Theorem s8eqd 15020

Description: Equality theorem for a length 8 word. (Contributed by Mario Carneiro, 27-Feb-2016.)

Hypotheses

Ref	Expression
s2eqd.1	⊢ (𝜑 → 𝐴 = 𝑁)
s2eqd.2	⊢ (𝜑 → 𝐵 = 𝑂)
s3eqd.3	⊢ (𝜑 → 𝐶 = 𝑃)
s4eqd.4	⊢ (𝜑 → 𝐷 = 𝑄)
s5eqd.5	⊢ (𝜑 → 𝐸 = 𝑅)
s6eqd.6	⊢ (𝜑 → 𝐹 = 𝑆)
s7eqd.6	⊢ (𝜑 → 𝐺 = 𝑇)
s8eqd.6	⊢ (𝜑 → 𝐻 = 𝑈)

Assertion

Ref	Expression
s8eqd	⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈”⟩)

Proof of Theorem s8eqd

Step	Hyp	Ref	Expression
1		s2eqd.1	. . . 4 ⊢ (𝜑 → 𝐴 = 𝑁)
2		s2eqd.2	. . . 4 ⊢ (𝜑 → 𝐵 = 𝑂)
3		s3eqd.3	. . . 4 ⊢ (𝜑 → 𝐶 = 𝑃)
4		s4eqd.4	. . . 4 ⊢ (𝜑 → 𝐷 = 𝑄)
5		s5eqd.5	. . . 4 ⊢ (𝜑 → 𝐸 = 𝑅)
6		s6eqd.6	. . . 4 ⊢ (𝜑 → 𝐹 = 𝑆)
7		s7eqd.6	. . . 4 ⊢ (𝜑 → 𝐺 = 𝑇)
8	1, 2, 3, 4, 5, 6, 7	s7eqd 15019	. . 3 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩)
9		s8eqd.6	. . . 4 ⊢ (𝜑 → 𝐻 = 𝑈)
10	9	s1eqd 14748	. . 3 ⊢ (𝜑 → ⟨“𝐻”⟩ = ⟨“𝑈”⟩)
11	8, 10	oveq12d 7438	. 2 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ++ ⟨“𝐻”⟩) = (⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩ ++ ⟨“𝑈”⟩))
12		df-s8 15005	. 2 ⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ++ ⟨“𝐻”⟩)
13		df-s8 15005	. 2 ⊢ ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈”⟩ = (⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩ ++ ⟨“𝑈”⟩)
14	11, 12, 13	3eqtr4g 2821	1 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈”⟩)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  (class class class)co 7420   ++ cconcat 14715  ⟨“cs1 14742  ⟨“cs7 14997  ⟨“cs8 14998

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6494  df-fv 6546  df-ov 7423  df-s1 14743  df-s2 14999  df-s3 15000  df-s4 15001  df-s5 15002  df-s6 15003  df-s7 15004  df-s8 15005

This theorem is used by: (None)