Hilbert Space Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  HSE Home  >  Th. List  >  hvadd4i Structured version   Visualization version   GIF version

 Description: Hilbert vector space addition law. (Contributed by NM, 3-Sep-1999.) (New usage is discouraged.)
Hypotheses
Ref Expression
hvass.1 𝐴 ∈ ℋ
hvass.2 𝐵 ∈ ℋ
hvass.3 𝐶 ∈ ℋ
hvadd4.4 𝐷 ∈ ℋ
Assertion
Ref Expression
hvadd4i ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))

Proof of Theorem hvadd4i
StepHypRef Expression
1 hvass.1 . 2 𝐴 ∈ ℋ
2 hvass.2 . 2 𝐵 ∈ ℋ
3 hvass.3 . 2 𝐶 ∈ ℋ
4 hvadd4.4 . 2 𝐷 ∈ ℋ
5 hvadd4 28817 . 2 (((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) ∧ (𝐶 ∈ ℋ ∧ 𝐷 ∈ ℋ)) → ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷)))
61, 2, 3, 4, 5mp4an 692 1 ((𝐴 + 𝐵) + (𝐶 + 𝐷)) = ((𝐴 + 𝐶) + (𝐵 + 𝐷))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2114  (class class class)co 7140   ℋchba 28700   +ℎ cva 28701 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-hfvadd 28781  ax-hvcom 28782  ax-hvass 28783 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-ov 7143 This theorem is referenced by:  hvsubsub4i  28840  hvsubcan2i  28845  pjaddii  29456
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