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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 30808 | . 2 ⊢ ·ℎ :(ℂ × ℋ)⟶ ℋ | |
2 | cnex 11213 | . . 3 ⊢ ℂ ∈ V | |
3 | ax-hilex 30802 | . . 3 ⊢ ℋ ∈ V | |
4 | 2, 3 | xpex 7749 | . 2 ⊢ (ℂ × ℋ) ∈ V |
5 | fex 7232 | . 2 ⊢ (( ·ℎ :(ℂ × ℋ)⟶ ℋ ∧ (ℂ × ℋ) ∈ V) → ·ℎ ∈ V) | |
6 | 1, 4, 5 | mp2an 691 | 1 ⊢ ·ℎ ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2099 Vcvv 3469 × cxp 5670 ⟶wf 6538 ℂcc 11130 ℋchba 30722 ·ℎ csm 30724 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-hilex 30802 ax-hfvmul 30808 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 |
This theorem is referenced by: hhph 30981 hhssva 31060 hhsssm 31061 hhshsslem1 31070 |
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