Theorem s8eqd 14831

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 14830	. . 3 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩)
9		s8eqd.6	. . . 4 ⊢ (𝜑 → 𝐻 = 𝑈)
10	9	s1eqd 14564	. . 3 ⊢ (𝜑 → ⟨“𝐻”⟩ = ⟨“𝑈”⟩)
11	8, 10	oveq12d 7385	. 2 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ++ ⟨“𝐻”⟩) = (⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩ ++ ⟨“𝑈”⟩))
12		df-s8 14816	. 2 ⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ ++ ⟨“𝐻”⟩)
13		df-s8 14816	. 2 ⊢ ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈”⟩ = (⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩ ++ ⟨“𝑈”⟩)
14	11, 12, 13	3eqtr4g 2796	1 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺𝐻”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇𝑈”⟩)

Syntax hints:   → wi 4   = wceq 1542  (class class class)co 7367   ++ cconcat 14532  ⟨“cs1 14558  ⟨“cs7 14808  ⟨“cs8 14809

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-iota 6454  df-fv 6506  df-ov 7370  df-s1 14559  df-s2 14810  df-s3 14811  df-s4 14812  df-s5 14813  df-s6 14814  df-s7 14815  df-s8 14816

This theorem is referenced by: (None)