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 Description: Rearrangement of 4 terms in a sum. (Contributed by NM, 13-Nov-1999.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Assertion
Ref Expression
add4 (((A B ℂ) (𝐶 𝐷 ℂ)) → ((A + B) + (𝐶 + 𝐷)) = ((A + 𝐶) + (B + 𝐷)))

StepHypRef Expression
1 add12 6771 . . . . 5 ((B 𝐶 𝐷 ℂ) → (B + (𝐶 + 𝐷)) = (𝐶 + (B + 𝐷)))
213expb 1089 . . . 4 ((B (𝐶 𝐷 ℂ)) → (B + (𝐶 + 𝐷)) = (𝐶 + (B + 𝐷)))
32oveq2d 5448 . . 3 ((B (𝐶 𝐷 ℂ)) → (A + (B + (𝐶 + 𝐷))) = (A + (𝐶 + (B + 𝐷))))
43adantll 448 . 2 (((A B ℂ) (𝐶 𝐷 ℂ)) → (A + (B + (𝐶 + 𝐷))) = (A + (𝐶 + (B + 𝐷))))
5 addcl 6609 . . 3 ((𝐶 𝐷 ℂ) → (𝐶 + 𝐷) ℂ)
6 addass 6614 . . . 4 ((A B (𝐶 + 𝐷) ℂ) → ((A + B) + (𝐶 + 𝐷)) = (A + (B + (𝐶 + 𝐷))))
763expa 1088 . . 3 (((A B ℂ) (𝐶 + 𝐷) ℂ) → ((A + B) + (𝐶 + 𝐷)) = (A + (B + (𝐶 + 𝐷))))
85, 7sylan2 270 . 2 (((A B ℂ) (𝐶 𝐷 ℂ)) → ((A + B) + (𝐶 + 𝐷)) = (A + (B + (𝐶 + 𝐷))))
9 addcl 6609 . . . 4 ((B 𝐷 ℂ) → (B + 𝐷) ℂ)
10 addass 6614 . . . . 5 ((A 𝐶 (B + 𝐷) ℂ) → ((A + 𝐶) + (B + 𝐷)) = (A + (𝐶 + (B + 𝐷))))
11103expa 1088 . . . 4 (((A 𝐶 ℂ) (B + 𝐷) ℂ) → ((A + 𝐶) + (B + 𝐷)) = (A + (𝐶 + (B + 𝐷))))
129, 11sylan2 270 . . 3 (((A 𝐶 ℂ) (B 𝐷 ℂ)) → ((A + 𝐶) + (B + 𝐷)) = (A + (𝐶 + (B + 𝐷))))
1312an4s 509 . 2 (((A B ℂ) (𝐶 𝐷 ℂ)) → ((A + 𝐶) + (B + 𝐷)) = (A + (𝐶 + (B + 𝐷))))
144, 8, 133eqtr4d 2060 1 (((A B ℂ) (𝐶 𝐷 ℂ)) → ((A + B) + (𝐶 + 𝐷)) = ((A + 𝐶) + (B + 𝐷)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1226   ∈ wcel 1370  (class class class)co 5432  ℂcc 6522   + caddc 6527 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-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-addcl 6586  ax-addcom 6590  ax-addass 6592 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-rex 2286  df-v 2533  df-un 2895  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-iota 4790  df-fv 4833  df-ov 5435 This theorem is referenced by:  add42  6775  add4i  6778  add4d  6782
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