Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sspba | Structured version Visualization version GIF version |
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 2821 | . . . . . 6 ⊢ ( +𝑣 ‘𝑈) = ( +𝑣 ‘𝑈) | |
2 | eqid 2821 | . . . . . 6 ⊢ ( +𝑣 ‘𝑊) = ( +𝑣 ‘𝑊) | |
3 | eqid 2821 | . . . . . 6 ⊢ ( ·𝑠OLD ‘𝑈) = ( ·𝑠OLD ‘𝑈) | |
4 | eqid 2821 | . . . . . 6 ⊢ ( ·𝑠OLD ‘𝑊) = ( ·𝑠OLD ‘𝑊) | |
5 | eqid 2821 | . . . . . 6 ⊢ (normCV‘𝑈) = (normCV‘𝑈) | |
6 | eqid 2821 | . . . . . 6 ⊢ (normCV‘𝑊) = (normCV‘𝑊) | |
7 | sspba.h | . . . . . 6 ⊢ 𝐻 = (SubSp‘𝑈) | |
8 | 1, 2, 3, 4, 5, 6, 7 | isssp 28495 | . . . . 5 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈))))) |
9 | 8 | simplbda 502 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈) ∧ ( ·𝑠OLD ‘𝑊) ⊆ ( ·𝑠OLD ‘𝑈) ∧ (normCV‘𝑊) ⊆ (normCV‘𝑈))) |
10 | 9 | simp1d 1138 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → ( +𝑣 ‘𝑊) ⊆ ( +𝑣 ‘𝑈)) |
11 | rnss 5803 | . . 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 28375 | . 2 ⊢ 𝑌 = ran ( +𝑣 ‘𝑊) |
15 | sspba.x | . . 3 ⊢ 𝑋 = (BaseSet‘𝑈) | |
16 | 15, 1 | bafval 28375 | . 2 ⊢ 𝑋 = ran ( +𝑣 ‘𝑈) |
17 | 12, 14, 16 | 3sstr4g 4011 | 1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑌 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 ran crn 5550 ‘cfv 6349 NrmCVeccnv 28355 +𝑣 cpv 28356 BaseSetcba 28357 ·𝑠OLD cns 28358 normCVcnmcv 28361 SubSpcss 28492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fo 6355 df-fv 6357 df-oprab 7154 df-1st 7683 df-2nd 7684 df-vc 28330 df-nv 28363 df-va 28366 df-ba 28367 df-sm 28368 df-nmcv 28371 df-ssp 28493 |
This theorem is referenced by: sspg 28499 ssps 28501 sspmlem 28503 sspmval 28504 sspz 28506 sspn 28507 sspimsval 28509 minvecolem1 28645 minvecolem2 28646 minvecolem3 28647 minvecolem4b 28649 minvecolem4 28651 minvecolem5 28652 minvecolem6 28653 minvecolem7 28654 |
Copyright terms: Public domain | W3C validator |