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| 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 30743 | . . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐻 = {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)}) | 
| 6 | 5 | eleq2d 2826 | . 2 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ 𝑊 ∈ {𝑤 ∈ NrmCVec ∣ (( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁)})) | 
| 7 | fveq2 6905 | . . . . . 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 4014 | . . . 4 ⊢ (𝑤 = 𝑊 → (( +𝑣 ‘𝑤) ⊆ 𝐺 ↔ 𝐹 ⊆ 𝐺)) | 
| 11 | fveq2 6905 | . . . . . 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 4014 | . . . 4 ⊢ (𝑤 = 𝑊 → (( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ↔ 𝑅 ⊆ 𝑆)) | 
| 15 | fveq2 6905 | . . . . . 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 4014 | . . . 4 ⊢ (𝑤 = 𝑊 → ((normCV‘𝑤) ⊆ 𝑁 ↔ 𝑀 ⊆ 𝑁)) | 
| 19 | 10, 14, 18 | 3anbi123d 1437 | . . 3 ⊢ (𝑤 = 𝑊 → ((( +𝑣 ‘𝑤) ⊆ 𝐺 ∧ ( ·𝑠OLD ‘𝑤) ⊆ 𝑆 ∧ (normCV‘𝑤) ⊆ 𝑁) ↔ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁))) | 
| 20 | 19 | elrab 3691 | . 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 287 | 1 ⊢ (𝑈 ∈ NrmCVec → (𝑊 ∈ 𝐻 ↔ (𝑊 ∈ NrmCVec ∧ (𝐹 ⊆ 𝐺 ∧ 𝑅 ⊆ 𝑆 ∧ 𝑀 ⊆ 𝑁)))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 {crab 3435 ⊆ wss 3950 ‘cfv 6560 NrmCVeccnv 30604 +𝑣 cpv 30605 ·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-sm 30617 df-nmcv 30620 df-ssp 30742 | 
| This theorem is referenced by: sspid 30745 sspnv 30746 sspba 30747 sspg 30748 ssps 30750 sspn 30756 hhsst 31286 hhsssh2 31290 | 
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