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Theorem ecovidi 6154
Description: Lemma used to transfer a distributive law via an equivalence relation. (Contributed by Jim Kingdon, 17-Sep-2019.)
Hypotheses
Ref Expression
ecovidi.1 𝐷 = ((𝑆 × 𝑆) / )
ecovidi.2 (((z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → ([⟨z, w⟩] + [⟨v, u⟩] ) = [⟨𝑀, 𝑁⟩] )
ecovidi.3 (((x 𝑆 y 𝑆) (𝑀 𝑆 𝑁 𝑆)) → ([⟨x, y⟩] · [⟨𝑀, 𝑁⟩] ) = [⟨𝐻, 𝐽⟩] )
ecovidi.4 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → ([⟨x, y⟩] · [⟨z, w⟩] ) = [⟨𝑊, 𝑋⟩] )
ecovidi.5 (((x 𝑆 y 𝑆) (v 𝑆 u 𝑆)) → ([⟨x, y⟩] · [⟨v, u⟩] ) = [⟨𝑌, 𝑍⟩] )
ecovidi.6 (((𝑊 𝑆 𝑋 𝑆) (𝑌 𝑆 𝑍 𝑆)) → ([⟨𝑊, 𝑋⟩] + [⟨𝑌, 𝑍⟩] ) = [⟨𝐾, 𝐿⟩] )
ecovidi.7 (((z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → (𝑀 𝑆 𝑁 𝑆))
ecovidi.8 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → (𝑊 𝑆 𝑋 𝑆))
ecovidi.9 (((x 𝑆 y 𝑆) (v 𝑆 u 𝑆)) → (𝑌 𝑆 𝑍 𝑆))
ecovidi.10 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → 𝐻 = 𝐾)
ecovidi.11 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → 𝐽 = 𝐿)
Assertion
Ref Expression
ecovidi ((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 ecovidi
StepHypRef Expression
1 ecovidi.1 . 2 𝐷 = ((𝑆 × 𝑆) / )
2 oveq1 5462 . . 3 ([⟨x, y⟩] = A → ([⟨x, y⟩] · ([⟨z, w⟩] + [⟨v, u⟩] )) = (A · ([⟨z, w⟩] + [⟨v, u⟩] )))
3 oveq1 5462 . . . 4 ([⟨x, y⟩] = A → ([⟨x, y⟩] · [⟨z, w⟩] ) = (A · [⟨z, w⟩] ))
4 oveq1 5462 . . . 4 ([⟨x, y⟩] = A → ([⟨x, y⟩] · [⟨v, u⟩] ) = (A · [⟨v, u⟩] ))
53, 4oveq12d 5473 . . 3 ([⟨x, y⟩] = A → (([⟨x, y⟩] · [⟨z, w⟩] ) + ([⟨x, y⟩] · [⟨v, u⟩] )) = ((A · [⟨z, w⟩] ) + (A · [⟨v, u⟩] )))
62, 5eqeq12d 2051 . 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 5462 . . . 4 ([⟨z, w⟩] = B → ([⟨z, w⟩] + [⟨v, u⟩] ) = (B + [⟨v, u⟩] ))
87oveq2d 5471 . . 3 ([⟨z, w⟩] = B → (A · ([⟨z, w⟩] + [⟨v, u⟩] )) = (A · (B + [⟨v, u⟩] )))
9 oveq2 5463 . . . 4 ([⟨z, w⟩] = B → (A · [⟨z, w⟩] ) = (A · B))
109oveq1d 5470 . . 3 ([⟨z, w⟩] = B → ((A · [⟨z, w⟩] ) + (A · [⟨v, u⟩] )) = ((A · B) + (A · [⟨v, u⟩] )))
118, 10eqeq12d 2051 . 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 5463 . . . 4 ([⟨v, u⟩] = 𝐶 → (B + [⟨v, u⟩] ) = (B + 𝐶))
1312oveq2d 5471 . . 3 ([⟨v, u⟩] = 𝐶 → (A · (B + [⟨v, u⟩] )) = (A · (B + 𝐶)))
14 oveq2 5463 . . . 4 ([⟨v, u⟩] = 𝐶 → (A · [⟨v, u⟩] ) = (A · 𝐶))
1514oveq2d 5471 . . 3 ([⟨v, u⟩] = 𝐶 → ((A · B) + (A · [⟨v, u⟩] )) = ((A · B) + (A · 𝐶)))
1613, 15eqeq12d 2051 . 2 ([⟨v, u⟩] = 𝐶 → ((A · (B + [⟨v, u⟩] )) = ((A · B) + (A · [⟨v, u⟩] )) ↔ (A · (B + 𝐶)) = ((A · B) + (A · 𝐶))))
17 ecovidi.10 . . . 4 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → 𝐻 = 𝐾)
18 ecovidi.11 . . . 4 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → 𝐽 = 𝐿)
19 opeq12 3542 . . . . 5 ((𝐻 = 𝐾 𝐽 = 𝐿) → ⟨𝐻, 𝐽⟩ = ⟨𝐾, 𝐿⟩)
2019eceq1d 6078 . . . 4 ((𝐻 = 𝐾 𝐽 = 𝐿) → [⟨𝐻, 𝐽⟩] = [⟨𝐾, 𝐿⟩] )
2117, 18, 20syl2anc 391 . . 3 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → [⟨𝐻, 𝐽⟩] = [⟨𝐾, 𝐿⟩] )
22 ecovidi.2 . . . . . . 7 (((z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → ([⟨z, w⟩] + [⟨v, u⟩] ) = [⟨𝑀, 𝑁⟩] )
2322oveq2d 5471 . . . . . 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 ecovidi.7 . . . . . 6 (((z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → (𝑀 𝑆 𝑁 𝑆))
26 ecovidi.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 2069 . . . 4 (((x 𝑆 y 𝑆) ((z 𝑆 w 𝑆) (v 𝑆 u 𝑆))) → ([⟨x, y⟩] · ([⟨z, w⟩] + [⟨v, u⟩] )) = [⟨𝐻, 𝐽⟩] )
29283impb 1099 . . 3 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆) (v 𝑆 u 𝑆)) → ([⟨x, y⟩] · ([⟨z, w⟩] + [⟨v, u⟩] )) = [⟨𝐻, 𝐽⟩] )
30 ecovidi.4 . . . . . 6 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → ([⟨x, y⟩] · [⟨z, w⟩] ) = [⟨𝑊, 𝑋⟩] )
31 ecovidi.5 . . . . . 6 (((x 𝑆 y 𝑆) (v 𝑆 u 𝑆)) → ([⟨x, y⟩] · [⟨v, u⟩] ) = [⟨𝑌, 𝑍⟩] )
3230, 31oveqan12d 5474 . . . . 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 ecovidi.8 . . . . . 6 (((x 𝑆 y 𝑆) (z 𝑆 w 𝑆)) → (𝑊 𝑆 𝑋 𝑆))
34 ecovidi.9 . . . . . 6 (((x 𝑆 y 𝑆) (v 𝑆 u 𝑆)) → (𝑌 𝑆 𝑍 𝑆))
35 ecovidi.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 2069 . . . 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 1189 . . 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, 383eqtr4d 2079 . 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 6131 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 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:  distrnqg  6371  distrsrg  6647  axdistr  6718
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