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Mirrors > Home > MPE Home > Th. List > hladdf | Structured version Visualization version GIF version |
Description: Mapping for Hilbert space vector addition. (Contributed by NM, 7-Sep-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hladdf.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
hladdf.2 | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
Ref | Expression |
---|---|
hladdf | ⊢ (𝑈 ∈ CHilOLD → 𝐺:(𝑋 × 𝑋)⟶𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlnv 27131 | . 2 ⊢ (𝑈 ∈ CHilOLD → 𝑈 ∈ NrmCVec) | |
2 | hladdf.1 | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
3 | hladdf.2 | . . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈) | |
4 | 2, 3 | nvgf 26857 | . 2 ⊢ (𝑈 ∈ NrmCVec → 𝐺:(𝑋 × 𝑋)⟶𝑋) |
5 | 1, 4 | syl 17 | 1 ⊢ (𝑈 ∈ CHilOLD → 𝐺:(𝑋 × 𝑋)⟶𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 × cxp 5036 ⟶wf 5800 ‘cfv 5804 NrmCVeccnv 26823 +𝑣 cpv 26824 BaseSetcba 26825 CHilOLDchlo 27125 |
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 |
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-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 df-ov 6552 df-oprab 6553 df-1st 7059 df-2nd 7060 df-grpo 26731 df-ablo 26783 df-vc 26798 df-nv 26831 df-va 26834 df-ba 26835 df-sm 26836 df-0v 26837 df-nmcv 26839 df-cbn 27103 df-hlo 27126 |
This theorem is referenced by: axhfvadd-zf 27223 |
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