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Theorem ecovass 6151
Description: Lemma used to transfer an associative law via an equivalence relation. In most cases ecoviass 6152 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
StepHypRef Expression
1 ecovass.1 . 2 𝐷 = ((𝑆 × 𝑆) / )
2 oveq1 5462 . . . 4 ([⟨x, y⟩] = A → ([⟨x, y⟩] + [⟨z, w⟩] ) = (A + [⟨z, w⟩] ))
32oveq1d 5470 . . 3 ([⟨x, y⟩] = A → (([⟨x, y⟩] + [⟨z, w⟩] ) + [⟨v, u⟩] ) = ((A + [⟨z, w⟩] ) + [⟨v, u⟩] ))
4 oveq1 5462 . . 3 ([⟨x, y⟩] = A → ([⟨x, y⟩] + ([⟨z, w⟩] + [⟨v, u⟩] )) = (A + ([⟨z, w⟩] + [⟨v, u⟩] )))
53, 4eqeq12d 2051 . 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 5463 . . . 4 ([⟨z, w⟩] = B → (A + [⟨z, w⟩] ) = (A + B))
76oveq1d 5470 . . 3 ([⟨z, w⟩] = B → ((A + [⟨z, w⟩] ) + [⟨v, u⟩] ) = ((A + B) + [⟨v, u⟩] ))
8 oveq1 5462 . . . 4 ([⟨z, w⟩] = B → ([⟨z, w⟩] + [⟨v, u⟩] ) = (B + [⟨v, u⟩] ))
98oveq2d 5471 . . 3 ([⟨z, w⟩] = B → (A + ([⟨z, w⟩] + [⟨v, u⟩] )) = (A + (B + [⟨v, u⟩] )))
107, 9eqeq12d 2051 . 2 ([⟨z, w⟩] = B → (((A + [⟨z, w⟩] ) + [⟨v, u⟩] ) = (A + ([⟨z, w⟩] + [⟨v, u⟩] )) ↔ ((A + B) + [⟨v, u⟩] ) = (A + (B + [⟨v, u⟩] ))))
11 oveq2 5463 . . 3 ([⟨v, u⟩] = 𝐶 → ((A + B) + [⟨v, u⟩] ) = ((A + B) + 𝐶))
12 oveq2 5463 . . . 4 ([⟨v, u⟩] = 𝐶 → (B + [⟨v, u⟩] ) = (B + 𝐶))
1312oveq2d 5471 . . 3 ([⟨v, u⟩] = 𝐶 → (A + (B + [⟨v, u⟩] )) = (A + (B + 𝐶)))
1411, 13eqeq12d 2051 . 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 3542 . . . . 5 ((𝐽 = 𝐿 𝐾 = 𝑀) → ⟨𝐽, 𝐾⟩ = ⟨𝐿, 𝑀⟩)
1817eceq1d 6078 . . . 4 ((𝐽 = 𝐿 𝐾 = 𝑀) → [⟨𝐽, 𝐾⟩] = [⟨𝐿, 𝑀⟩] )
1915, 16, 18mp2an 402 . . 3 [⟨𝐽, 𝐾⟩] = [⟨𝐿, 𝑀⟩]
20 ecovass.2 . . . . . . 7 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → ([⟨x, y⟩] + [⟨z, w⟩] ) = [⟨𝐺, 𝐻⟩] )
2120oveq1d 5470 . . . . . 6 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → (([⟨x, y⟩] + [⟨z, w⟩] ) + [⟨v, u⟩] ) = ([⟨𝐺, 𝐻⟩] + [⟨v, u⟩] ))
2221adantr 261 . . . . 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⟩] )
2523, 24sylan 267 . . . . 5 ((((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) (v 𝑆 u 𝑆)) → ([⟨𝐺, 𝐻⟩] + [⟨v, u⟩] ) = [⟨𝐽, 𝐾⟩] )
2622, 25eqtrd 2069 . . . 4 ((((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) (v 𝑆 u 𝑆)) → (([⟨x, y⟩] + [⟨z, w⟩] ) + [⟨v, u⟩] ) = [⟨𝐽, 𝐾⟩] )
27263impa 1098 . . 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⟩] )
2928oveq2d 5471 . . . . . 6 (((z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → ([⟨x, y⟩] + ([⟨z, w⟩] + [⟨v, u⟩] )) = ([⟨x, y⟩] + [⟨𝑁, 𝑄⟩] ))
3029adantl 262 . . . . 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⟩] )
3331, 32sylan2 270 . . . . 5 (((x 𝑆 y 𝑆) ((z 𝑆 w 𝑆) (v 𝑆 u 𝑆))) → ([⟨x, y⟩] + [⟨𝑁, 𝑄⟩] ) = [⟨𝐿, 𝑀⟩] )
3430, 33eqtrd 2069 . . . 4 (((x 𝑆 y 𝑆) ((z 𝑆 w 𝑆) (v 𝑆 u 𝑆))) → ([⟨x, y⟩] + ([⟨z, w⟩] + [⟨v, u⟩] )) = [⟨𝐿, 𝑀⟩] )
35343impb 1099 . . 3 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → ([⟨x, y⟩] + ([⟨z, w⟩] + [⟨v, u⟩] )) = [⟨𝐿, 𝑀⟩] )
3619, 27, 353eqtr4a 2095 . 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⟩] )))
371, 5, 10, 14, 363ecoptocl 6131 1 ((A 𝐷 B 𝐷 𝐶 𝐷) → ((A + B) + 𝐶) = (A + (B + 𝐶)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242   wcel 1390  cop 3370   × cxp 4286  (class class class)co 5455  [cec 6040   / cqs 6041
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fv 4853  df-ov 5458  df-ec 6044  df-qs 6048
This theorem is referenced by: (None)
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