![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > isssp | Structured version Visualization version GIF version |
Description: The predicate "is a subspace." (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.) |
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) |
Ref | Expression |
---|---|
isssp | ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁)))) |
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 29663 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)}) |
6 | 5 | eleq2d 2823 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ 𝑊 ∈ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)})) |
7 | fveq2 6842 | . . . . . 6 ⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = ( +𝑣 ‘𝑊)) | |
8 | isssp.f | . . . . . 6 ⊢ 𝐹 = ( +𝑣 ‘𝑊) | |
9 | 7, 8 | eqtr4di 2794 | . . . . 5 ⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = 𝐹) |
10 | 9 | sseq1d 3975 | . . . 4 ⊢ (𝑤 = 𝑊 → (( +𝑣 ‘𝑤) ⊆ 𝐺 ↔ 𝐹 ⊆ 𝐺)) |
11 | fveq2 6842 | . . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·𝑠OLD ‘𝑤) = ( ·𝑠OLD ‘𝑊)) | |
12 | isssp.r | . . . . . 6 ⊢ 𝑅 = ( ·𝑠OLD ‘𝑊) | |
13 | 11, 12 | eqtr4di 2794 | . . . . 5 ⊢ (𝑤 = 𝑊 → ( ·𝑠OLD ‘𝑤) = 𝑅) |
14 | 13 | sseq1d 3975 | . . . 4 ⊢ (𝑤 = 𝑊 → (( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ↔ 𝑅 ⊆ 𝑆)) |
15 | fveq2 6842 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (normCV‘𝑤) = (normCV‘𝑊)) | |
16 | isssp.m | . . . . . 6 ⊢ 𝑀 = (normCV‘𝑊) | |
17 | 15, 16 | eqtr4di 2794 | . . . . 5 ⊢ (𝑤 = 𝑊 → (normCV‘𝑤) = 𝑀) |
18 | 17 | sseq1d 3975 | . . . 4 ⊢ (𝑤 = 𝑊 → ((normCV‘𝑤) ⊆ 𝑁 ↔ 𝑀 ⊆ 𝑁)) |
19 | 10, 14, 18 | 3anbi123d 1436 | . . 3 ⊢ (𝑤 = 𝑊 → ((( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁) ↔ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁))) |
20 | 19 | elrab 3645 | . 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 286 | 1 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 {crab 3407 ⊆ wss 3910 ‘cfv 6496 NrmCVeccnv 29524 +𝑣 cpv 29525 ·𝑠OLD cns 29527 normCVcnmcv 29530 SubSpcss 29661 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-fo 6502 df-fv 6504 df-oprab 7360 df-1st 7920 df-2nd 7921 df-vc 29499 df-nv 29532 df-va 29535 df-sm 29537 df-nmcv 29540 df-ssp 29662 |
This theorem is referenced by: sspid 29665 sspnv 29666 sspba 29667 sspg 29668 ssps 29670 sspn 29676 hhsst 30206 hhsssh2 30210 |
Copyright terms: Public domain | W3C validator |