Theorem hladdid 29265

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 29253	. 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 28997	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴)
6	1, 5	sylan 580	1 ⊢ ((𝑈 ∈ CHilOLD ∧ 𝐴 ∈ 𝑋) → (𝐴𝐺𝑍) = 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ‘cfv 6433  (class class class)co 7275  NrmCVeccnv 28946   +_𝑣 cpv 28947  BaseSetcba 28948  0_veccn0v 28950  CHil_OLDchlo 29247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-1st 7831  df-2nd 7832  df-grpo 28855  df-gid 28856  df-ablo 28907  df-vc 28921  df-nv 28954  df-va 28957  df-ba 28958  df-sm 28959  df-0v 28960  df-nmcv 28962  df-cbn 29225  df-hlo 29248

This theorem is referenced by:  axhvaddid-zf  29348