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 = ( +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 26962	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)})
6	5	eleq2d 2673	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ 𝑊 ∈ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)}))
7		fveq2 6103	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = ( +𝑣 ‘𝑊))
8		isssp.f	. . . . . 6 ⊢ 𝐹 = ( +𝑣 ‘𝑊)
9	7, 8	syl6eqr 2662	. . . . 5 ⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = 𝐹)
10	9	sseq1d 3595	. . . 4 ⊢ (𝑤 = 𝑊 → (( +𝑣 ‘𝑤) ⊆ 𝐺 ↔ 𝐹 ⊆ 𝐺))
11		fveq2 6103	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·𝑠OLD ‘𝑤) = ( ·𝑠OLD ‘𝑊))
12		isssp.r	. . . . . 6 ⊢ 𝑅 = ( ·𝑠OLD ‘𝑊)
13	11, 12	syl6eqr 2662	. . . . 5 ⊢ (𝑤 = 𝑊 → ( ·𝑠OLD ‘𝑤) = 𝑅)
14	13	sseq1d 3595	. . . 4 ⊢ (𝑤 = 𝑊 → (( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ↔ 𝑅 ⊆ 𝑆))
15		fveq2 6103	. . . . . 6 ⊢ (𝑤 = 𝑊 → (normCV‘𝑤) = (normCV‘𝑊))
16		isssp.m	. . . . . 6 ⊢ 𝑀 = (normCV‘𝑊)
17	15, 16	syl6eqr 2662	. . . . 5 ⊢ (𝑤 = 𝑊 → (normCV‘𝑤) = 𝑀)
18	17	sseq1d 3595	. . . 4 ⊢ (𝑤 = 𝑊 → ((normCV‘𝑤) ⊆ 𝑁 ↔ 𝑀 ⊆ 𝑁))
19	10, 14, 18	3anbi123d 1391	. . 3 ⊢ (𝑤 = 𝑊 → ((( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁) ↔ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁)))
20	19	elrab 3331	. 2 ⊢ (𝑊 ∈ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)} ↔ (𝑊 ∈ NrmCVec ∧ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁)))
21	6, 20	syl6bb 275	1 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁))))

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  norm_CVcnmcv 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