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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 27990 | . 2 ⊢ ·ℎ :(ℂ × ℋ)⟶ ℋ | |
2 | cnex 10055 | . . 3 ⊢ ℂ ∈ V | |
3 | ax-hilex 27984 | . . 3 ⊢ ℋ ∈ V | |
4 | 2, 3 | xpex 7004 | . 2 ⊢ (ℂ × ℋ) ∈ V |
5 | fex 6530 | . 2 ⊢ (( ·ℎ :(ℂ × ℋ)⟶ ℋ ∧ (ℂ × ℋ) ∈ V) → ·ℎ ∈ V) | |
6 | 1, 4, 5 | mp2an 708 | 1 ⊢ ·ℎ ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2030 Vcvv 3231 × cxp 5141 ⟶wf 5922 ℂcc 9972 ℋchil 27904 ·ℎ csm 27906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-hilex 27984 ax-hfvmul 27990 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 |
This theorem is referenced by: hhph 28163 hhssva 28242 hhsssm 28243 hhshsslem1 28252 |
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