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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 3989 | . . . 4 ⊢ ( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈) | |
2 | ssid 3989 | . . . 4 ⊢ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈) | |
3 | ssid 3989 | . . . 4 ⊢ (normCV‘𝑈) ⊆ (normCV‘𝑈) | |
4 | 1, 2, 3 | 3pm3.2i 1335 | . . 3 ⊢ (( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑈) ⊆ (normCV‘𝑈)) |
5 | 4 | jctr 527 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ NrmCVec ∧ (( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑈) ⊆ (normCV‘𝑈)))) |
6 | eqid 2821 | . . 3 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
7 | eqid 2821 | . . 3 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
8 | eqid 2821 | . . 3 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
9 | sspid.h | . . 3 ⊢ 𝐻 = (SubSp‘𝑈) | |
10 | 6, 6, 7, 7, 8, 8, 9 | isssp 28501 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑈 ∈ 𝐻 ↔ (𝑈 ∈ NrmCVec ∧ (( +𝑣 ‘𝑈) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑈) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑈) ⊆ (normCV‘𝑈))))) |
11 | 5, 10 | mpbird 259 | 1 ⊢ (𝑈 ∈ NrmCVec → 𝑈 ∈ 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ‘cfv 6355 NrmCVeccnv 28361 +𝑣 cpv 28362 ·𝑠OLD cns 28364 normCVcnmcv 28367 SubSpcss 28498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fo 6361 df-fv 6363 df-oprab 7160 df-1st 7689 df-2nd 7690 df-vc 28336 df-nv 28369 df-va 28372 df-sm 28374 df-nmcv 28377 df-ssp 28499 |
This theorem is referenced by: hhsssh 29046 |
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