Theorem ecovass 6754

Description: Lemma used to transfer an associative law via an equivalence relation. In most cases ecoviass 6755 will be more useful. (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 5974	. . . 4 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) = (𝐴 + [⟨𝑧, 𝑤⟩] ∼ ))
3	2	oveq1d 5982	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → (([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) + [⟨𝑣, 𝑢⟩] ∼ ) = ((𝐴 + [⟨𝑧, 𝑤⟩] ∼ ) + [⟨𝑣, 𝑢⟩] ∼ ))
4		oveq1 5974	. . 3 ⊢ ([⟨𝑥, 𝑦⟩] ∼ = 𝐴 → ([⟨𝑥, 𝑦⟩] ∼ + ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = (𝐴 + ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )))
5	3, 4	eqeq12d 2222	. 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 5975	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → (𝐴 + [⟨𝑧, 𝑤⟩] ∼ ) = (𝐴 + 𝐵))
7	6	oveq1d 5982	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → ((𝐴 + [⟨𝑧, 𝑤⟩] ∼ ) + [⟨𝑣, 𝑢⟩] ∼ ) = ((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ∼ ))
8		oveq1 5974	. . . 4 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ ) = (𝐵 + [⟨𝑣, 𝑢⟩] ∼ ))
9	8	oveq2d 5983	. . 3 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → (𝐴 + ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] ∼ )))
10	7, 9	eqeq12d 2222	. 2 ⊢ ([⟨𝑧, 𝑤⟩] ∼ = 𝐵 → (((𝐴 + [⟨𝑧, 𝑤⟩] ∼ ) + [⟨𝑣, 𝑢⟩] ∼ ) = (𝐴 + ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) ↔ ((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ∼ ) = (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] ∼ ))))
11		oveq2 5975	. . 3 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → ((𝐴 + 𝐵) + [⟨𝑣, 𝑢⟩] ∼ ) = ((𝐴 + 𝐵) + 𝐶))
12		oveq2 5975	. . . 4 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → (𝐵 + [⟨𝑣, 𝑢⟩] ∼ ) = (𝐵 + 𝐶))
13	12	oveq2d 5983	. . 3 ⊢ ([⟨𝑣, 𝑢⟩] ∼ = 𝐶 → (𝐴 + (𝐵 + [⟨𝑣, 𝑢⟩] ∼ )) = (𝐴 + (𝐵 + 𝐶)))
14	11, 13	eqeq12d 2222	. 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 3835	. . . . 5 ⊢ ((𝐽 = 𝐿 ∧ 𝐾 = 𝑀) → ⟨𝐽, 𝐾⟩ = ⟨𝐿, 𝑀⟩)
18	17	eceq1d 6679	. . . 4 ⊢ ((𝐽 = 𝐿 ∧ 𝐾 = 𝑀) → [⟨𝐽, 𝐾⟩] ∼ = [⟨𝐿, 𝑀⟩] ∼ )
19	15, 16, 18	mp2an 426	. . 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 5982	. . . . . 6 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) → (([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) + [⟨𝑣, 𝑢⟩] ∼ ) = ([⟨𝐺, 𝐻⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ ))
22	21	adantr 276	. . . . 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 283	. . . . 5 ⊢ ((((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([⟨𝐺, 𝐻⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ ) = [⟨𝐽, 𝐾⟩] ∼ )
26	22, 25	eqtrd 2240	. . . 4 ⊢ ((((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ (𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆)) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → (([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑧, 𝑤⟩] ∼ ) + [⟨𝑣, 𝑢⟩] ∼ ) = [⟨𝐽, 𝐾⟩] ∼ )
27	26	3impa 1197	. . 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 5983	. . . . . 6 ⊢ (((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆)) → ([⟨𝑥, 𝑦⟩] ∼ + ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = ([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑁, 𝑄⟩] ∼ ))
30	29	adantl 277	. . . . 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 286	. . . . 5 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆))) → ([⟨𝑥, 𝑦⟩] ∼ + [⟨𝑁, 𝑄⟩] ∼ ) = [⟨𝐿, 𝑀⟩] ∼ )
34	30, 33	eqtrd 2240	. . . 4 ⊢ (((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) ∧ ((𝑧 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ 𝑢 ∈ 𝑆))) → ([⟨𝑥, 𝑦⟩] ∼ + ([⟨𝑧, 𝑤⟩] ∼ + [⟨𝑣, 𝑢⟩] ∼ )) = [⟨𝐿, 𝑀⟩] ∼ )
35	34	3impb 1202	. . 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 2266	. 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 6734	1 ⊢ ((𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝐷 ∧ 𝐶 ∈ 𝐷) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 981   = wceq 1373   ∈ wcel 2178  ⟨cop 3646   × cxp 4691  (class class class)co 5967  [cec 6641   / cqs 6642

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-xp 4699  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fv 5298  df-ov 5970  df-ec 6645  df-qs 6649

This theorem is referenced by: (None)