Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  s7eqd Structured version   Visualization version   GIF version

Theorem s7eqd 14091
 Description: Equality theorem for a length 7 word. (Contributed by Mario Carneiro, 27-Feb-2016.)
Hypotheses
Ref Expression
s2eqd.1 (𝜑𝐴 = 𝑁)
s2eqd.2 (𝜑𝐵 = 𝑂)
s3eqd.3 (𝜑𝐶 = 𝑃)
s4eqd.4 (𝜑𝐷 = 𝑄)
s5eqd.5 (𝜑𝐸 = 𝑅)
s6eqd.6 (𝜑𝐹 = 𝑆)
s7eqd.6 (𝜑𝐺 = 𝑇)
Assertion
Ref Expression
s7eqd (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩)

Proof of Theorem s7eqd
StepHypRef Expression
1 s2eqd.1 . . . 4 (𝜑𝐴 = 𝑁)
2 s2eqd.2 . . . 4 (𝜑𝐵 = 𝑂)
3 s3eqd.3 . . . 4 (𝜑𝐶 = 𝑃)
4 s4eqd.4 . . . 4 (𝜑𝐷 = 𝑄)
5 s5eqd.5 . . . 4 (𝜑𝐸 = 𝑅)
6 s6eqd.6 . . . 4 (𝜑𝐹 = 𝑆)
71, 2, 3, 4, 5, 6s6eqd 14090 . . 3 (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆”⟩)
8 s7eqd.6 . . . 4 (𝜑𝐺 = 𝑇)
98s1eqd 13763 . . 3 (𝜑 → ⟨“𝐺”⟩ = ⟨“𝑇”⟩)
107, 9oveq12d 6993 . 2 (𝜑 → (⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ++ ⟨“𝐺”⟩) = (⟨“𝑁𝑂𝑃𝑄𝑅𝑆”⟩ ++ ⟨“𝑇”⟩))
11 df-s7 14076 . 2 ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ++ ⟨“𝐺”⟩)
12 df-s7 14076 . 2 ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩ = (⟨“𝑁𝑂𝑃𝑄𝑅𝑆”⟩ ++ ⟨“𝑇”⟩)
1310, 11, 123eqtr4g 2834 1 (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1508  (class class class)co 6975   ++ cconcat 13732  ⟨“cs1 13757  ⟨“cs6 14068  ⟨“cs7 14069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2745 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-rex 3089  df-rab 3092  df-v 3412  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-br 4927  df-iota 6150  df-fv 6194  df-ov 6978  df-s1 13758  df-s2 14071  df-s3 14072  df-s4 14073  df-s5 14074  df-s6 14075  df-s7 14076 This theorem is referenced by:  s8eqd  14092
 Copyright terms: Public domain W3C validator