1-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
ccatopthOLD (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Proof of Theorem ccatopthOLD
StepHypRef Expression
1 oveq1 6885 . . . 4 ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → ((𝐴 ++ 𝐵) substr ⟨0, (♯‘𝐴)⟩) = ((𝐶 ++ 𝐷) substr ⟨0, (♯‘𝐴)⟩))
2 swrdccat1OLD 13711 . . . . . 6 ((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) → ((𝐴 ++ 𝐵) substr ⟨0, (♯‘𝐴)⟩) = 𝐴)
323ad2ant1 1164 . . . . 5 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) substr ⟨0, (♯‘𝐴)⟩) = 𝐴)
4 simp3 1169 . . . . . . . 8 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → (♯‘𝐴) = (♯‘𝐶))
54opeq2d 4600 . . . . . . 7 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ⟨0, (♯‘𝐴)⟩ = ⟨0, (♯‘𝐶)⟩)
65oveq2d 6894 . . . . . 6 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐶 ++ 𝐷) substr ⟨0, (♯‘𝐴)⟩) = ((𝐶 ++ 𝐷) substr ⟨0, (♯‘𝐶)⟩))
7 swrdccat1OLD 13711 . . . . . . 7 ((𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) → ((𝐶 ++ 𝐷) substr ⟨0, (♯‘𝐶)⟩) = 𝐶)
873ad2ant2 1165 . . . . . 6 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐶 ++ 𝐷) substr ⟨0, (♯‘𝐶)⟩) = 𝐶)
96, 8eqtrd 2833 . . . . 5 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐶 ++ 𝐷) substr ⟨0, (♯‘𝐴)⟩) = 𝐶)
103, 9eqeq12d 2814 . . . 4 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → (((𝐴 ++ 𝐵) substr ⟨0, (♯‘𝐴)⟩) = ((𝐶 ++ 𝐷) substr ⟨0, (♯‘𝐴)⟩) ↔ 𝐴 = 𝐶))
111, 10syl5ib 236 . . 3 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → 𝐴 = 𝐶))
12 simpr 478 . . . . . 6 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷))
13 simpl3 1247 . . . . . . 7 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (♯‘𝐴) = (♯‘𝐶))
1412fveq2d 6415 . . . . . . . 8 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (♯‘(𝐴 ++ 𝐵)) = (♯‘(𝐶 ++ 𝐷)))
15 simpl1 1243 . . . . . . . . 9 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋))
16 ccatlen 13595 . . . . . . . . 9 ((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
1715, 16syl 17 . . . . . . . 8 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (♯‘(𝐴 ++ 𝐵)) = ((♯‘𝐴) + (♯‘𝐵)))
18 simpl2 1245 . . . . . . . . 9 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋))
19 ccatlen 13595 . . . . . . . . 9 ((𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) → (♯‘(𝐶 ++ 𝐷)) = ((♯‘𝐶) + (♯‘𝐷)))
2018, 19syl 17 . . . . . . . 8 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → (♯‘(𝐶 ++ 𝐷)) = ((♯‘𝐶) + (♯‘𝐷)))
2114, 17, 203eqtr3d 2841 . . . . . . 7 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → ((♯‘𝐴) + (♯‘𝐵)) = ((♯‘𝐶) + (♯‘𝐷)))
2213, 21opeq12d 4601 . . . . . 6 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → ⟨(♯‘𝐴), ((♯‘𝐴) + (♯‘𝐵))⟩ = ⟨(♯‘𝐶), ((♯‘𝐶) + (♯‘𝐷))⟩)
2312, 22oveq12d 6896 . . . . 5 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → ((𝐴 ++ 𝐵) substr ⟨(♯‘𝐴), ((♯‘𝐴) + (♯‘𝐵))⟩) = ((𝐶 ++ 𝐷) substr ⟨(♯‘𝐶), ((♯‘𝐶) + (♯‘𝐷))⟩))
24 swrdccat2 13712 . . . . . 6 ((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) → ((𝐴 ++ 𝐵) substr ⟨(♯‘𝐴), ((♯‘𝐴) + (♯‘𝐵))⟩) = 𝐵)
2515, 24syl 17 . . . . 5 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → ((𝐴 ++ 𝐵) substr ⟨(♯‘𝐴), ((♯‘𝐴) + (♯‘𝐵))⟩) = 𝐵)
26 swrdccat2 13712 . . . . . 6 ((𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) → ((𝐶 ++ 𝐷) substr ⟨(♯‘𝐶), ((♯‘𝐶) + (♯‘𝐷))⟩) = 𝐷)
2718, 26syl 17 . . . . 5 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → ((𝐶 ++ 𝐷) substr ⟨(♯‘𝐶), ((♯‘𝐶) + (♯‘𝐷))⟩) = 𝐷)
2823, 25, 273eqtr3d 2841 . . . 4 ((((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) ∧ (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷)) → 𝐵 = 𝐷)
2928ex 402 . . 3 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → 𝐵 = 𝐷))
3011, 29jcad 509 . 2 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷)))
31 oveq12 6887 . 2 ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴 ++ 𝐵) = (𝐶 ++ 𝐷))
3230, 31impbid1 217 1 (((𝐴 ∈ Word 𝑋𝐵 ∈ Word 𝑋) ∧ (𝐶 ∈ Word 𝑋𝐷 ∈ Word 𝑋) ∧ (♯‘𝐴) = (♯‘𝐶)) → ((𝐴 ++ 𝐵) = (𝐶 ++ 𝐷) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  cop 4374  cfv 6101  (class class class)co 6878  0cc0 10224   + caddc 10227  chash 13370  Word cword 13534   ++ cconcat 13590   substr csubstr 13664
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 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
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 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-int 4668  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-om 7300  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-oadd 7803  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-card 9051  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-sub 10558  df-neg 10559  df-nn 11313  df-n0 11581  df-z 11667  df-uz 11931  df-fz 12581  df-fzo 12721  df-hash 13371  df-word 13535  df-concat 13591  df-substr 13665
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator