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Theorem isssp 26963
 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 26962 . . 3 (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)})
65eleq2d 2673 . 2 (𝑈 ∈ NrmCVec → (𝑊𝐻𝑊 ∈ {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)}))
7 fveq2 6103 . . . . . 6 (𝑤 = 𝑊 → ( +𝑣𝑤) = ( +𝑣𝑊))
8 isssp.f . . . . . 6 𝐹 = ( +𝑣𝑊)
97, 8syl6eqr 2662 . . . . 5 (𝑤 = 𝑊 → ( +𝑣𝑤) = 𝐹)
109sseq1d 3595 . . . 4 (𝑤 = 𝑊 → (( +𝑣𝑤) ⊆ 𝐺𝐹𝐺))
11 fveq2 6103 . . . . . 6 (𝑤 = 𝑊 → ( ·𝑠OLD𝑤) = ( ·𝑠OLD𝑊))
12 isssp.r . . . . . 6 𝑅 = ( ·𝑠OLD𝑊)
1311, 12syl6eqr 2662 . . . . 5 (𝑤 = 𝑊 → ( ·𝑠OLD𝑤) = 𝑅)
1413sseq1d 3595 . . . 4 (𝑤 = 𝑊 → (( ·𝑠OLD𝑤) ⊆ 𝑆𝑅𝑆))
15 fveq2 6103 . . . . . 6 (𝑤 = 𝑊 → (normCV𝑤) = (normCV𝑊))
16 isssp.m . . . . . 6 𝑀 = (normCV𝑊)
1715, 16syl6eqr 2662 . . . . 5 (𝑤 = 𝑊 → (normCV𝑤) = 𝑀)
1817sseq1d 3595 . . . 4 (𝑤 = 𝑊 → ((normCV𝑤) ⊆ 𝑁𝑀𝑁))
1910, 14, 183anbi123d 1391 . . 3 (𝑤 = 𝑊 → ((( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁) ↔ (𝐹𝐺𝑅𝑆𝑀𝑁)))
2019elrab 3331 . 2 (𝑊 ∈ {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)} ↔ (𝑊 ∈ NrmCVec ∧ (𝐹𝐺𝑅𝑆𝑀𝑁)))
216, 20syl6bb 275 1 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹𝐺𝑅𝑆𝑀𝑁))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {crab 2900   ⊆ wss 3540  ‘cfv 5804  NrmCVeccnv 26823   +𝑣 cpv 26824   ·𝑠OLD cns 26826  normCVcnmcv 26829  SubSpcss 26960 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-oprab 6553  df-1st 7059  df-2nd 7060  df-vc 26798  df-nv 26831  df-va 26834  df-sm 26836  df-nmcv 26839  df-ssp 26961 This theorem is referenced by:  sspid  26964  sspnv  26965  sspba  26966  sspg  26967  ssps  26969  sspn  26975  hhsst  27507  hhsssh2  27511
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