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Mirrors > Home > MPE Home > Th. List > nvex | Structured version Visualization version Unicode version |
Description: The components of a normed complex vector space are sets. (Contributed by NM, 5-Jun-2008.) (Revised by Mario Carneiro, 1-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvvcop 27449 | . . 3 | |
2 | vcex 27433 | . . 3 | |
3 | 1, 2 | syl 17 | . 2 |
4 | nvss 27448 | . . . 4 | |
5 | 4 | sseli 3599 | . . 3 |
6 | opelxp2 5151 | . . 3 | |
7 | 5, 6 | syl 17 | . 2 |
8 | df-3an 1039 | . 2 | |
9 | 3, 7, 8 | sylanbrc 698 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wcel 1990 cvv 3200 cop 4183 cxp 5112 cvc 27413 cnv 27439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-xp 5120 df-rel 5121 df-oprab 6654 df-vc 27414 df-nv 27447 |
This theorem is referenced by: isnv 27467 h2hva 27831 h2hsm 27832 h2hnm 27833 |
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