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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 3944 | . . . 4 ⊢ ( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈) | |
2 | ssid 3944 | . . . 4 ⊢ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈) | |
3 | ssid 3944 | . . . 4 ⊢ (normCV‘𝑈) ⊆ (normCV‘𝑈) | |
4 | 1, 2, 3 | 3pm3.2i 1337 | . . 3 ⊢ (( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑈) ⊆ (normCV‘𝑈)) |
5 | 4 | jctr 524 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ∧ (( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑈) ⊆ (normCV‘𝑈)))) |
6 | eqid 2737 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
7 | eqid 2737 | . . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
8 | eqid 2737 | . . 3 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
9 | sspid.h | . . 3 ⊢ 𝐻 = (SubSp‘𝑈) | |
10 | 6, 6, 7, 7, 8, 8, 9 | isssp 29027 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ NrmCVec ∧ (( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑈) ⊆ (normCV‘𝑈))))) |
11 | 5, 10 | mpbird 256 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2107 ⊆ wss 3888 ‘cfv 6423 NrmCVeccnv 28887 +𝑣 cpv 28888 ·𝑠OLD cns 28890 normCVcnmcv 28893 SubSpcss 29024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7571 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3429 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-br 5076 df-opab 5138 df-mpt 5159 df-id 5485 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-iota 6381 df-fun 6425 df-fn 6426 df-f 6427 df-fo 6429 df-fv 6431 df-oprab 7264 df-1st 7809 df-2nd 7810 df-vc 28862 df-nv 28895 df-va 28898 df-sm 28900 df-nmcv 28903 df-ssp 29025 |
This theorem is referenced by: hhsssh 29572 |
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