HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  hvadd32 Structured version   Visualization version   GIF version

Theorem hvadd32 28225
Description: Commutative/associative law. (Contributed by NM, 16-Oct-1999.) (New usage is discouraged.)
Assertion
Ref Expression
hvadd32 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵))

Proof of Theorem hvadd32
StepHypRef Expression
1 ax-hvcom 28192 . . . 4 ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐵 + 𝐶) = (𝐶 + 𝐵))
21oveq2d 6808 . . 3 ((𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 + (𝐵 + 𝐶)) = (𝐴 + (𝐶 + 𝐵)))
323adant1 1123 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → (𝐴 + (𝐵 + 𝐶)) = (𝐴 + (𝐶 + 𝐵)))
4 ax-hvass 28193 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))
5 ax-hvass 28193 . . 3 ((𝐴 ∈ ℋ ∧ 𝐶 ∈ ℋ ∧ 𝐵 ∈ ℋ) → ((𝐴 + 𝐶) + 𝐵) = (𝐴 + (𝐶 + 𝐵)))
653com23 1119 . 2 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐶) + 𝐵) = (𝐴 + (𝐶 + 𝐵)))
73, 4, 63eqtr4d 2814 1 ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ ∧ 𝐶 ∈ ℋ) → ((𝐴 + 𝐵) + 𝐶) = ((𝐴 + 𝐶) + 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1070   = wceq 1630  wcel 2144  (class class class)co 6792  chil 28110   + cva 28111
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-hvcom 28192  ax-hvass 28193
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-iota 5994  df-fv 6039  df-ov 6795
This theorem is referenced by:  hvadd4  28227  hvadd32i  28245
  Copyright terms: Public domain W3C validator