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Theorem isssp 28104
Description: The predicate "is a subspace." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
isssp.g 𝐺 = ( +𝑣𝑈)
isssp.f 𝐹 = ( +𝑣𝑊)
isssp.s 𝑆 = ( ·𝑠OLD𝑈)
isssp.r 𝑅 = ( ·𝑠OLD𝑊)
isssp.n 𝑁 = (normCV𝑈)
isssp.m 𝑀 = (normCV𝑊)
isssp.h 𝐻 = (SubSp‘𝑈)
Assertion
Ref Expression
isssp (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹𝐺𝑅𝑆𝑀𝑁))))

Proof of Theorem isssp
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 isssp.g . . . 4 𝐺 = ( +𝑣𝑈)
2 isssp.s . . . 4 𝑆 = ( ·𝑠OLD𝑈)
3 isssp.n . . . 4 𝑁 = (normCV𝑈)
4 isssp.h . . . 4 𝐻 = (SubSp‘𝑈)
51, 2, 3, 4sspval 28103 . . 3 (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)})
65eleq2d 2864 . 2 (𝑈 ∈ NrmCVec → (𝑊𝐻𝑊 ∈ {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)}))
7 fveq2 6411 . . . . . 6 (𝑤 = 𝑊 → ( +𝑣𝑤) = ( +𝑣𝑊))
8 isssp.f . . . . . 6 𝐹 = ( +𝑣𝑊)
97, 8syl6eqr 2851 . . . . 5 (𝑤 = 𝑊 → ( +𝑣𝑤) = 𝐹)
109sseq1d 3828 . . . 4 (𝑤 = 𝑊 → (( +𝑣𝑤) ⊆ 𝐺𝐹𝐺))
11 fveq2 6411 . . . . . 6 (𝑤 = 𝑊 → ( ·𝑠OLD𝑤) = ( ·𝑠OLD𝑊))
12 isssp.r . . . . . 6 𝑅 = ( ·𝑠OLD𝑊)
1311, 12syl6eqr 2851 . . . . 5 (𝑤 = 𝑊 → ( ·𝑠OLD𝑤) = 𝑅)
1413sseq1d 3828 . . . 4 (𝑤 = 𝑊 → (( ·𝑠OLD𝑤) ⊆ 𝑆𝑅𝑆))
15 fveq2 6411 . . . . . 6 (𝑤 = 𝑊 → (normCV𝑤) = (normCV𝑊))
16 isssp.m . . . . . 6 𝑀 = (normCV𝑊)
1715, 16syl6eqr 2851 . . . . 5 (𝑤 = 𝑊 → (normCV𝑤) = 𝑀)
1817sseq1d 3828 . . . 4 (𝑤 = 𝑊 → ((normCV𝑤) ⊆ 𝑁𝑀𝑁))
1910, 14, 183anbi123d 1561 . . 3 (𝑤 = 𝑊 → ((( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁) ↔ (𝐹𝐺𝑅𝑆𝑀𝑁)))
2019elrab 3556 . 2 (𝑊 ∈ {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)} ↔ (𝑊 ∈ NrmCVec ∧ (𝐹𝐺𝑅𝑆𝑀𝑁)))
216, 20syl6bb 279 1 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹𝐺𝑅𝑆𝑀𝑁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  w3a 1108   = wceq 1653  wcel 2157  {crab 3093  wss 3769  cfv 6101  NrmCVeccnv 27964   +𝑣 cpv 27965   ·𝑠OLD cns 27967  normCVcnmcv 27970  SubSpcss 28101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fo 6107  df-fv 6109  df-oprab 6882  df-1st 7401  df-2nd 7402  df-vc 27939  df-nv 27972  df-va 27975  df-sm 27977  df-nmcv 27980  df-ssp 28102
This theorem is referenced by:  sspid  28105  sspnv  28106  sspba  28107  sspg  28108  ssps  28110  sspn  28116  hhsst  28648  hhsssh2  28652
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