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Mirrors > Home > HSE Home > Th. List > hvmulex | Structured version Visualization version GIF version |
Description: The Hilbert space scalar product operation is a set. (Contributed by NM, 17-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hvmulex | ⊢ ·ℎ ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-hfvmul 27246 | . 2 ⊢ ·ℎ :(ℂ × ℋ)⟶ ℋ | |
2 | cnex 9896 | . . 3 ⊢ ℂ ∈ V | |
3 | ax-hilex 27240 | . . 3 ⊢ ℋ ∈ V | |
4 | 2, 3 | xpex 6860 | . 2 ⊢ (ℂ × ℋ) ∈ V |
5 | fex 6394 | . 2 ⊢ (( ·ℎ :(ℂ × ℋ)⟶ ℋ ∧ (ℂ × ℋ) ∈ V) → ·ℎ ∈ V) | |
6 | 1, 4, 5 | mp2an 704 | 1 ⊢ ·ℎ ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 Vcvv 3173 × cxp 5036 ⟶wf 5800 ℂcc 9813 ℋchil 27160 ·ℎ csm 27162 |
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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-hilex 27240 ax-hfvmul 27246 |
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-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 |
This theorem is referenced by: hhph 27419 hhssva 27498 hhsssm 27499 hhshsslem1 27508 |
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