Theorem s7eqd 11357

Description: Equality theorem for a length 7 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)

Assertion

Ref	Expression
s7eqd	⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩)

Proof of Theorem s7eqd

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	1, 2, 3, 4, 5, 6	s6eqd 11356	. . 3 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆”⟩)
8		s7eqd.6	. . . 4 ⊢ (𝜑 → 𝐺 = 𝑇)
9	8	s1eqd 11198	. . 3 ⊢ (𝜑 → ⟨“𝐺”⟩ = ⟨“𝑇”⟩)
10	7, 9	oveq12d 6036	. 2 ⊢ (𝜑 → (⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ++ ⟨“𝐺”⟩) = (⟨“𝑁𝑂𝑃𝑄𝑅𝑆”⟩ ++ ⟨“𝑇”⟩))
11		df-s7 11343	. 2 ⊢ ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = (⟨“𝐴𝐵𝐶𝐷𝐸𝐹”⟩ ++ ⟨“𝐺”⟩)
12		df-s7 11343	. 2 ⊢ ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩ = (⟨“𝑁𝑂𝑃𝑄𝑅𝑆”⟩ ++ ⟨“𝑇”⟩)
13	10, 11, 12	3eqtr4g 2289	1 ⊢ (𝜑 → ⟨“𝐴𝐵𝐶𝐷𝐸𝐹𝐺”⟩ = ⟨“𝑁𝑂𝑃𝑄𝑅𝑆𝑇”⟩)

Syntax hints:   → wi 4   = wceq 1397  (class class class)co 6018   ++ cconcat 11168  ⟨“cs1 11193  ⟨“cs6 11335  ⟨“cs7 11336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6021  df-s1 11194  df-s2 11338  df-s3 11339  df-s4 11340  df-s5 11341  df-s6 11342  df-s7 11343

This theorem is referenced by:  s8eqd  11358