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Mirrors > Home > MPE Home > Th. List > sspid | Structured version Visualization version GIF version |
Description: A normed complex vector space is a subspace of itself. (Contributed by NM, 8-Apr-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sspid.h | ⊢ 𝐻 = (SubSp‘𝑈) |
Ref | Expression |
---|---|
sspid | ⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3937 | . . . 4 ⊢ ( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈) | |
2 | ssid 3937 | . . . 4 ⊢ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈) | |
3 | ssid 3937 | . . . 4 ⊢ (normCV‘𝑈) ⊆ (normCV‘𝑈) | |
4 | 1, 2, 3 | 3pm3.2i 1336 | . . 3 ⊢ (( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑈) ⊆ (normCV‘𝑈)) |
5 | 4 | jctr 528 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ∧ (( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑈) ⊆ (normCV‘𝑈)))) |
6 | eqid 2798 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
7 | eqid 2798 | . . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
8 | eqid 2798 | . . 3 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
9 | sspid.h | . . 3 ⊢ 𝐻 = (SubSp‘𝑈) | |
10 | 6, 6, 7, 7, 8, 8, 9 | isssp 28507 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ NrmCVec ∧ (( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑈) ⊆ (normCV‘𝑈))))) |
11 | 5, 10 | mpbird 260 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ‘cfv 6324 NrmCVeccnv 28367 +𝑣 cpv 28368 ·𝑠OLD cns 28370 normCVcnmcv 28373 SubSpcss 28504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 df-oprab 7139 df-1st 7671 df-2nd 7672 df-vc 28342 df-nv 28375 df-va 28378 df-sm 28380 df-nmcv 28383 df-ssp 28505 |
This theorem is referenced by: hhsssh 29052 |
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