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Theorem isssp 30744
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 30743 . . 3 (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)})
65eleq2d 2826 . 2 (𝑈 ∈ NrmCVec → (𝑊𝐻𝑊 ∈ {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)}))
7 fveq2 6905 . . . . . 6 (𝑤 = 𝑊 → ( +𝑣𝑤) = ( +𝑣𝑊))
8 isssp.f . . . . . 6 𝐹 = ( +𝑣𝑊)
97, 8eqtr4di 2794 . . . . 5 (𝑤 = 𝑊 → ( +𝑣𝑤) = 𝐹)
109sseq1d 4014 . . . 4 (𝑤 = 𝑊 → (( +𝑣𝑤) ⊆ 𝐺𝐹𝐺))
11 fveq2 6905 . . . . . 6 (𝑤 = 𝑊 → ( ·𝑠OLD𝑤) = ( ·𝑠OLD𝑊))
12 isssp.r . . . . . 6 𝑅 = ( ·𝑠OLD𝑊)
1311, 12eqtr4di 2794 . . . . 5 (𝑤 = 𝑊 → ( ·𝑠OLD𝑤) = 𝑅)
1413sseq1d 4014 . . . 4 (𝑤 = 𝑊 → (( ·𝑠OLD𝑤) ⊆ 𝑆𝑅𝑆))
15 fveq2 6905 . . . . . 6 (𝑤 = 𝑊 → (normCV𝑤) = (normCV𝑊))
16 isssp.m . . . . . 6 𝑀 = (normCV𝑊)
1715, 16eqtr4di 2794 . . . . 5 (𝑤 = 𝑊 → (normCV𝑤) = 𝑀)
1817sseq1d 4014 . . . 4 (𝑤 = 𝑊 → ((normCV𝑤) ⊆ 𝑁𝑀𝑁))
1910, 14, 183anbi123d 1437 . . 3 (𝑤 = 𝑊 → ((( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁) ↔ (𝐹𝐺𝑅𝑆𝑀𝑁)))
2019elrab 3691 . 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 1539  wcel 2107  {crab 3435  wss 3950  cfv 6560  NrmCVeccnv 30604   +𝑣 cpv 30605   ·𝑠OLD cns 30607  normCVcnmcv 30610  SubSpcss 30741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566  df-fv 6568  df-oprab 7436  df-1st 8015  df-2nd 8016  df-vc 30579  df-nv 30612  df-va 30615  df-sm 30617  df-nmcv 30620  df-ssp 30742
This theorem is referenced by:  sspid  30745  sspnv  30746  sspba  30747  sspg  30748  ssps  30750  sspn  30756  hhsst  31286  hhsssh2  31290
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