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Mirrors > Home > MPE Home > Th. List > hlphl | Structured version Visualization version GIF version |
Description: Every complex Hilbert space is an inner product space (also called a pre-Hilbert space). (Contributed by NM, 28-Apr-2007.) (Revised by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
hlphl | ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlcph 22968 | . 2 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil) | |
2 | cphphl 22779 | . 2 ⊢ (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ PreHil) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 PreHilcphl 19788 ℂPreHilccph 22774 ℂHilchl 22939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fv 5812 df-ov 6552 df-cph 22776 df-hl 22942 |
This theorem is referenced by: pjthlem1 23016 pjth 23018 pjth2 23019 cldcss 23020 hlhil 23022 |
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