Theorem isssp 28503

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 28502	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)})
6	5	eleq2d 2900	. 2 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ 𝑊 ∈ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)}))
7		fveq2 6672	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = ( +𝑣 ‘𝑊))
8		isssp.f	. . . . . 6 ⊢ 𝐹 = ( +𝑣 ‘𝑊)
9	7, 8	syl6eqr 2876	. . . . 5 ⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = 𝐹)
10	9	sseq1d 4000	. . . 4 ⊢ (𝑤 = 𝑊 → (( +𝑣 ‘𝑤) ⊆ 𝐺 ↔ 𝐹 ⊆ 𝐺))
11		fveq2 6672	. . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·𝑠OLD ‘𝑤) = ( ·𝑠OLD ‘𝑊))
12		isssp.r	. . . . . 6 ⊢ 𝑅 = ( ·𝑠OLD ‘𝑊)
13	11, 12	syl6eqr 2876	. . . . 5 ⊢ (𝑤 = 𝑊 → ( ·𝑠OLD ‘𝑤) = 𝑅)
14	13	sseq1d 4000	. . . 4 ⊢ (𝑤 = 𝑊 → (( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ↔ 𝑅 ⊆ 𝑆))
15		fveq2 6672	. . . . . 6 ⊢ (𝑤 = 𝑊 → (normCV‘𝑤) = (normCV‘𝑊))
16		isssp.m	. . . . . 6 ⊢ 𝑀 = (normCV‘𝑊)
17	15, 16	syl6eqr 2876	. . . . 5 ⊢ (𝑤 = 𝑊 → (normCV‘𝑤) = 𝑀)
18	17	sseq1d 4000	. . . 4 ⊢ (𝑤 = 𝑊 → ((normCV‘𝑤) ⊆ 𝑁 ↔ 𝑀 ⊆ 𝑁))
19	10, 14, 18	3anbi123d 1432	. . 3 ⊢ (𝑤 = 𝑊 → ((( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁) ↔ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁)))
20	19	elrab 3682	. 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 289	1 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  {crab 3144   ⊆ wss 3938  ‘cfv 6357  NrmCVeccnv 28363   +_𝑣 cpv 28364   ·_𝑠OLD cns 28366  norm_CVcnmcv 28369  SubSpcss 28500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fo 6363  df-fv 6365  df-oprab 7162  df-1st 7691  df-2nd 7692  df-vc 28338  df-nv 28371  df-va 28374  df-sm 28376  df-nmcv 28379  df-ssp 28501

This theorem is referenced by:  sspid  28504  sspnv  28505  sspba  28506  sspg  28507  ssps  28509  sspn  28515  hhsst  29045  hhsssh2  29049