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Theorem isssp 27909
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 27908 . . 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 6353 . . . . . 6 (𝑤 = 𝑊 → ( +𝑣𝑤) = ( +𝑣𝑊))
8 isssp.f . . . . . 6 𝐹 = ( +𝑣𝑊)
97, 8syl6eqr 2812 . . . . 5 (𝑤 = 𝑊 → ( +𝑣𝑤) = 𝐹)
109sseq1d 3773 . . . 4 (𝑤 = 𝑊 → (( +𝑣𝑤) ⊆ 𝐺𝐹𝐺))
11 fveq2 6353 . . . . . 6 (𝑤 = 𝑊 → ( ·𝑠OLD𝑤) = ( ·𝑠OLD𝑊))
12 isssp.r . . . . . 6 𝑅 = ( ·𝑠OLD𝑊)
1311, 12syl6eqr 2812 . . . . 5 (𝑤 = 𝑊 → ( ·𝑠OLD𝑤) = 𝑅)
1413sseq1d 3773 . . . 4 (𝑤 = 𝑊 → (( ·𝑠OLD𝑤) ⊆ 𝑆𝑅𝑆))
15 fveq2 6353 . . . . . 6 (𝑤 = 𝑊 → (normCV𝑤) = (normCV𝑊))
16 isssp.m . . . . . 6 𝑀 = (normCV𝑊)
1715, 16syl6eqr 2812 . . . . 5 (𝑤 = 𝑊 → (normCV𝑤) = 𝑀)
1817sseq1d 3773 . . . 4 (𝑤 = 𝑊 → ((normCV𝑤) ⊆ 𝑁𝑀𝑁))
1910, 14, 183anbi123d 1548 . . 3 (𝑤 = 𝑊 → ((( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁) ↔ (𝐹𝐺𝑅𝑆𝑀𝑁)))
2019elrab 3504 . 2 (𝑊 ∈ {𝑤 ∈ NrmCVec ∣ (( +𝑣𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD𝑤) ⊆ 𝑆 ∧ (normCV𝑤) ⊆ 𝑁)} ↔ (𝑊 ∈ NrmCVec ∧ (𝐹𝐺𝑅𝑆𝑀𝑁)))
216, 20syl6bb 276 1 (𝑈 ∈ NrmCVec → (𝑊𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹𝐺𝑅𝑆𝑀𝑁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  {crab 3054  wss 3715  cfv 6049  NrmCVeccnv 27769   +𝑣 cpv 27770   ·𝑠OLD cns 27772  normCVcnmcv 27775  SubSpcss 27906
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057  df-oprab 6818  df-1st 7334  df-2nd 7335  df-vc 27744  df-nv 27777  df-va 27780  df-sm 27782  df-nmcv 27785  df-ssp 27907
This theorem is referenced by:  sspid  27910  sspnv  27911  sspba  27912  sspg  27913  ssps  27915  sspn  27921  hhsst  28453  hhsssh2  28457
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