Theorem isssp 30884

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 30883	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)})
6	5	eleq2d 2847	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ 𝑊 ∈ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)}))
7		fveq2 6862	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = ( +𝑣 ‘𝑊))
8		isssp.f	. . . . . 6 ⊢ 𝐹 = ( +𝑣 ‘𝑊)
9	7, 8	eqtr4di 2814	. . . . 5 ⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = 𝐹)
10	9	sseq1d 3965	. . . 4 ⊢ (𝑤 = 𝑊 → (( +𝑣 ‘𝑤) ⊆ 𝐺 ↔ 𝐹 ⊆ 𝐺))
11		fveq2 6862	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·𝑠OLD ‘𝑤) = ( ·𝑠OLD ‘𝑊))
12		isssp.r	. . . . . 6 ⊢ 𝑅 = ( ·𝑠OLD ‘𝑊)
13	11, 12	eqtr4di 2814	. . . . 5 ⊢ (𝑤 = 𝑊 → ( ·𝑠OLD ‘𝑤) = 𝑅)
14	13	sseq1d 3965	. . . 4 ⊢ (𝑤 = 𝑊 → (( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ↔ 𝑅 ⊆ 𝑆))
15		fveq2 6862	. . . . . 6 ⊢ (𝑤 = 𝑊 → (normCV‘𝑤) = (normCV‘𝑊))
16		isssp.m	. . . . . 6 ⊢ 𝑀 = (normCV‘𝑊)
17	15, 16	eqtr4di 2814	. . . . 5 ⊢ (𝑤 = 𝑊 → (normCV‘𝑤) = 𝑀)
18	17	sseq1d 3965	. . . 4 ⊢ (𝑤 = 𝑊 → ((normCV‘𝑤) ⊆ 𝑁 ↔ 𝑀 ⊆ 𝑁))
19	10, 14, 18	3anbi123d 1456	. . 3 ⊢ (𝑤 = 𝑊 → ((( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁) ↔ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁)))
20	19	elrab 3649	. 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 289	1 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  {crab 3413   ⊆ wss 3902  ‘cfv 6516  NrmCVeccnv 30744   +_𝑣 cpv 30745   ·_𝑠OLD cns 30747  norm_CVcnmcv 30750  SubSpcss 30881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-fo 6522  df-fv 6524  df-oprab 7395  df-1st 7965  df-2nd 7966  df-vc 30719  df-nv 30752  df-va 30755  df-sm 30757  df-nmcv 30760  df-ssp 30882

This theorem is referenced by:  sspid  30885  sspnv  30886  sspba  30887  sspg  30888  ssps  30890  sspn  30896  hhsst  31426  hhsssh2  31430