Theorem hicli 27938

Description: Closure inference for inner product. (Contributed by NM, 1-Aug-1999.) (New usage is discouraged.)

Hypotheses

Ref	Expression
hicl.1	⊢ 𝐴 ∈ ℋ
hicl.2	⊢ 𝐵 ∈ ℋ

Assertion

Ref	Expression
hicli	⊢ (𝐴 ·ih 𝐵) ∈ ℂ

Proof of Theorem hicli

Step	Hyp	Ref	Expression
1		hicl.1	. 2 ⊢ 𝐴 ∈ ℋ
2		hicl.2	. 2 ⊢ 𝐵 ∈ ℋ
3		hicl 27937	. 2 ⊢ ((𝐴 ∈ ℋ ∧ 𝐵 ∈ ℋ) → (𝐴 ·ih 𝐵) ∈ ℂ)
4	1, 2, 3	mp2an 708	1 ⊢ (𝐴 ·ih 𝐵) ∈ ℂ

Syntax hints:   ∈ wcel 1990  (class class class)co 6650  ℂcc 9934   ℋchil 27776   ·_ih csp 27779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-hfi 27936

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653

This theorem is referenced by:  hisubcomi  27961  normlem0  27966  normlem2  27968  normlem3  27969  normlem7  27973  normlem8  27974  normlem9  27975  bcseqi  27977  norm-ii-i  27994  normpythi  27999  normpari  28011  polid2i  28014  bcsiALT  28036  h1de2i  28412  h1de2bi  28413  h1de2ctlem  28414  eigrei  28693  eigorthi  28696  lnopunilem1  28869  lnopunilem2  28870