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Theorem isssp 30753
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 30752 . . 3 (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)})
65eleq2d 2825 . 2 (𝑈 ∈ NrmCVec → (𝑊𝐻𝑊 ∈ {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)}))
7 fveq2 6907 . . . . . 6 (𝑤 = 𝑊 → ( +𝑣𝑤) = ( +𝑣𝑊))
8 isssp.f . . . . . 6 𝐹 = ( +𝑣𝑊)
97, 8eqtr4di 2793 . . . . 5 (𝑤 = 𝑊 → ( +𝑣𝑤) = 𝐹)
109sseq1d 4027 . . . 4 (𝑤 = 𝑊 → (( +𝑣𝑤) ⊆ 𝐺𝐹𝐺))
11 fveq2 6907 . . . . . 6 (𝑤 = 𝑊 → ( ·𝑠OLD𝑤) = ( ·𝑠OLD𝑊))
12 isssp.r . . . . . 6 𝑅 = ( ·𝑠OLD𝑊)
1311, 12eqtr4di 2793 . . . . 5 (𝑤 = 𝑊 → ( ·𝑠OLD𝑤) = 𝑅)
1413sseq1d 4027 . . . 4 (𝑤 = 𝑊 → (( ·𝑠OLD𝑤) ⊆ 𝑆𝑅𝑆))
15 fveq2 6907 . . . . . 6 (𝑤 = 𝑊 → (normCV𝑤) = (normCV𝑊))
16 isssp.m . . . . . 6 𝑀 = (normCV𝑊)
1715, 16eqtr4di 2793 . . . . 5 (𝑤 = 𝑊 → (normCV𝑤) = 𝑀)
1817sseq1d 4027 . . . 4 (𝑤 = 𝑊 → ((normCV𝑤) ⊆ 𝑁𝑀𝑁))
1910, 14, 183anbi123d 1435 . . 3 (𝑤 = 𝑊 → ((( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁) ↔ (𝐹𝐺𝑅𝑆𝑀𝑁)))
2019elrab 3695 . 2 (𝑊 ∈ {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)} ↔ (𝑊 ∈ NrmCVec ∧ (𝐹𝐺𝑅𝑆𝑀𝑁)))
216, 20bitrdi 287 1 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹𝐺𝑅𝑆𝑀𝑁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  {crab 3433  wss 3963  cfv 6563  NrmCVeccnv 30613   +𝑣 cpv 30614   ·𝑠OLD cns 30616  normCVcnmcv 30619  SubSpcss 30750
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-oprab 7435  df-1st 8013  df-2nd 8014  df-vc 30588  df-nv 30621  df-va 30624  df-sm 30626  df-nmcv 30629  df-ssp 30751
This theorem is referenced by:  sspid  30754  sspnv  30755  sspba  30756  sspg  30757  ssps  30759  sspn  30765  hhsst  31295  hhsssh2  31299
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