Step | Hyp | Ref
| Expression |
1 | | simp3 1056 |
. 2
⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) |
2 | | simp2 1055 |
. . . 4
⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑇 ⊊ 𝑈) |
3 | | pssss 3664 |
. . . 4
⊢ (𝑇 ⊊ 𝑈 → 𝑇 ⊆ 𝑈) |
4 | 2, 3 | syl 17 |
. . 3
⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑇 ⊆ 𝑈) |
5 | | pssnel 3991 |
. . . . 5
⊢ (𝑇 ⊊ 𝑈 → ∃𝑥(𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) |
6 | 2, 5 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → ∃𝑥(𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) |
7 | | simpl3 1059 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) |
8 | | simprl 790 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → 𝑥 ∈ 𝑈) |
9 | 7, 8 | sseldd 3569 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → 𝑥 ∈ (𝑇 ⊕ (𝑁‘{𝑋}))) |
10 | | lsmcv.w |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑊 ∈ LVec) |
11 | | lveclmod 18927 |
. . . . . . . . . . . 12
⊢ (𝑊 ∈ LVec → 𝑊 ∈ LMod) |
12 | 10, 11 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑊 ∈ LMod) |
13 | | lsmcv.s |
. . . . . . . . . . . 12
⊢ 𝑆 = (LSubSp‘𝑊) |
14 | 13 | lsssssubg 18779 |
. . . . . . . . . . 11
⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
15 | 12, 14 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊)) |
16 | | lsmcv.t |
. . . . . . . . . 10
⊢ (𝜑 → 𝑇 ∈ 𝑆) |
17 | 15, 16 | sseldd 3569 |
. . . . . . . . 9
⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊)) |
18 | | lsmcv.x |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑋 ∈ 𝑉) |
19 | | lsmcv.v |
. . . . . . . . . . . 12
⊢ 𝑉 = (Base‘𝑊) |
20 | | lsmcv.n |
. . . . . . . . . . . 12
⊢ 𝑁 = (LSpan‘𝑊) |
21 | 19, 13, 20 | lspsncl 18798 |
. . . . . . . . . . 11
⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ 𝑆) |
22 | 12, 18, 21 | syl2anc 691 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑁‘{𝑋}) ∈ 𝑆) |
23 | 15, 22 | sseldd 3569 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) |
24 | | eqid 2610 |
. . . . . . . . . 10
⊢
(+g‘𝑊) = (+g‘𝑊) |
25 | | lsmcv.p |
. . . . . . . . . 10
⊢ ⊕ =
(LSSum‘𝑊) |
26 | 24, 25 | lsmelval 17887 |
. . . . . . . . 9
⊢ ((𝑇 ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) → (𝑥 ∈ (𝑇 ⊕ (𝑁‘{𝑋})) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ (𝑁‘{𝑋})𝑥 = (𝑦(+g‘𝑊)𝑧))) |
27 | 17, 23, 26 | syl2anc 691 |
. . . . . . . 8
⊢ (𝜑 → (𝑥 ∈ (𝑇 ⊕ (𝑁‘{𝑋})) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ (𝑁‘{𝑋})𝑥 = (𝑦(+g‘𝑊)𝑧))) |
28 | 27 | 3ad2ant1 1075 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → (𝑥 ∈ (𝑇 ⊕ (𝑁‘{𝑋})) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ (𝑁‘{𝑋})𝑥 = (𝑦(+g‘𝑊)𝑧))) |
29 | 28 | adantr 480 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → (𝑥 ∈ (𝑇 ⊕ (𝑁‘{𝑋})) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ (𝑁‘{𝑋})𝑥 = (𝑦(+g‘𝑊)𝑧))) |
30 | 9, 29 | mpbid 221 |
. . . . 5
⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ (𝑁‘{𝑋})𝑥 = (𝑦(+g‘𝑊)𝑧)) |
31 | | simp1rr 1120 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → ¬ 𝑥 ∈ 𝑇) |
32 | | simp2l 1080 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → 𝑦 ∈ 𝑇) |
33 | | oveq2 6557 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = (0g‘𝑊) → (𝑦(+g‘𝑊)𝑧) = (𝑦(+g‘𝑊)(0g‘𝑊))) |
34 | 33 | eqeq2d 2620 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = (0g‘𝑊) → (𝑥 = (𝑦(+g‘𝑊)𝑧) ↔ 𝑥 = (𝑦(+g‘𝑊)(0g‘𝑊)))) |
35 | 34 | biimpac 502 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 = (𝑦(+g‘𝑊)𝑧) ∧ 𝑧 = (0g‘𝑊)) → 𝑥 = (𝑦(+g‘𝑊)(0g‘𝑊))) |
36 | 12 | 3ad2ant1 1075 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑊 ∈ LMod) |
37 | 36 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋}))) → 𝑊 ∈ LMod) |
38 | 16 | 3ad2ant1 1075 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑇 ∈ 𝑆) |
39 | 38 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋}))) → 𝑇 ∈ 𝑆) |
40 | | simprl 790 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋}))) → 𝑦 ∈ 𝑇) |
41 | 19, 13 | lssel 18759 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑇 ∈ 𝑆 ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑉) |
42 | 39, 40, 41 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋}))) → 𝑦 ∈ 𝑉) |
43 | | eqid 2610 |
. . . . . . . . . . . . . . . . . . 19
⊢
(0g‘𝑊) = (0g‘𝑊) |
44 | 19, 24, 43 | lmod0vrid 18717 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑊 ∈ LMod ∧ 𝑦 ∈ 𝑉) → (𝑦(+g‘𝑊)(0g‘𝑊)) = 𝑦) |
45 | 37, 42, 44 | syl2anc 691 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋}))) → (𝑦(+g‘𝑊)(0g‘𝑊)) = 𝑦) |
46 | 45 | eqeq2d 2620 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋}))) → (𝑥 = (𝑦(+g‘𝑊)(0g‘𝑊)) ↔ 𝑥 = 𝑦)) |
47 | 46 | biimpd 218 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋}))) → (𝑥 = (𝑦(+g‘𝑊)(0g‘𝑊)) → 𝑥 = 𝑦)) |
48 | 47 | ex 449 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → ((𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) → (𝑥 = (𝑦(+g‘𝑊)(0g‘𝑊)) → 𝑥 = 𝑦))) |
49 | 35, 48 | syl7 72 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → ((𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) → ((𝑥 = (𝑦(+g‘𝑊)𝑧) ∧ 𝑧 = (0g‘𝑊)) → 𝑥 = 𝑦))) |
50 | 49 | exp4a 631 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → ((𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) → (𝑥 = (𝑦(+g‘𝑊)𝑧) → (𝑧 = (0g‘𝑊) → 𝑥 = 𝑦)))) |
51 | 50 | 3imp 1249 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → (𝑧 = (0g‘𝑊) → 𝑥 = 𝑦)) |
52 | | eleq1 2676 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝑇 ↔ 𝑦 ∈ 𝑇)) |
53 | 52 | biimparc 503 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ 𝑇 ∧ 𝑥 = 𝑦) → 𝑥 ∈ 𝑇) |
54 | 32, 51, 53 | syl6an 566 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → (𝑧 = (0g‘𝑊) → 𝑥 ∈ 𝑇)) |
55 | 54 | necon3bd 2796 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → (¬ 𝑥 ∈ 𝑇 → 𝑧 ≠ (0g‘𝑊))) |
56 | 31, 55 | mpd 15 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → 𝑧 ≠ (0g‘𝑊)) |
57 | 10 | 3ad2ant1 1075 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑊 ∈ LVec) |
58 | 57 | adantr 480 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → 𝑊 ∈ LVec) |
59 | 58 | 3ad2ant1 1075 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → 𝑊 ∈ LVec) |
60 | | lmodabl 18733 |
. . . . . . . . . . . 12
⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) |
61 | 11, 60 | syl 17 |
. . . . . . . . . . 11
⊢ (𝑊 ∈ LVec → 𝑊 ∈ Abel) |
62 | 59, 61 | syl 17 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → 𝑊 ∈ Abel) |
63 | | simp1l1 1147 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → 𝜑) |
64 | 63, 16 | syl 17 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → 𝑇 ∈ 𝑆) |
65 | 64, 32, 41 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → 𝑦 ∈ 𝑉) |
66 | 59, 11 | syl 17 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → 𝑊 ∈ LMod) |
67 | 63, 18 | syl 17 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → 𝑋 ∈ 𝑉) |
68 | 66, 67, 21 | syl2anc 691 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → (𝑁‘{𝑋}) ∈ 𝑆) |
69 | | simp2r 1081 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → 𝑧 ∈ (𝑁‘{𝑋})) |
70 | 19, 13 | lssel 18759 |
. . . . . . . . . . 11
⊢ (((𝑁‘{𝑋}) ∈ 𝑆 ∧ 𝑧 ∈ (𝑁‘{𝑋})) → 𝑧 ∈ 𝑉) |
71 | 68, 69, 70 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → 𝑧 ∈ 𝑉) |
72 | | eqid 2610 |
. . . . . . . . . . 11
⊢
(-g‘𝑊) = (-g‘𝑊) |
73 | 19, 24, 72 | ablpncan2 18044 |
. . . . . . . . . 10
⊢ ((𝑊 ∈ Abel ∧ 𝑦 ∈ 𝑉 ∧ 𝑧 ∈ 𝑉) → ((𝑦(+g‘𝑊)𝑧)(-g‘𝑊)𝑦) = 𝑧) |
74 | 62, 65, 71, 73 | syl3anc 1318 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → ((𝑦(+g‘𝑊)𝑧)(-g‘𝑊)𝑦) = 𝑧) |
75 | | lsmcv.u |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑈 ∈ 𝑆) |
76 | 63, 75 | syl 17 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → 𝑈 ∈ 𝑆) |
77 | | simp3 1056 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → 𝑥 = (𝑦(+g‘𝑊)𝑧)) |
78 | | simp1rl 1119 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → 𝑥 ∈ 𝑈) |
79 | 77, 78 | eqeltrrd 2689 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → (𝑦(+g‘𝑊)𝑧) ∈ 𝑈) |
80 | | simp1l2 1148 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → 𝑇 ⊊ 𝑈) |
81 | 3 | sselda 3568 |
. . . . . . . . . . 11
⊢ ((𝑇 ⊊ 𝑈 ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ 𝑈) |
82 | 80, 32, 81 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → 𝑦 ∈ 𝑈) |
83 | 72, 13 | lssvsubcl 18765 |
. . . . . . . . . 10
⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ ((𝑦(+g‘𝑊)𝑧) ∈ 𝑈 ∧ 𝑦 ∈ 𝑈)) → ((𝑦(+g‘𝑊)𝑧)(-g‘𝑊)𝑦) ∈ 𝑈) |
84 | 66, 76, 79, 82, 83 | syl22anc 1319 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → ((𝑦(+g‘𝑊)𝑧)(-g‘𝑊)𝑦) ∈ 𝑈) |
85 | 74, 84 | eqeltrrd 2689 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → 𝑧 ∈ 𝑈) |
86 | 59 | 3ad2ant1 1075 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0g‘𝑊) ∧ 𝑧 ∈ 𝑈) → 𝑊 ∈ LVec) |
87 | 63 | 3ad2ant1 1075 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0g‘𝑊) ∧ 𝑧 ∈ 𝑈) → 𝜑) |
88 | 87, 18 | syl 17 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0g‘𝑊) ∧ 𝑧 ∈ 𝑈) → 𝑋 ∈ 𝑉) |
89 | | simp12r 1168 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0g‘𝑊) ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ (𝑁‘{𝑋})) |
90 | | simp2 1055 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0g‘𝑊) ∧ 𝑧 ∈ 𝑈) → 𝑧 ≠ (0g‘𝑊)) |
91 | 19, 43, 20, 86, 88, 89, 90 | lspsneleq 18936 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0g‘𝑊) ∧ 𝑧 ∈ 𝑈) → (𝑁‘{𝑧}) = (𝑁‘{𝑋})) |
92 | 86, 11 | syl 17 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0g‘𝑊) ∧ 𝑧 ∈ 𝑈) → 𝑊 ∈ LMod) |
93 | 87, 75 | syl 17 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0g‘𝑊) ∧ 𝑧 ∈ 𝑈) → 𝑈 ∈ 𝑆) |
94 | | simp3 1056 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0g‘𝑊) ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ 𝑈) |
95 | 13, 20, 92, 93, 94 | lspsnel5a 18817 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0g‘𝑊) ∧ 𝑧 ∈ 𝑈) → (𝑁‘{𝑧}) ⊆ 𝑈) |
96 | 91, 95 | eqsstr3d 3603 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) ∧ 𝑧 ≠ (0g‘𝑊) ∧ 𝑧 ∈ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈) |
97 | 56, 85, 96 | mpd3an23 1418 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) ∧ (𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) ∧ 𝑥 = (𝑦(+g‘𝑊)𝑧)) → (𝑁‘{𝑋}) ⊆ 𝑈) |
98 | 97 | 3exp 1256 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → ((𝑦 ∈ 𝑇 ∧ 𝑧 ∈ (𝑁‘{𝑋})) → (𝑥 = (𝑦(+g‘𝑊)𝑧) → (𝑁‘{𝑋}) ⊆ 𝑈))) |
99 | 98 | rexlimdvv 3019 |
. . . . 5
⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → (∃𝑦 ∈ 𝑇 ∃𝑧 ∈ (𝑁‘{𝑋})𝑥 = (𝑦(+g‘𝑊)𝑧) → (𝑁‘{𝑋}) ⊆ 𝑈)) |
100 | 30, 99 | mpd 15 |
. . . 4
⊢ (((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) ∧ (𝑥 ∈ 𝑈 ∧ ¬ 𝑥 ∈ 𝑇)) → (𝑁‘{𝑋}) ⊆ 𝑈) |
101 | 6, 100 | exlimddv 1850 |
. . 3
⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → (𝑁‘{𝑋}) ⊆ 𝑈) |
102 | 15, 75 | sseldd 3569 |
. . . . 5
⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
103 | 25 | lsmlub 17901 |
. . . . 5
⊢ ((𝑇 ∈ (SubGrp‘𝑊) ∧ (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊) ∧ 𝑈 ∈ (SubGrp‘𝑊)) → ((𝑇 ⊆ 𝑈 ∧ (𝑁‘{𝑋}) ⊆ 𝑈) ↔ (𝑇 ⊕ (𝑁‘{𝑋})) ⊆ 𝑈)) |
104 | 17, 23, 102, 103 | syl3anc 1318 |
. . . 4
⊢ (𝜑 → ((𝑇 ⊆ 𝑈 ∧ (𝑁‘{𝑋}) ⊆ 𝑈) ↔ (𝑇 ⊕ (𝑁‘{𝑋})) ⊆ 𝑈)) |
105 | 104 | 3ad2ant1 1075 |
. . 3
⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → ((𝑇 ⊆ 𝑈 ∧ (𝑁‘{𝑋}) ⊆ 𝑈) ↔ (𝑇 ⊕ (𝑁‘{𝑋})) ⊆ 𝑈)) |
106 | 4, 101, 105 | mpbi2and 958 |
. 2
⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → (𝑇 ⊕ (𝑁‘{𝑋})) ⊆ 𝑈) |
107 | 1, 106 | eqssd 3585 |
1
⊢ ((𝜑 ∧ 𝑇 ⊊ 𝑈 ∧ 𝑈 ⊆ (𝑇 ⊕ (𝑁‘{𝑋}))) → 𝑈 = (𝑇 ⊕ (𝑁‘{𝑋}))) |