Proof of Theorem lcfrlem1
Step | Hyp | Ref
| Expression |
1 | | lcfrlem1.h |
. . 3
⊢ 𝐻 = (𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)) |
2 | 1 | fveq1i 6739 |
. 2
⊢ (𝐻‘𝑋) = ((𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺))‘𝑋) |
3 | | lcfrlem1.v |
. . . 4
⊢ 𝑉 = (Base‘𝑈) |
4 | | lcfrlem1.s |
. . . 4
⊢ 𝑆 = (Scalar‘𝑈) |
5 | | eqid 2739 |
. . . 4
⊢
(-g‘𝑆) = (-g‘𝑆) |
6 | | lcfrlem1.f |
. . . 4
⊢ 𝐹 = (LFnl‘𝑈) |
7 | | lcfrlem1.d |
. . . 4
⊢ 𝐷 = (LDual‘𝑈) |
8 | | lcfrlem1.m |
. . . 4
⊢ − =
(-g‘𝐷) |
9 | | lcfrlem1.u |
. . . . 5
⊢ (𝜑 → 𝑈 ∈ LVec) |
10 | | lveclmod 20173 |
. . . . 5
⊢ (𝑈 ∈ LVec → 𝑈 ∈ LMod) |
11 | 9, 10 | syl 17 |
. . . 4
⊢ (𝜑 → 𝑈 ∈ LMod) |
12 | | lcfrlem1.e |
. . . 4
⊢ (𝜑 → 𝐸 ∈ 𝐹) |
13 | | eqid 2739 |
. . . . 5
⊢
(Base‘𝑆) =
(Base‘𝑆) |
14 | | lcfrlem1.t |
. . . . 5
⊢ · = (
·𝑠 ‘𝐷) |
15 | 4 | lvecdrng 20172 |
. . . . . . . 8
⊢ (𝑈 ∈ LVec → 𝑆 ∈
DivRing) |
16 | 9, 15 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ DivRing) |
17 | | lcfrlem1.g |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ 𝐹) |
18 | | lcfrlem1.x |
. . . . . . . 8
⊢ (𝜑 → 𝑋 ∈ 𝑉) |
19 | 4, 13, 3, 6 | lflcl 36847 |
. . . . . . . 8
⊢ ((𝑈 ∈ LVec ∧ 𝐺 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐺‘𝑋) ∈ (Base‘𝑆)) |
20 | 9, 17, 18, 19 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 → (𝐺‘𝑋) ∈ (Base‘𝑆)) |
21 | | lcfrlem1.n |
. . . . . . 7
⊢ (𝜑 → (𝐺‘𝑋) ≠ 0 ) |
22 | | lcfrlem1.z |
. . . . . . . 8
⊢ 0 =
(0g‘𝑆) |
23 | | lcfrlem1.i |
. . . . . . . 8
⊢ 𝐼 = (invr‘𝑆) |
24 | 13, 22, 23 | drnginvrcl 19814 |
. . . . . . 7
⊢ ((𝑆 ∈ DivRing ∧ (𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐺‘𝑋) ≠ 0 ) → (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆)) |
25 | 16, 20, 21, 24 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆)) |
26 | 4, 13, 3, 6 | lflcl 36847 |
. . . . . . 7
⊢ ((𝑈 ∈ LVec ∧ 𝐸 ∈ 𝐹 ∧ 𝑋 ∈ 𝑉) → (𝐸‘𝑋) ∈ (Base‘𝑆)) |
27 | 9, 12, 18, 26 | syl3anc 1373 |
. . . . . 6
⊢ (𝜑 → (𝐸‘𝑋) ∈ (Base‘𝑆)) |
28 | | lcfrlem1.q |
. . . . . . 7
⊢ × =
(.r‘𝑆) |
29 | 4, 13, 28 | lmodmcl 19941 |
. . . . . 6
⊢ ((𝑈 ∈ LMod ∧ (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆) ∧ (𝐸‘𝑋) ∈ (Base‘𝑆)) → ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) ∈ (Base‘𝑆)) |
30 | 11, 25, 27, 29 | syl3anc 1373 |
. . . . 5
⊢ (𝜑 → ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) ∈ (Base‘𝑆)) |
31 | 6, 4, 13, 7, 14, 11, 30, 17 | ldualvscl 36922 |
. . . 4
⊢ (𝜑 → (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺) ∈ 𝐹) |
32 | 3, 4, 5, 6, 7, 8, 11, 12, 31, 18 | ldualvsubval 36940 |
. . 3
⊢ (𝜑 → ((𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺))‘𝑋) = ((𝐸‘𝑋)(-g‘𝑆)((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋))) |
33 | 6, 3, 4, 13, 28, 7, 14, 9, 30, 17, 18 | ldualvsval 36921 |
. . . . 5
⊢ (𝜑 → ((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋) = ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)))) |
34 | | eqid 2739 |
. . . . . . . . 9
⊢
(1r‘𝑆) = (1r‘𝑆) |
35 | 13, 22, 28, 34, 23 | drnginvrr 19817 |
. . . . . . . 8
⊢ ((𝑆 ∈ DivRing ∧ (𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐺‘𝑋) ≠ 0 ) → ((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) = (1r‘𝑆)) |
36 | 16, 20, 21, 35 | syl3anc 1373 |
. . . . . . 7
⊢ (𝜑 → ((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) = (1r‘𝑆)) |
37 | 36 | oveq1d 7249 |
. . . . . 6
⊢ (𝜑 → (((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) × (𝐸‘𝑋)) = ((1r‘𝑆) × (𝐸‘𝑋))) |
38 | 4 | lmodring 19937 |
. . . . . . . 8
⊢ (𝑈 ∈ LMod → 𝑆 ∈ Ring) |
39 | 11, 38 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ Ring) |
40 | 13, 28 | ringass 19612 |
. . . . . . 7
⊢ ((𝑆 ∈ Ring ∧ ((𝐺‘𝑋) ∈ (Base‘𝑆) ∧ (𝐼‘(𝐺‘𝑋)) ∈ (Base‘𝑆) ∧ (𝐸‘𝑋) ∈ (Base‘𝑆))) → (((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) × (𝐸‘𝑋)) = ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)))) |
41 | 39, 20, 25, 27, 40 | syl13anc 1374 |
. . . . . 6
⊢ (𝜑 → (((𝐺‘𝑋) × (𝐼‘(𝐺‘𝑋))) × (𝐸‘𝑋)) = ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)))) |
42 | 13, 28, 34 | ringlidm 19619 |
. . . . . . 7
⊢ ((𝑆 ∈ Ring ∧ (𝐸‘𝑋) ∈ (Base‘𝑆)) → ((1r‘𝑆) × (𝐸‘𝑋)) = (𝐸‘𝑋)) |
43 | 39, 27, 42 | syl2anc 587 |
. . . . . 6
⊢ (𝜑 →
((1r‘𝑆)
×
(𝐸‘𝑋)) = (𝐸‘𝑋)) |
44 | 37, 41, 43 | 3eqtr3d 2787 |
. . . . 5
⊢ (𝜑 → ((𝐺‘𝑋) × ((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋))) = (𝐸‘𝑋)) |
45 | 33, 44 | eqtrd 2779 |
. . . 4
⊢ (𝜑 → ((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋) = (𝐸‘𝑋)) |
46 | 45 | oveq2d 7250 |
. . 3
⊢ (𝜑 → ((𝐸‘𝑋)(-g‘𝑆)((((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺)‘𝑋)) = ((𝐸‘𝑋)(-g‘𝑆)(𝐸‘𝑋))) |
47 | 4 | lmodfgrp 19938 |
. . . . 5
⊢ (𝑈 ∈ LMod → 𝑆 ∈ Grp) |
48 | 11, 47 | syl 17 |
. . . 4
⊢ (𝜑 → 𝑆 ∈ Grp) |
49 | 13, 22, 5 | grpsubid 18477 |
. . . 4
⊢ ((𝑆 ∈ Grp ∧ (𝐸‘𝑋) ∈ (Base‘𝑆)) → ((𝐸‘𝑋)(-g‘𝑆)(𝐸‘𝑋)) = 0 ) |
50 | 48, 27, 49 | syl2anc 587 |
. . 3
⊢ (𝜑 → ((𝐸‘𝑋)(-g‘𝑆)(𝐸‘𝑋)) = 0 ) |
51 | 32, 46, 50 | 3eqtrd 2783 |
. 2
⊢ (𝜑 → ((𝐸 − (((𝐼‘(𝐺‘𝑋)) × (𝐸‘𝑋)) · 𝐺))‘𝑋) = 0 ) |
52 | 2, 51 | syl5eq 2792 |
1
⊢ (𝜑 → (𝐻‘𝑋) = 0 ) |