Theorem add32d 8187

Description: Commutative/associative law that swaps the last two terms in a triple sum. (Contributed by Mario Carneiro, 27-May-2016.)

Hypotheses

Ref	Expression
addd.1	⊢ (𝜑 → 𝐴 ∈ ℂ)
addd.2	⊢ (𝜑 → 𝐵 ∈ ℂ)
addd.3	⊢ (𝜑 → 𝐶 ∈ ℂ)

Assertion

Ref	Expression
add32d	⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵))

Proof of Theorem add32d

Step	Hyp	Ref	Expression
1		addd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ ℂ)
2		addd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ ℂ)
3		addd.3	. 2 ⊢ (𝜑 → 𝐶 ∈ ℂ)
4		add32 8178	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵))
5	1, 2, 3, 4	syl3anc 1249	1 ⊢ (𝜑 → ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵))

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2164  (class class class)co 5918  ℂcc 7870   + caddc 7875

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-addcom 7972  ax-addass 7974

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-iota 5215  df-fv 5262  df-ov 5921

This theorem is referenced by:  nppcan  8241  muladd  8403  peano5uzti  9425  flqaddz  10366  seq3shft2  10552  zesq  10729  abstri  11248  bdtrilem  11382  pythagtriplem1  12403  pythagtriplem12  12413  gsumfzconst  13411