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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 28427 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)}) |
6 | 5 | eleq2d 2895 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ 𝑊 ∈ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)})) |
7 | fveq2 6663 | . . . . . 6 ⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = ( +𝑣 ‘𝑊)) | |
8 | isssp.f | . . . . . 6 ⊢ 𝐹 = ( +𝑣 ‘𝑊) | |
9 | 7, 8 | syl6eqr 2871 | . . . . 5 ⊢ (𝑤 = 𝑊 → ( +𝑣 ‘𝑤) = 𝐹) |
10 | 9 | sseq1d 3995 | . . . 4 ⊢ (𝑤 = 𝑊 → (( +𝑣 ‘𝑤) ⊆ 𝐺 ↔ 𝐹 ⊆ 𝐺)) |
11 | fveq2 6663 | . . . . . 6 ⊢ (𝑤 = 𝑊 → ( ·𝑠OLD ‘𝑤) = ( ·𝑠OLD ‘𝑊)) | |
12 | isssp.r | . . . . . 6 ⊢ 𝑅 = ( ·𝑠OLD ‘𝑊) | |
13 | 11, 12 | syl6eqr 2871 | . . . . 5 ⊢ (𝑤 = 𝑊 → ( ·𝑠OLD ‘𝑤) = 𝑅) |
14 | 13 | sseq1d 3995 | . . . 4 ⊢ (𝑤 = 𝑊 → (( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ↔ 𝑅 ⊆ 𝑆)) |
15 | fveq2 6663 | . . . . . 6 ⊢ (𝑤 = 𝑊 → (normCV‘𝑤) = (normCV‘𝑊)) | |
16 | isssp.m | . . . . . 6 ⊢ 𝑀 = (normCV‘𝑊) | |
17 | 15, 16 | syl6eqr 2871 | . . . . 5 ⊢ (𝑤 = 𝑊 → (normCV‘𝑤) = 𝑀) |
18 | 17 | sseq1d 3995 | . . . 4 ⊢ (𝑤 = 𝑊 → ((normCV‘𝑤) ⊆ 𝑁 ↔ 𝑀 ⊆ 𝑁)) |
19 | 10, 14, 18 | 3anbi123d 1427 | . . 3 ⊢ (𝑤 = 𝑊 → ((( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁) ↔ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁))) |
20 | 19 | elrab 3677 | . 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 288 | 1 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 {crab 3139 ⊆ wss 3933 ‘cfv 6348 NrmCVeccnv 28288 +𝑣 cpv 28289 ·𝑠OLD cns 28291 normCVcnmcv 28294 SubSpcss 28425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fo 6354 df-fv 6356 df-oprab 7149 df-1st 7678 df-2nd 7679 df-vc 28263 df-nv 28296 df-va 28299 df-sm 28301 df-nmcv 28304 df-ssp 28426 |
This theorem is referenced by: sspid 28429 sspnv 28430 sspba 28431 sspg 28432 ssps 28434 sspn 28440 hhsst 28970 hhsssh2 28974 |
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