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Theorem s7eqd 14221
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 14220 . . 3 (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆”⟩)
8 s7eqd.6 . . . 4 (𝜑𝐺 = 𝑇)
98s1eqd 13946 . . 3 (𝜑 → ⟨“𝐺”⟩ = ⟨“𝑇”⟩)
107, 9oveq12d 7153 . 2 (𝜑 → (⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ++ ⟨“𝐺”⟩) = (⟨“𝑁𝑂𝑃𝑄𝑅𝑆”⟩ ++ ⟨“𝑇”⟩))
11 df-s7 14206 . 2 ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ++ ⟨“𝐺”⟩)
12 df-s7 14206 . 2 ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩ = (⟨“𝑁𝑂𝑃𝑄𝑅𝑆”⟩ ++ ⟨“𝑇”⟩)
1310, 11, 123eqtr4g 2858 1 (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  (class class class)co 7135   ++ cconcat 13913  ⟨“cs1 13940  ⟨“cs6 14198  ⟨“cs7 14199
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138  df-s1 13941  df-s2 14201  df-s3 14202  df-s4 14203  df-s5 14204  df-s6 14205  df-s7 14206
This theorem is referenced by:  s8eqd  14222
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