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Mirrors > Home > MPE Home > Th. List > hlphl | Structured version Visualization version GIF version |
Description: Every subcomplex 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 23660 | . 2 ⊢ (𝑊 ∈ ℂHil → 𝑊 ∈ ℂPreHil) | |
2 | cphphl 23468 | . 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 2048 PreHilcphl 20460 ℂPreHilccph 23463 ℂHilchl 23630 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-ext 2745 ax-nul 5061 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3678 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-nul 4174 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-mpt 5003 df-xp 5406 df-cnv 5408 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-iota 6146 df-fv 6190 df-ov 6973 df-cph 23465 df-hl 23633 |
This theorem is referenced by: chlcsschl 23674 pjthlem1 23733 pjth 23735 pjth2 23736 cldcss 23737 hlhil 23739 |
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