Theorem ecovass 8800

Description: Lemma used to transfer an associative law via an equivalence relation. (Contributed by NM, 31-Aug-1995.) (Revised by David Abernethy, 4-Jun-2013.)

Hypotheses

Ref	Expression
ecovass.1	⊢ 𝐷 = ((𝑆 × 𝑆) / ∼ )
ecovass.2	⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) = [⟨𝐺, 𝐻⟩] ∼ )
ecovass.3	⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ ) = [⟨𝑁, 𝑄⟩] ∼ )
ecovass.4	⊢ (((𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([⟨𝐺, 𝐻⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ ) = [⟨𝐽, 𝐾⟩] ∼ )
ecovass.5	⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑁 ∈ 𝑆 ∧ 𝑄 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑁, 𝑄⟩] ∼ ) = [⟨𝐿, 𝑀⟩] ∼ )
ecovass.6	⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆))
ecovass.7	⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (𝑁 ∈ 𝑆 ∧ 𝑄 ∈ 𝑆))
ecovass.8	⊢ 𝐽 = 𝐿
ecovass.9	⊢ 𝐾 = 𝑀

Assertion

Ref	Expression
ecovass	⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝐴   𝑧,𝐵,𝑤,𝑣,𝑢   𝑥,𝐶,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥, + ,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥, ∼ ,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥,𝑆,𝑦,𝑧,𝑤,𝑣,𝑢   𝑧,𝐷,𝑤,𝑣,𝑢

Allowed substitution hints:   𝐵(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑄(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐺(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐻(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐽(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐾(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝐿(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝑀(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝑁(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)

Proof of Theorem ecovass

Step	Hyp	Ref	Expression
1		ecovass.1	. 2 ⊢ 𝐷 = ((𝑆 × 𝑆) / ∼ )
2		oveq1 7397	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) = (𝐴 + [⟨𝑧, 𝑤⟩] ∼ ))
3	2	oveq1d 7405	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → (([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) + [⟨𝑣, 𝑢⟩] ∼ ) = ((𝐴 + [⟨𝑧, 𝑤⟩] ∼ ) + [⟨𝑣, 𝑢⟩] ∼ ))
4		oveq1 7397	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑥, 𝑦⟩] ∼ + ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = (𝐴 + ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )))
5	3, 4	eqeq12d 2746	. 2 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ((([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) + [⟨𝑣, 𝑢⟩] ∼ ) = ([⟨𝑥, 𝑦⟩] ∼ + ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) ↔ ((𝐴 + [⟨𝑧, 𝑤⟩] ∼ ) + [⟨𝑣, 𝑢⟩] ∼ ) = (𝐴 + ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ ))))
6		oveq2 7398	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → (𝐴 + [⟨𝑧, 𝑤⟩] ∼ ) = (𝐴 + 𝐵))
7	6	oveq1d 7405	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → ((𝐴 + [⟨𝑧, 𝑤⟩] ∼ ) + [⟨𝑣, 𝑢⟩] ∼ ) = ((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ∼ ))
8		oveq1 7397	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ ) = (𝐵 + [⟨𝑣, 𝑢⟩] ∼ ))
9	8	oveq2d 7406	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → (𝐴 + ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] ∼ )))
10	7, 9	eqeq12d 2746	. 2 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → (((𝐴 + [⟨𝑧, 𝑤⟩] ∼ ) + [⟨𝑣, 𝑢⟩] ∼ ) = (𝐴 + ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) ↔ ((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ∼ ) = (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] ∼ ))))
11		oveq2 7398	. . 3 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → ((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ∼ ) = ((𝐴 + 𝐵) + 𝐶))
12		oveq2 7398	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → (𝐵 + [⟨𝑣, 𝑢⟩] ∼ ) = (𝐵 + 𝐶))
13	12	oveq2d 7406	. . 3 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] ∼ )) = (𝐴 + (𝐵 + 𝐶)))
14	11, 13	eqeq12d 2746	. 2 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → (((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ∼ ) = (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] ∼ )) ↔ ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶))))
15		ecovass.8	. . . 4 ⊢ 𝐽 = 𝐿
16		ecovass.9	. . . 4 ⊢ 𝐾 = 𝑀
17		opeq12 4842	. . . . 5 ⊢ ((𝐽 = 𝐿 ∧ 𝐾 = 𝑀) → ⟨𝐽, 𝐾⟩ = ⟨𝐿, 𝑀⟩)
18	17	eceq1d 8714	. . . 4 ⊢ ((𝐽 = 𝐿 ∧ 𝐾 = 𝑀) → [⟨𝐽, 𝐾⟩] ∼ = [⟨𝐿, 𝑀⟩] ∼ )
19	15, 16, 18	mp2an 692	. . 3 ⊢ [⟨𝐽, 𝐾⟩] ∼ = [⟨𝐿, 𝑀⟩] ∼
20		ecovass.2	. . . . . . 7 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) = [⟨𝐺, 𝐻⟩] ∼ )
21	20	oveq1d 7405	. . . . . 6 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) + [⟨𝑣, 𝑢⟩] ∼ ) = ([⟨𝐺, 𝐻⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ ))
22	21	adantr 480	. . . . 5 ⊢ ((((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) + [⟨𝑣, 𝑢⟩] ∼ ) = ([⟨𝐺, 𝐻⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ ))
23		ecovass.6	. . . . . 6 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆))
24		ecovass.4	. . . . . 6 ⊢ (((𝐺 ∈ 𝑆 ∧ 𝐻 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([⟨𝐺, 𝐻⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ ) = [⟨𝐽, 𝐾⟩] ∼ )
25	23, 24	sylan 580	. . . . 5 ⊢ ((((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([⟨𝐺, 𝐻⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ ) = [⟨𝐽, 𝐾⟩] ∼ )
26	22, 25	eqtrd 2765	. . . 4 ⊢ ((((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) + [⟨𝑣, 𝑢⟩] ∼ ) = [⟨𝐽, 𝐾⟩] ∼ )
27	26	3impa 1109	. . 3 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) + [⟨𝑣, 𝑢⟩] ∼ ) = [⟨𝐽, 𝐾⟩] ∼ )
28		ecovass.3	. . . . . . 7 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ ) = [⟨𝑁, 𝑄⟩] ∼ )
29	28	oveq2d 7406	. . . . . 6 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ + ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = ([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑁, 𝑄⟩] ∼ ))
30	29	adantl 481	. . . . 5 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆))) → ([⟨𝑥, 𝑦⟩] ∼ + ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = ([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑁, 𝑄⟩] ∼ ))
31		ecovass.7	. . . . . 6 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (𝑁 ∈ 𝑆 ∧ 𝑄 ∈ 𝑆))
32		ecovass.5	. . . . . 6 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑁 ∈ 𝑆 ∧ 𝑄 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑁, 𝑄⟩] ∼ ) = [⟨𝐿, 𝑀⟩] ∼ )
33	31, 32	sylan2 593	. . . . 5 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆))) → ([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑁, 𝑄⟩] ∼ ) = [⟨𝐿, 𝑀⟩] ∼ )
34	30, 33	eqtrd 2765	. . . 4 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆))) → ([⟨𝑥, 𝑦⟩] ∼ + ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = [⟨𝐿, 𝑀⟩] ∼ )
35	34	3impb 1114	. . 3 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ + ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = [⟨𝐿, 𝑀⟩] ∼ )
36	19, 27, 35	3eqtr4a 2791	. 2 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) + [⟨𝑣, 𝑢⟩] ∼ ) = ([⟨𝑥, 𝑦⟩] ∼ + ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )))
37	1, 5, 10, 14, 36	3ecoptocl 8785	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ⟨cop 4598   × cxp 5639  (class class class)co 7390  [cec 8672   / cqs 8673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-xp 5647  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fv 6522  df-ov 7393  df-ec 8676  df-qs 8680

This theorem is referenced by:  addasssr  11048  mulasssr  11050  axaddass  11116  axmulass  11117