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Theorem ecovdi 6124
 Description: Lemma used to transfer a distributive law via an equivalence relation. Most likely ecovidi 6125 will be more helpful. (Contributed by NM, 2-Sep-1995.) (Revised by David Abernethy, 4-Jun-2013.)
Hypotheses
Ref Expression
ecovdi.1 𝐷 = ((𝑆 × 𝑆) / )
ecovdi.2 (((z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → ([⟨z, w⟩] + [⟨v, u⟩] ) = [⟨𝑀, 𝑁⟩] )
ecovdi.3 (((x 𝑆 y 𝑆) (𝑀 𝑆 𝑁 𝑆)) → ([⟨x, y⟩] · [⟨𝑀, 𝑁⟩] ) = [⟨𝐻, 𝐽⟩] )
ecovdi.4 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → ([⟨x, y⟩] · [⟨z, w⟩] ) = [⟨𝑊, 𝑋⟩] )
ecovdi.5 (((x 𝑆 y 𝑆) (v 𝑆 u 𝑆)) → ([⟨x, y⟩] · [⟨v, u⟩] ) = [⟨𝑌, 𝑍⟩] )
ecovdi.6 (((𝑊 𝑆 𝑋 𝑆) (𝑌 𝑆 𝑍 𝑆)) → ([⟨𝑊, 𝑋⟩] + [⟨𝑌, 𝑍⟩] ) = [⟨𝐾, 𝐿⟩] )
ecovdi.7 (((z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → (𝑀 𝑆 𝑁 𝑆))
ecovdi.8 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → (𝑊 𝑆 𝑋 𝑆))
ecovdi.9 (((x 𝑆 y 𝑆) (v 𝑆 u 𝑆)) → (𝑌 𝑆 𝑍 𝑆))
ecovdi.10 𝐻 = 𝐾
ecovdi.11 𝐽 = 𝐿
Assertion
Ref Expression
ecovdi ((A 𝐷 B 𝐷 𝐶 𝐷) → (A · (B + 𝐶)) = ((A · B) + (A · 𝐶)))
Distinct variable groups:   x,y,z,w,v,u,A   z,B,w,v,u   w,𝐶,v,u   x, + ,y,z,w,v,u   x, ,y,z,w,v,u   x,𝑆,y,z,w,v,u   x, · ,y,z,w,v,u   z,𝐷,w,v,u
Allowed substitution hints:   B(x,y)   𝐶(x,y,z)   𝐷(x,y)   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)   𝑊(x,y,z,w,v,u)   𝑋(x,y,z,w,v,u)   𝑌(x,y,z,w,v,u)   𝑍(x,y,z,w,v,u)

Proof of Theorem ecovdi
StepHypRef Expression
1 ecovdi.1 . 2 𝐷 = ((𝑆 × 𝑆) / )
2 oveq1 5439 . . 3 ([⟨x, y⟩] = A → ([⟨x, y⟩] · ([⟨z, w⟩] + [⟨v, u⟩] )) = (A · ([⟨z, w⟩] + [⟨v, u⟩] )))
3 oveq1 5439 . . . 4 ([⟨x, y⟩] = A → ([⟨x, y⟩] · [⟨z, w⟩] ) = (A · [⟨z, w⟩] ))
4 oveq1 5439 . . . 4 ([⟨x, y⟩] = A → ([⟨x, y⟩] · [⟨v, u⟩] ) = (A · [⟨v, u⟩] ))
53, 4oveq12d 5450 . . 3 ([⟨x, y⟩] = A → (([⟨x, y⟩] · [⟨z, w⟩] ) + ([⟨x, y⟩] · [⟨v, u⟩] )) = ((A · [⟨z, w⟩] ) + (A · [⟨v, u⟩] )))
62, 5eqeq12d 2032 . 2 ([⟨x, y⟩] = A → (([⟨x, y⟩] · ([⟨z, w⟩] + [⟨v, u⟩] )) = (([⟨x, y⟩] · [⟨z, w⟩] ) + ([⟨x, y⟩] · [⟨v, u⟩] )) ↔ (A · ([⟨z, w⟩] + [⟨v, u⟩] )) = ((A · [⟨z, w⟩] ) + (A · [⟨v, u⟩] ))))
7 oveq1 5439 . . . 4 ([⟨z, w⟩] = B → ([⟨z, w⟩] + [⟨v, u⟩] ) = (B + [⟨v, u⟩] ))
87oveq2d 5448 . . 3 ([⟨z, w⟩] = B → (A · ([⟨z, w⟩] + [⟨v, u⟩] )) = (A · (B + [⟨v, u⟩] )))
9 oveq2 5440 . . . 4 ([⟨z, w⟩] = B → (A · [⟨z, w⟩] ) = (A · B))
109oveq1d 5447 . . 3 ([⟨z, w⟩] = B → ((A · [⟨z, w⟩] ) + (A · [⟨v, u⟩] )) = ((A · B) + (A · [⟨v, u⟩] )))
118, 10eqeq12d 2032 . 2 ([⟨z, w⟩] = B → ((A · ([⟨z, w⟩] + [⟨v, u⟩] )) = ((A · [⟨z, w⟩] ) + (A · [⟨v, u⟩] )) ↔ (A · (B + [⟨v, u⟩] )) = ((A · B) + (A · [⟨v, u⟩] ))))
12 oveq2 5440 . . . 4 ([⟨v, u⟩] = 𝐶 → (B + [⟨v, u⟩] ) = (B + 𝐶))
1312oveq2d 5448 . . 3 ([⟨v, u⟩] = 𝐶 → (A · (B + [⟨v, u⟩] )) = (A · (B + 𝐶)))
14 oveq2 5440 . . . 4 ([⟨v, u⟩] = 𝐶 → (A · [⟨v, u⟩] ) = (A · 𝐶))
1514oveq2d 5448 . . 3 ([⟨v, u⟩] = 𝐶 → ((A · B) + (A · [⟨v, u⟩] )) = ((A · B) + (A · 𝐶)))
1613, 15eqeq12d 2032 . 2 ([⟨v, u⟩] = 𝐶 → ((A · (B + [⟨v, u⟩] )) = ((A · B) + (A · [⟨v, u⟩] )) ↔ (A · (B + 𝐶)) = ((A · B) + (A · 𝐶))))
17 ecovdi.10 . . . 4 𝐻 = 𝐾
18 ecovdi.11 . . . 4 𝐽 = 𝐿
19 opeq12 3521 . . . . 5 ((𝐻 = 𝐾 𝐽 = 𝐿) → ⟨𝐻, 𝐽⟩ = ⟨𝐾, 𝐿⟩)
2019eceq1d 6049 . . . 4 ((𝐻 = 𝐾 𝐽 = 𝐿) → [⟨𝐻, 𝐽⟩] = [⟨𝐾, 𝐿⟩] )
2117, 18, 20mp2an 404 . . 3 [⟨𝐻, 𝐽⟩] = [⟨𝐾, 𝐿⟩]
22 ecovdi.2 . . . . . . 7 (((z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → ([⟨z, w⟩] + [⟨v, u⟩] ) = [⟨𝑀, 𝑁⟩] )
2322oveq2d 5448 . . . . . 6 (((z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → ([⟨x, y⟩] · ([⟨z, w⟩] + [⟨v, u⟩] )) = ([⟨x, y⟩] · [⟨𝑀, 𝑁⟩] ))
2423adantl 262 . . . . 5 (((x 𝑆 y 𝑆) ((z 𝑆 w 𝑆) (v 𝑆 u 𝑆))) → ([⟨x, y⟩] · ([⟨z, w⟩] + [⟨v, u⟩] )) = ([⟨x, y⟩] · [⟨𝑀, 𝑁⟩] ))
25 ecovdi.7 . . . . . 6 (((z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → (𝑀 𝑆 𝑁 𝑆))
26 ecovdi.3 . . . . . 6 (((x 𝑆 y 𝑆) (𝑀 𝑆 𝑁 𝑆)) → ([⟨x, y⟩] · [⟨𝑀, 𝑁⟩] ) = [⟨𝐻, 𝐽⟩] )
2725, 26sylan2 270 . . . . 5 (((x 𝑆 y 𝑆) ((z 𝑆 w 𝑆) (v 𝑆 u 𝑆))) → ([⟨x, y⟩] · [⟨𝑀, 𝑁⟩] ) = [⟨𝐻, 𝐽⟩] )
2824, 27eqtrd 2050 . . . 4 (((x 𝑆 y 𝑆) ((z 𝑆 w 𝑆) (v 𝑆 u 𝑆))) → ([⟨x, y⟩] · ([⟨z, w⟩] + [⟨v, u⟩] )) = [⟨𝐻, 𝐽⟩] )
29283impb 1084 . . 3 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → ([⟨x, y⟩] · ([⟨z, w⟩] + [⟨v, u⟩] )) = [⟨𝐻, 𝐽⟩] )
30 ecovdi.4 . . . . . 6 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → ([⟨x, y⟩] · [⟨z, w⟩] ) = [⟨𝑊, 𝑋⟩] )
31 ecovdi.5 . . . . . 6 (((x 𝑆 y 𝑆) (v 𝑆 u 𝑆)) → ([⟨x, y⟩] · [⟨v, u⟩] ) = [⟨𝑌, 𝑍⟩] )
3230, 31oveqan12d 5451 . . . . 5 ((((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) ((x 𝑆 y 𝑆) (v 𝑆 u 𝑆))) → (([⟨x, y⟩] · [⟨z, w⟩] ) + ([⟨x, y⟩] · [⟨v, u⟩] )) = ([⟨𝑊, 𝑋⟩] + [⟨𝑌, 𝑍⟩] ))
33 ecovdi.8 . . . . . 6 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → (𝑊 𝑆 𝑋 𝑆))
34 ecovdi.9 . . . . . 6 (((x 𝑆 y 𝑆) (v 𝑆 u 𝑆)) → (𝑌 𝑆 𝑍 𝑆))
35 ecovdi.6 . . . . . 6 (((𝑊 𝑆 𝑋 𝑆) (𝑌 𝑆 𝑍 𝑆)) → ([⟨𝑊, 𝑋⟩] + [⟨𝑌, 𝑍⟩] ) = [⟨𝐾, 𝐿⟩] )
3633, 34, 35syl2an 273 . . . . 5 ((((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) ((x 𝑆 y 𝑆) (v 𝑆 u 𝑆))) → ([⟨𝑊, 𝑋⟩] + [⟨𝑌, 𝑍⟩] ) = [⟨𝐾, 𝐿⟩] )
3732, 36eqtrd 2050 . . . 4 ((((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) ((x 𝑆 y 𝑆) (v 𝑆 u 𝑆))) → (([⟨x, y⟩] · [⟨z, w⟩] ) + ([⟨x, y⟩] · [⟨v, u⟩] )) = [⟨𝐾, 𝐿⟩] )
38373impdi 1174 . . 3 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → (([⟨x, y⟩] · [⟨z, w⟩] ) + ([⟨x, y⟩] · [⟨v, u⟩] )) = [⟨𝐾, 𝐿⟩] )
3921, 29, 383eqtr4a 2076 . 2 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → ([⟨x, y⟩] · ([⟨z, w⟩] + [⟨v, u⟩] )) = (([⟨x, y⟩] · [⟨z, w⟩] ) + ([⟨x, y⟩] · [⟨v, u⟩] )))
401, 6, 11, 16, 393ecoptocl 6102 1 ((A 𝐷 B 𝐷 𝐶 𝐷) → (A · (B + 𝐶)) = ((A · B) + (A · 𝐶)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∧ w3a 871   = wceq 1226   ∈ wcel 1370  ⟨cop 3349   × cxp 4266  (class class class)co 5432  [cec 6011   / cqs 6012 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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-xp 4274  df-cnv 4276  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fv 4833  df-ov 5435  df-ec 6015  df-qs 6019 This theorem is referenced by: (None)
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