Theorem hladdid 28683

Description: Hilbert space addition with the zero vector. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hladdid.1	⊢ 𝑋 = (BaseSet‘𝑈)
hladdid.2	⊢ 𝐺 = ( +𝑣 ‘𝑈)
hladdid.5	⊢ 𝑍 = (0vec‘𝑈)

Assertion

Ref	Expression
hladdid	⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴)

Proof of Theorem hladdid

Step	Hyp	Ref	Expression
1		hlnv 28671	. 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec)
2		hladdid.1	. . 3 ⊢ 𝑋 = (BaseSet‘𝑈)
3		hladdid.2	. . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
4		hladdid.5	. . 3 ⊢ 𝑍 = (0vec‘𝑈)
5	2, 3, 4	nv0rid 28415	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴)
6	1, 5	sylan 582	1 ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113  ‘cfv 6358  (class class class)co 7159  NrmCVeccnv 28364   +_𝑣 cpv 28365  BaseSetcba 28366  0_veccn0v 28368  CHil_OLDchlo 28665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-1st 7692  df-2nd 7693  df-grpo 28273  df-gid 28274  df-ablo 28325  df-vc 28339  df-nv 28372  df-va 28375  df-ba 28376  df-sm 28377  df-0v 28378  df-nmcv 28380  df-cbn 28643  df-hlo 28666

This theorem is referenced by:  axhvaddid-zf  28766