Theorem isssp 30812

Description: The predicate "is a subspace." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
isssp.g	⊢ 𝐺 = ( +𝑣 ‘𝑈)
isssp.f	⊢ 𝐹 = ( +𝑣 ‘𝑊)
isssp.s	⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
isssp.r	⊢ 𝑅 = ( ·𝑠OLD ‘𝑊)
isssp.n	⊢ 𝑁 = (normCV‘𝑈)
isssp.m	⊢ 𝑀 = (normCV‘𝑊)
isssp.h	⊢ 𝐻 = (SubSp‘𝑈)

Assertion

Ref	Expression
isssp	⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁))))

Proof of Theorem isssp

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isssp.g	. . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
2		isssp.s	. . . 4 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
3		isssp.n	. . . 4 ⊢ 𝑁 = (normCV‘𝑈)
4		isssp.h	. . . 4 ⊢ 𝐻 = (SubSp‘𝑈)
5	1, 2, 3, 4	sspval 30811	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)})
6	5	eleq2d 2823	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ 𝑊 ∈ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)}))
7		fveq2 6842	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = ( +𝑣 ‘𝑊))
8		isssp.f	. . . . . 6 ⊢ 𝐹 = ( +𝑣 ‘𝑊)
9	7, 8	eqtr4di 2790	. . . . 5 ⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = 𝐹)
10	9	sseq1d 3967	. . . 4 ⊢ (𝑤 = 𝑊 → (( +𝑣 ‘𝑤) ⊆ 𝐺 ↔ 𝐹 ⊆ 𝐺))
11		fveq2 6842	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·𝑠OLD ‘𝑤) = ( ·𝑠OLD ‘𝑊))
12		isssp.r	. . . . . 6 ⊢ 𝑅 = ( ·𝑠OLD ‘𝑊)
13	11, 12	eqtr4di 2790	. . . . 5 ⊢ (𝑤 = 𝑊 → ( ·𝑠OLD ‘𝑤) = 𝑅)
14	13	sseq1d 3967	. . . 4 ⊢ (𝑤 = 𝑊 → (( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ↔ 𝑅 ⊆ 𝑆))
15		fveq2 6842	. . . . . 6 ⊢ (𝑤 = 𝑊 → (normCV‘𝑤) = (normCV‘𝑊))
16		isssp.m	. . . . . 6 ⊢ 𝑀 = (normCV‘𝑊)
17	15, 16	eqtr4di 2790	. . . . 5 ⊢ (𝑤 = 𝑊 → (normCV‘𝑤) = 𝑀)
18	17	sseq1d 3967	. . . 4 ⊢ (𝑤 = 𝑊 → ((normCV‘𝑤) ⊆ 𝑁 ↔ 𝑀 ⊆ 𝑁))
19	10, 14, 18	3anbi123d 1439	. . 3 ⊢ (𝑤 = 𝑊 → ((( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁) ↔ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁)))
20	19	elrab 3648	. 2 ⊢ (𝑊 ∈ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)} ↔ (𝑊 ∈ NrmCVec ∧ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁)))
21	6, 20	bitrdi 287	1 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {crab 3401   ⊆ wss 3903  ‘cfv 6500  NrmCVeccnv 30672   +_𝑣 cpv 30673   ·_𝑠OLD cns 30675  norm_CVcnmcv 30678  SubSpcss 30809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-oprab 7372  df-1st 7943  df-2nd 7944  df-vc 30647  df-nv 30680  df-va 30683  df-sm 30685  df-nmcv 30688  df-ssp 30810

This theorem is referenced by:  sspid  30813  sspnv  30814  sspba  30815  sspg  30816  ssps  30818  sspn  30824  hhsst  31354  hhsssh2  31358