| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | eqid 2736 | . . . . . . 7
⊢
(0g‘𝑊) = (0g‘𝑊) | 
| 2 | 1 | obsne0 21746 | . . . . . 6
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → 𝐴 ≠ (0g‘𝑊)) | 
| 3 | 2 | 3adant2 1131 | . . . . 5
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ≠ (0g‘𝑊)) | 
| 4 |  | elin 3966 | . . . . . . . 8
⊢ (𝐴 ∈ (𝐶 ∩ ( ⊥ ‘𝐶)) ↔ (𝐴 ∈ 𝐶 ∧ 𝐴 ∈ ( ⊥ ‘𝐶))) | 
| 5 |  | obsrcl 21744 | . . . . . . . . . . . . . 14
⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil) | 
| 6 | 5 | 3ad2ant1 1133 | . . . . . . . . . . . . 13
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝑊 ∈ PreHil) | 
| 7 |  | phllmod 21649 | . . . . . . . . . . . . 13
⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | 
| 8 | 6, 7 | syl 17 | . . . . . . . . . . . 12
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝑊 ∈ LMod) | 
| 9 |  | simp2 1137 | . . . . . . . . . . . . 13
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐶 ⊆ 𝐵) | 
| 10 |  | eqid 2736 | . . . . . . . . . . . . . . 15
⊢
(Base‘𝑊) =
(Base‘𝑊) | 
| 11 | 10 | obsss 21745 | . . . . . . . . . . . . . 14
⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝐵 ⊆ (Base‘𝑊)) | 
| 12 | 11 | 3ad2ant1 1133 | . . . . . . . . . . . . 13
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐵 ⊆ (Base‘𝑊)) | 
| 13 | 9, 12 | sstrd 3993 | . . . . . . . . . . . 12
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐶 ⊆ (Base‘𝑊)) | 
| 14 |  | eqid 2736 | . . . . . . . . . . . . 13
⊢
(LSpan‘𝑊) =
(LSpan‘𝑊) | 
| 15 | 10, 14 | lspssid 20984 | . . . . . . . . . . . 12
⊢ ((𝑊 ∈ LMod ∧ 𝐶 ⊆ (Base‘𝑊)) → 𝐶 ⊆ ((LSpan‘𝑊)‘𝐶)) | 
| 16 | 8, 13, 15 | syl2anc 584 | . . . . . . . . . . 11
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐶 ⊆ ((LSpan‘𝑊)‘𝐶)) | 
| 17 | 16 | ssrind 4243 | . . . . . . . . . 10
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝐶 ∩ ( ⊥ ‘𝐶)) ⊆ (((LSpan‘𝑊)‘𝐶) ∩ ( ⊥ ‘𝐶))) | 
| 18 |  | obselocv.o | . . . . . . . . . . . . . 14
⊢  ⊥ =
(ocv‘𝑊) | 
| 19 | 10, 18, 14 | ocvlsp 21695 | . . . . . . . . . . . . 13
⊢ ((𝑊 ∈ PreHil ∧ 𝐶 ⊆ (Base‘𝑊)) → ( ⊥
‘((LSpan‘𝑊)‘𝐶)) = ( ⊥ ‘𝐶)) | 
| 20 | 6, 13, 19 | syl2anc 584 | . . . . . . . . . . . 12
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → ( ⊥
‘((LSpan‘𝑊)‘𝐶)) = ( ⊥ ‘𝐶)) | 
| 21 | 20 | ineq2d 4219 | . . . . . . . . . . 11
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → (((LSpan‘𝑊)‘𝐶) ∩ ( ⊥
‘((LSpan‘𝑊)‘𝐶))) = (((LSpan‘𝑊)‘𝐶) ∩ ( ⊥ ‘𝐶))) | 
| 22 |  | eqid 2736 | . . . . . . . . . . . . . 14
⊢
(LSubSp‘𝑊) =
(LSubSp‘𝑊) | 
| 23 | 10, 22, 14 | lspcl 20975 | . . . . . . . . . . . . 13
⊢ ((𝑊 ∈ LMod ∧ 𝐶 ⊆ (Base‘𝑊)) → ((LSpan‘𝑊)‘𝐶) ∈ (LSubSp‘𝑊)) | 
| 24 | 8, 13, 23 | syl2anc 584 | . . . . . . . . . . . 12
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → ((LSpan‘𝑊)‘𝐶) ∈ (LSubSp‘𝑊)) | 
| 25 | 18, 22, 1 | ocvin 21693 | . . . . . . . . . . . 12
⊢ ((𝑊 ∈ PreHil ∧
((LSpan‘𝑊)‘𝐶) ∈ (LSubSp‘𝑊)) → (((LSpan‘𝑊)‘𝐶) ∩ ( ⊥
‘((LSpan‘𝑊)‘𝐶))) = {(0g‘𝑊)}) | 
| 26 | 6, 24, 25 | syl2anc 584 | . . . . . . . . . . 11
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → (((LSpan‘𝑊)‘𝐶) ∩ ( ⊥
‘((LSpan‘𝑊)‘𝐶))) = {(0g‘𝑊)}) | 
| 27 | 21, 26 | eqtr3d 2778 | . . . . . . . . . 10
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → (((LSpan‘𝑊)‘𝐶) ∩ ( ⊥ ‘𝐶)) = {(0g‘𝑊)}) | 
| 28 | 17, 27 | sseqtrd 4019 | . . . . . . . . 9
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝐶 ∩ ( ⊥ ‘𝐶)) ⊆
{(0g‘𝑊)}) | 
| 29 | 28 | sseld 3981 | . . . . . . . 8
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ (𝐶 ∩ ( ⊥ ‘𝐶)) → 𝐴 ∈ {(0g‘𝑊)})) | 
| 30 | 4, 29 | biimtrrid 243 | . . . . . . 7
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → ((𝐴 ∈ 𝐶 ∧ 𝐴 ∈ ( ⊥ ‘𝐶)) → 𝐴 ∈ {(0g‘𝑊)})) | 
| 31 |  | elsni 4642 | . . . . . . 7
⊢ (𝐴 ∈
{(0g‘𝑊)}
→ 𝐴 =
(0g‘𝑊)) | 
| 32 | 30, 31 | syl6 35 | . . . . . 6
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → ((𝐴 ∈ 𝐶 ∧ 𝐴 ∈ ( ⊥ ‘𝐶)) → 𝐴 = (0g‘𝑊))) | 
| 33 | 32 | necon3ad 2952 | . . . . 5
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝐴 ≠ (0g‘𝑊) → ¬ (𝐴 ∈ 𝐶 ∧ 𝐴 ∈ ( ⊥ ‘𝐶)))) | 
| 34 | 3, 33 | mpd 15 | . . . 4
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → ¬ (𝐴 ∈ 𝐶 ∧ 𝐴 ∈ ( ⊥ ‘𝐶))) | 
| 35 |  | imnan 399 | . . . 4
⊢ ((𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ ( ⊥ ‘𝐶)) ↔ ¬ (𝐴 ∈ 𝐶 ∧ 𝐴 ∈ ( ⊥ ‘𝐶))) | 
| 36 | 34, 35 | sylibr 234 | . . 3
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ 𝐶 → ¬ 𝐴 ∈ ( ⊥ ‘𝐶))) | 
| 37 | 36 | con2d 134 | . 2
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ ( ⊥ ‘𝐶) → ¬ 𝐴 ∈ 𝐶)) | 
| 38 |  | simpr 484 | . . . . . . 7
⊢ (((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶) | 
| 39 |  | eleq1 2828 | . . . . . . 7
⊢ (𝐴 = 𝑥 → (𝐴 ∈ 𝐶 ↔ 𝑥 ∈ 𝐶)) | 
| 40 | 38, 39 | syl5ibrcom 247 | . . . . . 6
⊢ (((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) ∧ 𝑥 ∈ 𝐶) → (𝐴 = 𝑥 → 𝐴 ∈ 𝐶)) | 
| 41 | 40 | con3d 152 | . . . . 5
⊢ (((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) ∧ 𝑥 ∈ 𝐶) → (¬ 𝐴 ∈ 𝐶 → ¬ 𝐴 = 𝑥)) | 
| 42 |  | simpl1 1191 | . . . . . . 7
⊢ (((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) ∧ 𝑥 ∈ 𝐶) → 𝐵 ∈ (OBasis‘𝑊)) | 
| 43 |  | simpl3 1193 | . . . . . . 7
⊢ (((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) ∧ 𝑥 ∈ 𝐶) → 𝐴 ∈ 𝐵) | 
| 44 | 9 | sselda 3982 | . . . . . . 7
⊢ (((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐵) | 
| 45 |  | eqid 2736 | . . . . . . . 8
⊢
(·𝑖‘𝑊) =
(·𝑖‘𝑊) | 
| 46 |  | eqid 2736 | . . . . . . . 8
⊢
(Scalar‘𝑊) =
(Scalar‘𝑊) | 
| 47 |  | eqid 2736 | . . . . . . . 8
⊢
(1r‘(Scalar‘𝑊)) =
(1r‘(Scalar‘𝑊)) | 
| 48 |  | eqid 2736 | . . . . . . . 8
⊢
(0g‘(Scalar‘𝑊)) =
(0g‘(Scalar‘𝑊)) | 
| 49 | 10, 45, 46, 47, 48 | obsip 21742 | . . . . . . 7
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐴(·𝑖‘𝑊)𝑥) = if(𝐴 = 𝑥, (1r‘(Scalar‘𝑊)),
(0g‘(Scalar‘𝑊)))) | 
| 50 | 42, 43, 44, 49 | syl3anc 1372 | . . . . . 6
⊢ (((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) ∧ 𝑥 ∈ 𝐶) → (𝐴(·𝑖‘𝑊)𝑥) = if(𝐴 = 𝑥, (1r‘(Scalar‘𝑊)),
(0g‘(Scalar‘𝑊)))) | 
| 51 |  | iffalse 4533 | . . . . . . 7
⊢ (¬
𝐴 = 𝑥 → if(𝐴 = 𝑥, (1r‘(Scalar‘𝑊)),
(0g‘(Scalar‘𝑊))) =
(0g‘(Scalar‘𝑊))) | 
| 52 | 51 | eqeq2d 2747 | . . . . . 6
⊢ (¬
𝐴 = 𝑥 → ((𝐴(·𝑖‘𝑊)𝑥) = if(𝐴 = 𝑥, (1r‘(Scalar‘𝑊)),
(0g‘(Scalar‘𝑊))) ↔ (𝐴(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)))) | 
| 53 | 50, 52 | syl5ibcom 245 | . . . . 5
⊢ (((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) ∧ 𝑥 ∈ 𝐶) → (¬ 𝐴 = 𝑥 → (𝐴(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)))) | 
| 54 | 41, 53 | syld 47 | . . . 4
⊢ (((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) ∧ 𝑥 ∈ 𝐶) → (¬ 𝐴 ∈ 𝐶 → (𝐴(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)))) | 
| 55 | 54 | ralrimdva 3153 | . . 3
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → (¬ 𝐴 ∈ 𝐶 → ∀𝑥 ∈ 𝐶 (𝐴(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)))) | 
| 56 |  | simp3 1138 | . . . . 5
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐵) | 
| 57 | 12, 56 | sseldd 3983 | . . . 4
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → 𝐴 ∈ (Base‘𝑊)) | 
| 58 | 10, 45, 46, 48, 18 | elocv 21687 | . . . . . 6
⊢ (𝐴 ∈ ( ⊥ ‘𝐶) ↔ (𝐶 ⊆ (Base‘𝑊) ∧ 𝐴 ∈ (Base‘𝑊) ∧ ∀𝑥 ∈ 𝐶 (𝐴(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)))) | 
| 59 |  | df-3an 1088 | . . . . . 6
⊢ ((𝐶 ⊆ (Base‘𝑊) ∧ 𝐴 ∈ (Base‘𝑊) ∧ ∀𝑥 ∈ 𝐶 (𝐴(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))) ↔ ((𝐶 ⊆ (Base‘𝑊) ∧ 𝐴 ∈ (Base‘𝑊)) ∧ ∀𝑥 ∈ 𝐶 (𝐴(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)))) | 
| 60 | 58, 59 | bitri 275 | . . . . 5
⊢ (𝐴 ∈ ( ⊥ ‘𝐶) ↔ ((𝐶 ⊆ (Base‘𝑊) ∧ 𝐴 ∈ (Base‘𝑊)) ∧ ∀𝑥 ∈ 𝐶 (𝐴(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)))) | 
| 61 | 60 | baib 535 | . . . 4
⊢ ((𝐶 ⊆ (Base‘𝑊) ∧ 𝐴 ∈ (Base‘𝑊)) → (𝐴 ∈ ( ⊥ ‘𝐶) ↔ ∀𝑥 ∈ 𝐶 (𝐴(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)))) | 
| 62 | 13, 57, 61 | syl2anc 584 | . . 3
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ ( ⊥ ‘𝐶) ↔ ∀𝑥 ∈ 𝐶 (𝐴(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)))) | 
| 63 | 55, 62 | sylibrd 259 | . 2
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → (¬ 𝐴 ∈ 𝐶 → 𝐴 ∈ ( ⊥ ‘𝐶))) | 
| 64 | 37, 63 | impbid 212 | 1
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ ( ⊥ ‘𝐶) ↔ ¬ 𝐴 ∈ 𝐶)) |