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| Description: The base set of a subspace is included in the parent base set. (Contributed by NM, 27-Jan-2008.) (New usage is discouraged.) | 
| Ref | Expression | 
|---|---|
| sspba.x | ⊢ 𝑋 = (BaseSet‘𝑈) | 
| sspba.y | ⊢ 𝑌 = (BaseSet‘𝑊) | 
| sspba.h | ⊢ 𝐻 = (SubSp‘𝑈) | 
| Ref | Expression | 
|---|---|
| sspba | ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ 𝑋) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2736 | . . . . . 6 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
| 2 | eqid 2736 | . . . . . 6 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | |
| 3 | eqid 2736 | . . . . . 6 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
| 4 | eqid 2736 | . . . . . 6 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊) | |
| 5 | eqid 2736 | . . . . . 6 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
| 6 | eqid 2736 | . . . . . 6 ⊢ (normCV‘𝑊) = (normCV‘𝑊) | |
| 7 | sspba.h | . . . . . 6 ⊢ 𝐻 = (SubSp‘𝑈) | |
| 8 | 1, 2, 3, 4, 5, 6, 7 | isssp 30744 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈))))) | 
| 9 | 8 | simplbda 499 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈))) | 
| 10 | 9 | simp1d 1142 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈)) | 
| 11 | rnss 5949 | . . 3 ⊢ (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) → ran ( +𝑣 ‘𝑊) ⊆ ran ( +𝑣 ‘𝑈)) | |
| 12 | 10, 11 | syl 17 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ran ( +𝑣 ‘𝑊) ⊆ ran ( +𝑣 ‘𝑈)) | 
| 13 | sspba.y | . . 3 ⊢ 𝑌 = (BaseSet‘𝑊) | |
| 14 | 13, 2 | bafval 30624 | . 2 ⊢ 𝑌 = ran ( +𝑣 ‘𝑊) | 
| 15 | sspba.x | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
| 16 | 15, 1 | bafval 30624 | . 2 ⊢ 𝑋 = ran ( +𝑣 ‘𝑈) | 
| 17 | 12, 14, 16 | 3sstr4g 4036 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 ran crn 5685 ‘cfv 6560 NrmCVeccnv 30604 +𝑣 cpv 30605 BaseSetcba 30606 ·𝑠OLD cns 30607 normCVcnmcv 30610 SubSpcss 30741 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fo 6566 df-fv 6568 df-oprab 7436 df-1st 8015 df-2nd 8016 df-vc 30579 df-nv 30612 df-va 30615 df-ba 30616 df-sm 30617 df-nmcv 30620 df-ssp 30742 | 
| This theorem is referenced by: sspg 30748 ssps 30750 sspmlem 30752 sspmval 30753 sspz 30755 sspn 30756 sspimsval 30758 minvecolem1 30894 minvecolem2 30895 minvecolem3 30896 minvecolem4b 30898 minvecolem4 30900 minvecolem5 30901 minvecolem6 30902 minvecolem7 30903 | 
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