Step | Hyp | Ref
| Expression |
1 | | simpr 488 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 = { 0 }) → 𝑋 = { 0 }) |
2 | 1 | oveq1d 7228 |
. . 3
⊢ ((𝜑 ∧ 𝑋 = { 0 }) → (𝑋 ∨ (𝑁‘{𝑇})) = ({ 0 } ∨ (𝑁‘{𝑇}))) |
3 | 1 | oveq1d 7228 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 = { 0 }) → (𝑋 ⊕ (𝑁‘{𝑇})) = ({ 0 } ⊕ (𝑁‘{𝑇}))) |
4 | | dihjat1.h |
. . . . . . 7
⊢ 𝐻 = (LHyp‘𝐾) |
5 | | dihjat1.u |
. . . . . . 7
⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
6 | | dihjat1.o |
. . . . . . 7
⊢ 0 =
(0g‘𝑈) |
7 | | dihjat1.i |
. . . . . . 7
⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
8 | | dihjat1.j |
. . . . . . 7
⊢ ∨ =
((joinH‘𝐾)‘𝑊) |
9 | | dihjat1.k |
. . . . . . 7
⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
10 | | dihjat1lem.q |
. . . . . . . . 9
⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) |
11 | | eldifi 4041 |
. . . . . . . . 9
⊢ (𝑇 ∈ (𝑉 ∖ { 0 }) → 𝑇 ∈ 𝑉) |
12 | 10, 11 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑇 ∈ 𝑉) |
13 | | dihjat1.v |
. . . . . . . . 9
⊢ 𝑉 = (Base‘𝑈) |
14 | | dihjat1.n |
. . . . . . . . 9
⊢ 𝑁 = (LSpan‘𝑈) |
15 | 4, 5, 13, 14, 7 | dihlsprn 39082 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑇 ∈ 𝑉) → (𝑁‘{𝑇}) ∈ ran 𝐼) |
16 | 9, 12, 15 | syl2anc 587 |
. . . . . . 7
⊢ (𝜑 → (𝑁‘{𝑇}) ∈ ran 𝐼) |
17 | 4, 5, 6, 7, 8, 9, 16 | djh02 39164 |
. . . . . 6
⊢ (𝜑 → ({ 0 } ∨ (𝑁‘{𝑇})) = (𝑁‘{𝑇})) |
18 | 4, 5, 9 | dvhlmod 38861 |
. . . . . . . 8
⊢ (𝜑 → 𝑈 ∈ LMod) |
19 | | eqid 2737 |
. . . . . . . . . 10
⊢
(LSubSp‘𝑈) =
(LSubSp‘𝑈) |
20 | 13, 19, 14 | lspsncl 20014 |
. . . . . . . . 9
⊢ ((𝑈 ∈ LMod ∧ 𝑇 ∈ 𝑉) → (𝑁‘{𝑇}) ∈ (LSubSp‘𝑈)) |
21 | 18, 12, 20 | syl2anc 587 |
. . . . . . . 8
⊢ (𝜑 → (𝑁‘{𝑇}) ∈ (LSubSp‘𝑈)) |
22 | 19 | lsssubg 19994 |
. . . . . . . 8
⊢ ((𝑈 ∈ LMod ∧ (𝑁‘{𝑇}) ∈ (LSubSp‘𝑈)) → (𝑁‘{𝑇}) ∈ (SubGrp‘𝑈)) |
23 | 18, 21, 22 | syl2anc 587 |
. . . . . . 7
⊢ (𝜑 → (𝑁‘{𝑇}) ∈ (SubGrp‘𝑈)) |
24 | | dihjat1.p |
. . . . . . . 8
⊢ ⊕ =
(LSSum‘𝑈) |
25 | 6, 24 | lsm02 19062 |
. . . . . . 7
⊢ ((𝑁‘{𝑇}) ∈ (SubGrp‘𝑈) → ({ 0 } ⊕ (𝑁‘{𝑇})) = (𝑁‘{𝑇})) |
26 | 23, 25 | syl 17 |
. . . . . 6
⊢ (𝜑 → ({ 0 } ⊕ (𝑁‘{𝑇})) = (𝑁‘{𝑇})) |
27 | 17, 26 | eqtr4d 2780 |
. . . . 5
⊢ (𝜑 → ({ 0 } ∨ (𝑁‘{𝑇})) = ({ 0 } ⊕ (𝑁‘{𝑇}))) |
28 | 27 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 = { 0 }) → ({ 0 } ∨ (𝑁‘{𝑇})) = ({ 0 } ⊕ (𝑁‘{𝑇}))) |
29 | 3, 28 | eqtr4d 2780 |
. . 3
⊢ ((𝜑 ∧ 𝑋 = { 0 }) → (𝑋 ⊕ (𝑁‘{𝑇})) = ({ 0 } ∨ (𝑁‘{𝑇}))) |
30 | 2, 29 | eqtr4d 2780 |
. 2
⊢ ((𝜑 ∧ 𝑋 = { 0 }) → (𝑋 ∨ (𝑁‘{𝑇})) = (𝑋 ⊕ (𝑁‘{𝑇}))) |
31 | 18 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 ≠ { 0 }) → 𝑈 ∈ LMod) |
32 | | dihjat1.x |
. . . . . . . 8
⊢ (𝜑 → 𝑋 ∈ ran 𝐼) |
33 | 4, 5, 7, 13 | dihrnss 39029 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → 𝑋 ⊆ 𝑉) |
34 | 9, 32, 33 | syl2anc 587 |
. . . . . . 7
⊢ (𝜑 → 𝑋 ⊆ 𝑉) |
35 | 13, 19 | lssss 19973 |
. . . . . . . 8
⊢ ((𝑁‘{𝑇}) ∈ (LSubSp‘𝑈) → (𝑁‘{𝑇}) ⊆ 𝑉) |
36 | 21, 35 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (𝑁‘{𝑇}) ⊆ 𝑉) |
37 | 4, 7, 5, 13, 8 | djhcl 39151 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ (𝑁‘{𝑇}) ⊆ 𝑉)) → (𝑋 ∨ (𝑁‘{𝑇})) ∈ ran 𝐼) |
38 | 9, 34, 36, 37 | syl12anc 837 |
. . . . . 6
⊢ (𝜑 → (𝑋 ∨ (𝑁‘{𝑇})) ∈ ran 𝐼) |
39 | 4, 5, 7, 13 | dihrnss 39029 |
. . . . . 6
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∨ (𝑁‘{𝑇})) ∈ ran 𝐼) → (𝑋 ∨ (𝑁‘{𝑇})) ⊆ 𝑉) |
40 | 9, 38, 39 | syl2anc 587 |
. . . . 5
⊢ (𝜑 → (𝑋 ∨ (𝑁‘{𝑇})) ⊆ 𝑉) |
41 | 40 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 ≠ { 0 }) → (𝑋 ∨ (𝑁‘{𝑇})) ⊆ 𝑉) |
42 | 4, 5, 7, 19 | dihrnlss 39028 |
. . . . . . 7
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → 𝑋 ∈ (LSubSp‘𝑈)) |
43 | 9, 32, 42 | syl2anc 587 |
. . . . . 6
⊢ (𝜑 → 𝑋 ∈ (LSubSp‘𝑈)) |
44 | 19, 24 | lsmcl 20120 |
. . . . . 6
⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ (LSubSp‘𝑈) ∧ (𝑁‘{𝑇}) ∈ (LSubSp‘𝑈)) → (𝑋 ⊕ (𝑁‘{𝑇})) ∈ (LSubSp‘𝑈)) |
45 | 18, 43, 21, 44 | syl3anc 1373 |
. . . . 5
⊢ (𝜑 → (𝑋 ⊕ (𝑁‘{𝑇})) ∈ (LSubSp‘𝑈)) |
46 | 45 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ 𝑋 ≠ { 0 }) → (𝑋 ⊕ (𝑁‘{𝑇})) ∈ (LSubSp‘𝑈)) |
47 | | simplr 769 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → 𝑋 ≠ { 0 }) |
48 | 9 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
49 | 32 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → 𝑋 ∈ ran 𝐼) |
50 | | simpr 488 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → 𝑥 ∈ (𝑉 ∖ { 0 })) |
51 | 10 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → 𝑇 ∈ (𝑉 ∖ { 0 })) |
52 | 4, 5, 13, 6, 14, 7, 8, 48, 49, 50, 51 | djhcvat42 39166 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → ((𝑋 ≠ { 0 } ∧ (𝑁‘{𝑥}) ⊆ (𝑋 ∨ (𝑁‘{𝑇}))) → ∃𝑦 ∈ (𝑉 ∖ { 0 })((𝑁‘{𝑦}) ⊆ 𝑋 ∧ (𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑇}))))) |
53 | 47, 52 | mpand 695 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → ((𝑁‘{𝑥}) ⊆ (𝑋 ∨ (𝑁‘{𝑇})) → ∃𝑦 ∈ (𝑉 ∖ { 0 })((𝑁‘{𝑦}) ⊆ 𝑋 ∧ (𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑇}))))) |
54 | | simprrl 781 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) ∧ (𝑦 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑦}) ⊆ 𝑋 ∧ (𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑇}))))) → (𝑁‘{𝑦}) ⊆ 𝑋) |
55 | 18 | ad3antrrr 730 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) ∧ (𝑦 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑦}) ⊆ 𝑋 ∧ (𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑇}))))) → 𝑈 ∈ LMod) |
56 | 43 | ad3antrrr 730 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) ∧ (𝑦 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑦}) ⊆ 𝑋 ∧ (𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑇}))))) → 𝑋 ∈ (LSubSp‘𝑈)) |
57 | | eldifi 4041 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ (𝑉 ∖ { 0 }) → 𝑦 ∈ 𝑉) |
58 | 57 | ad2antrl 728 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) ∧ (𝑦 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑦}) ⊆ 𝑋 ∧ (𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑇}))))) → 𝑦 ∈ 𝑉) |
59 | 13, 19, 14, 55, 56, 58 | lspsnel5 20032 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) ∧ (𝑦 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑦}) ⊆ 𝑋 ∧ (𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑇}))))) → (𝑦 ∈ 𝑋 ↔ (𝑁‘{𝑦}) ⊆ 𝑋)) |
60 | 54, 59 | mpbird 260 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) ∧ (𝑦 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑦}) ⊆ 𝑋 ∧ (𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑇}))))) → 𝑦 ∈ 𝑋) |
61 | 12 | ad3antrrr 730 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) ∧ (𝑦 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑦}) ⊆ 𝑋 ∧ (𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑇}))))) → 𝑇 ∈ 𝑉) |
62 | 13, 14 | lspsnid 20030 |
. . . . . . . . . . . 12
⊢ ((𝑈 ∈ LMod ∧ 𝑇 ∈ 𝑉) → 𝑇 ∈ (𝑁‘{𝑇})) |
63 | 55, 61, 62 | syl2anc 587 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) ∧ (𝑦 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑦}) ⊆ 𝑋 ∧ (𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑇}))))) → 𝑇 ∈ (𝑁‘{𝑇})) |
64 | | simprrr 782 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) ∧ (𝑦 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑦}) ⊆ 𝑋 ∧ (𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑇}))))) → (𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑇}))) |
65 | | sneq 4551 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = 𝑇 → {𝑧} = {𝑇}) |
66 | 65 | fveq2d 6721 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = 𝑇 → (𝑁‘{𝑧}) = (𝑁‘{𝑇})) |
67 | 66 | oveq2d 7229 |
. . . . . . . . . . . . 13
⊢ (𝑧 = 𝑇 → ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑧})) = ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑇}))) |
68 | 67 | sseq2d 3933 |
. . . . . . . . . . . 12
⊢ (𝑧 = 𝑇 → ((𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑧})) ↔ (𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑇})))) |
69 | 68 | rspcev 3537 |
. . . . . . . . . . 11
⊢ ((𝑇 ∈ (𝑁‘{𝑇}) ∧ (𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑇}))) → ∃𝑧 ∈ (𝑁‘{𝑇})(𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑧}))) |
70 | 63, 64, 69 | syl2anc 587 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) ∧ (𝑦 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑦}) ⊆ 𝑋 ∧ (𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑇}))))) → ∃𝑧 ∈ (𝑁‘{𝑇})(𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑧}))) |
71 | 60, 70 | jca 515 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) ∧ (𝑦 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑦}) ⊆ 𝑋 ∧ (𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑇}))))) → (𝑦 ∈ 𝑋 ∧ ∃𝑧 ∈ (𝑁‘{𝑇})(𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑧})))) |
72 | 71 | ex 416 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → ((𝑦 ∈ (𝑉 ∖ { 0 }) ∧ ((𝑁‘{𝑦}) ⊆ 𝑋 ∧ (𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑇})))) → (𝑦 ∈ 𝑋 ∧ ∃𝑧 ∈ (𝑁‘{𝑇})(𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑧}))))) |
73 | 72 | reximdv2 3190 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (∃𝑦 ∈ (𝑉 ∖ { 0 })((𝑁‘{𝑦}) ⊆ 𝑋 ∧ (𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑇}))) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ (𝑁‘{𝑇})(𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑧})))) |
74 | 53, 73 | syld 47 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → ((𝑁‘{𝑥}) ⊆ (𝑋 ∨ (𝑁‘{𝑇})) → ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ (𝑁‘{𝑇})(𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑧})))) |
75 | 74 | anim2d 615 |
. . . . 5
⊢ (((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → ((𝑥 ∈ 𝑉 ∧ (𝑁‘{𝑥}) ⊆ (𝑋 ∨ (𝑁‘{𝑇}))) → (𝑥 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ (𝑁‘{𝑇})(𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑧}))))) |
76 | 4, 5, 7, 19 | dihrnlss 39028 |
. . . . . . . 8
⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∨ (𝑁‘{𝑇})) ∈ ran 𝐼) → (𝑋 ∨ (𝑁‘{𝑇})) ∈ (LSubSp‘𝑈)) |
77 | 9, 38, 76 | syl2anc 587 |
. . . . . . 7
⊢ (𝜑 → (𝑋 ∨ (𝑁‘{𝑇})) ∈ (LSubSp‘𝑈)) |
78 | 13, 19, 14, 18, 77 | lspsnel6 20031 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (𝑋 ∨ (𝑁‘{𝑇})) ↔ (𝑥 ∈ 𝑉 ∧ (𝑁‘{𝑥}) ⊆ (𝑋 ∨ (𝑁‘{𝑇}))))) |
79 | 78 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝑥 ∈ (𝑋 ∨ (𝑁‘{𝑇})) ↔ (𝑥 ∈ 𝑉 ∧ (𝑁‘{𝑥}) ⊆ (𝑋 ∨ (𝑁‘{𝑇}))))) |
80 | 13, 19, 24, 14, 18, 43, 21 | lsmelval2 20122 |
. . . . . . 7
⊢ (𝜑 → (𝑥 ∈ (𝑋 ⊕ (𝑁‘{𝑇})) ↔ (𝑥 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ (𝑁‘{𝑇})(𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ⊕ (𝑁‘{𝑧}))))) |
81 | 9 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (𝑁‘{𝑇})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
82 | 43 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑋 ∈ (LSubSp‘𝑈)) |
83 | | simplr 769 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑦 ∈ 𝑋) |
84 | 13, 19 | lssel 19974 |
. . . . . . . . . . . . 13
⊢ ((𝑋 ∈ (LSubSp‘𝑈) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑉) |
85 | 82, 83, 84 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑦 ∈ 𝑉) |
86 | 21 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (𝑁‘{𝑇})) → (𝑁‘{𝑇}) ∈ (LSubSp‘𝑈)) |
87 | | simpr 488 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑧 ∈ (𝑁‘{𝑇})) |
88 | 13, 19 | lssel 19974 |
. . . . . . . . . . . . 13
⊢ (((𝑁‘{𝑇}) ∈ (LSubSp‘𝑈) ∧ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑧 ∈ 𝑉) |
89 | 86, 87, 88 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (𝑁‘{𝑇})) → 𝑧 ∈ 𝑉) |
90 | 4, 5, 13, 24, 14, 7, 8, 81, 85, 89 | djhlsmat 39178 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (𝑁‘{𝑇})) → ((𝑁‘{𝑦}) ⊕ (𝑁‘{𝑧})) = ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑧}))) |
91 | 90 | sseq2d 3933 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑋) ∧ 𝑧 ∈ (𝑁‘{𝑇})) → ((𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ⊕ (𝑁‘{𝑧})) ↔ (𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑧})))) |
92 | 91 | rexbidva 3215 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (∃𝑧 ∈ (𝑁‘{𝑇})(𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ⊕ (𝑁‘{𝑧})) ↔ ∃𝑧 ∈ (𝑁‘{𝑇})(𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑧})))) |
93 | 92 | rexbidva 3215 |
. . . . . . . 8
⊢ (𝜑 → (∃𝑦 ∈ 𝑋 ∃𝑧 ∈ (𝑁‘{𝑇})(𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ⊕ (𝑁‘{𝑧})) ↔ ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ (𝑁‘{𝑇})(𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑧})))) |
94 | 93 | anbi2d 632 |
. . . . . . 7
⊢ (𝜑 → ((𝑥 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ (𝑁‘{𝑇})(𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ⊕ (𝑁‘{𝑧}))) ↔ (𝑥 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ (𝑁‘{𝑇})(𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑧}))))) |
95 | 80, 94 | bitrd 282 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ (𝑋 ⊕ (𝑁‘{𝑇})) ↔ (𝑥 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ (𝑁‘{𝑇})(𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑧}))))) |
96 | 95 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝑥 ∈ (𝑋 ⊕ (𝑁‘{𝑇})) ↔ (𝑥 ∈ 𝑉 ∧ ∃𝑦 ∈ 𝑋 ∃𝑧 ∈ (𝑁‘{𝑇})(𝑁‘{𝑥}) ⊆ ((𝑁‘{𝑦}) ∨ (𝑁‘{𝑧}))))) |
97 | 75, 79, 96 | 3imtr4d 297 |
. . . 4
⊢ (((𝜑 ∧ 𝑋 ≠ { 0 }) ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝑥 ∈ (𝑋 ∨ (𝑁‘{𝑇})) → 𝑥 ∈ (𝑋 ⊕ (𝑁‘{𝑇})))) |
98 | 6, 19, 31, 41, 46, 97 | lssssr 19990 |
. . 3
⊢ ((𝜑 ∧ 𝑋 ≠ { 0 }) → (𝑋 ∨ (𝑁‘{𝑇})) ⊆ (𝑋 ⊕ (𝑁‘{𝑇}))) |
99 | 4, 5, 13, 24, 8, 9, 34, 36 | djhsumss 39158 |
. . . 4
⊢ (𝜑 → (𝑋 ⊕ (𝑁‘{𝑇})) ⊆ (𝑋 ∨ (𝑁‘{𝑇}))) |
100 | 99 | adantr 484 |
. . 3
⊢ ((𝜑 ∧ 𝑋 ≠ { 0 }) → (𝑋 ⊕ (𝑁‘{𝑇})) ⊆ (𝑋 ∨ (𝑁‘{𝑇}))) |
101 | 98, 100 | eqssd 3918 |
. 2
⊢ ((𝜑 ∧ 𝑋 ≠ { 0 }) → (𝑋 ∨ (𝑁‘{𝑇})) = (𝑋 ⊕ (𝑁‘{𝑇}))) |
102 | 30, 101 | pm2.61dane 3029 |
1
⊢ (𝜑 → (𝑋 ∨ (𝑁‘{𝑇})) = (𝑋 ⊕ (𝑁‘{𝑇}))) |