| Step | Hyp | Ref
| Expression |
| 1 | | lcvexch.s |
. . . 4
⊢ 𝑆 = (LSubSp‘𝑊) |
| 2 | | lcvexch.c |
. . . 4
⊢ 𝐶 = ( ⋖L
‘𝑊) |
| 3 | | lcvexch.w |
. . . 4
⊢ (𝜑 → 𝑊 ∈ LMod) |
| 4 | | lcvexch.t |
. . . . 5
⊢ (𝜑 → 𝑇 ∈ 𝑆) |
| 5 | | lcvexch.u |
. . . . 5
⊢ (𝜑 → 𝑈 ∈ 𝑆) |
| 6 | 1 | lssincl 20963 |
. . . . 5
⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ∩ 𝑈) ∈ 𝑆) |
| 7 | 3, 4, 5, 6 | syl3anc 1373 |
. . . 4
⊢ (𝜑 → (𝑇 ∩ 𝑈) ∈ 𝑆) |
| 8 | | lcvexch.g |
. . . 4
⊢ (𝜑 → (𝑇 ∩ 𝑈)𝐶𝑈) |
| 9 | 1, 2, 3, 7, 5, 8 | lcvpss 39025 |
. . 3
⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊊ 𝑈) |
| 10 | | lcvexch.p |
. . . 4
⊢ ⊕ =
(LSSum‘𝑊) |
| 11 | 1, 10, 2, 3, 4, 5 | lcvexchlem1 39035 |
. . 3
⊢ (𝜑 → (𝑇 ⊊ (𝑇 ⊕ 𝑈) ↔ (𝑇 ∩ 𝑈) ⊊ 𝑈)) |
| 12 | 9, 11 | mpbird 257 |
. 2
⊢ (𝜑 → 𝑇 ⊊ (𝑇 ⊕ 𝑈)) |
| 13 | | simp3l 1202 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑇 ⊆ 𝑠) |
| 14 | 13 | ssrind 4244 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈)) |
| 15 | | inss2 4238 |
. . . . . . 7
⊢ (𝑠 ∩ 𝑈) ⊆ 𝑈 |
| 16 | 14, 15 | jctir 520 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈)) |
| 17 | 8 | 3ad2ant1 1134 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (𝑇 ∩ 𝑈)𝐶𝑈) |
| 18 | 1, 2, 3, 7, 5 | lcvbr3 39024 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑇 ∩ 𝑈)𝐶𝑈 ↔ ((𝑇 ∩ 𝑈) ⊊ 𝑈 ∧ ∀𝑟 ∈ 𝑆 (((𝑇 ∩ 𝑈) ⊆ 𝑟 ∧ 𝑟 ⊆ 𝑈) → (𝑟 = (𝑇 ∩ 𝑈) ∨ 𝑟 = 𝑈))))) |
| 19 | 18 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((𝑇 ∩ 𝑈)𝐶𝑈 ↔ ((𝑇 ∩ 𝑈) ⊊ 𝑈 ∧ ∀𝑟 ∈ 𝑆 (((𝑇 ∩ 𝑈) ⊆ 𝑟 ∧ 𝑟 ⊆ 𝑈) → (𝑟 = (𝑇 ∩ 𝑈) ∨ 𝑟 = 𝑈))))) |
| 20 | 3 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑊 ∈ LMod) |
| 21 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑠 ∈ 𝑆) |
| 22 | 5 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → 𝑈 ∈ 𝑆) |
| 23 | 1 | lssincl 20963 |
. . . . . . . . . . . 12
⊢ ((𝑊 ∈ LMod ∧ 𝑠 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑠 ∩ 𝑈) ∈ 𝑆) |
| 24 | 20, 21, 22, 23 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (𝑠 ∩ 𝑈) ∈ 𝑆) |
| 25 | | sseq2 4010 |
. . . . . . . . . . . . . 14
⊢ (𝑟 = (𝑠 ∩ 𝑈) → ((𝑇 ∩ 𝑈) ⊆ 𝑟 ↔ (𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈))) |
| 26 | | sseq1 4009 |
. . . . . . . . . . . . . 14
⊢ (𝑟 = (𝑠 ∩ 𝑈) → (𝑟 ⊆ 𝑈 ↔ (𝑠 ∩ 𝑈) ⊆ 𝑈)) |
| 27 | 25, 26 | anbi12d 632 |
. . . . . . . . . . . . 13
⊢ (𝑟 = (𝑠 ∩ 𝑈) → (((𝑇 ∩ 𝑈) ⊆ 𝑟 ∧ 𝑟 ⊆ 𝑈) ↔ ((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈))) |
| 28 | | eqeq1 2741 |
. . . . . . . . . . . . . 14
⊢ (𝑟 = (𝑠 ∩ 𝑈) → (𝑟 = (𝑇 ∩ 𝑈) ↔ (𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈))) |
| 29 | | eqeq1 2741 |
. . . . . . . . . . . . . 14
⊢ (𝑟 = (𝑠 ∩ 𝑈) → (𝑟 = 𝑈 ↔ (𝑠 ∩ 𝑈) = 𝑈)) |
| 30 | 28, 29 | orbi12d 919 |
. . . . . . . . . . . . 13
⊢ (𝑟 = (𝑠 ∩ 𝑈) → ((𝑟 = (𝑇 ∩ 𝑈) ∨ 𝑟 = 𝑈) ↔ ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈))) |
| 31 | 27, 30 | imbi12d 344 |
. . . . . . . . . . . 12
⊢ (𝑟 = (𝑠 ∩ 𝑈) → ((((𝑇 ∩ 𝑈) ⊆ 𝑟 ∧ 𝑟 ⊆ 𝑈) → (𝑟 = (𝑇 ∩ 𝑈) ∨ 𝑟 = 𝑈)) ↔ (((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈)))) |
| 32 | 31 | rspcv 3618 |
. . . . . . . . . . 11
⊢ ((𝑠 ∩ 𝑈) ∈ 𝑆 → (∀𝑟 ∈ 𝑆 (((𝑇 ∩ 𝑈) ⊆ 𝑟 ∧ 𝑟 ⊆ 𝑈) → (𝑟 = (𝑇 ∩ 𝑈) ∨ 𝑟 = 𝑈)) → (((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈)))) |
| 33 | 24, 32 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (∀𝑟 ∈ 𝑆 (((𝑇 ∩ 𝑈) ⊆ 𝑟 ∧ 𝑟 ⊆ 𝑈) → (𝑟 = (𝑇 ∩ 𝑈) ∨ 𝑟 = 𝑈)) → (((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈)))) |
| 34 | 33 | adantld 490 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → (((𝑇 ∩ 𝑈) ⊊ 𝑈 ∧ ∀𝑟 ∈ 𝑆 (((𝑇 ∩ 𝑈) ⊆ 𝑟 ∧ 𝑟 ⊆ 𝑈) → (𝑟 = (𝑇 ∩ 𝑈) ∨ 𝑟 = 𝑈))) → (((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈)))) |
| 35 | 19, 34 | sylbid 240 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆) → ((𝑇 ∩ 𝑈)𝐶𝑈 → (((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈)))) |
| 36 | 35 | 3adant3 1133 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑇 ∩ 𝑈)𝐶𝑈 → (((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈)))) |
| 37 | 17, 36 | mpd 15 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (((𝑇 ∩ 𝑈) ⊆ (𝑠 ∩ 𝑈) ∧ (𝑠 ∩ 𝑈) ⊆ 𝑈) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈))) |
| 38 | 16, 37 | mpd 15 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈)) |
| 39 | | oveq1 7438 |
. . . . . . 7
⊢ ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) → ((𝑠 ∩ 𝑈) ⊕ 𝑇) = ((𝑇 ∩ 𝑈) ⊕ 𝑇)) |
| 40 | 3 | 3ad2ant1 1134 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑊 ∈ LMod) |
| 41 | 4 | 3ad2ant1 1134 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑇 ∈ 𝑆) |
| 42 | 5 | 3ad2ant1 1134 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑈 ∈ 𝑆) |
| 43 | | simp2 1138 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑠 ∈ 𝑆) |
| 44 | | simp3r 1203 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → 𝑠 ⊆ (𝑇 ⊕ 𝑈)) |
| 45 | 1, 10, 2, 40, 41, 42, 43, 13, 44 | lcvexchlem3 39037 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑠 ∩ 𝑈) ⊕ 𝑇) = 𝑠) |
| 46 | 1 | lsssssubg 20956 |
. . . . . . . . . . . 12
⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) |
| 47 | 3, 46 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑆 ⊆ (SubGrp‘𝑊)) |
| 48 | 47, 7 | sseldd 3984 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑇 ∩ 𝑈) ∈ (SubGrp‘𝑊)) |
| 49 | 47, 4 | sseldd 3984 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝑊)) |
| 50 | | inss1 4237 |
. . . . . . . . . . 11
⊢ (𝑇 ∩ 𝑈) ⊆ 𝑇 |
| 51 | 50 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑇 ∩ 𝑈) ⊆ 𝑇) |
| 52 | 10 | lsmss1 19683 |
. . . . . . . . . 10
⊢ (((𝑇 ∩ 𝑈) ∈ (SubGrp‘𝑊) ∧ 𝑇 ∈ (SubGrp‘𝑊) ∧ (𝑇 ∩ 𝑈) ⊆ 𝑇) → ((𝑇 ∩ 𝑈) ⊕ 𝑇) = 𝑇) |
| 53 | 48, 49, 51, 52 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑇 ∩ 𝑈) ⊕ 𝑇) = 𝑇) |
| 54 | 53 | 3ad2ant1 1134 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑇 ∩ 𝑈) ⊕ 𝑇) = 𝑇) |
| 55 | 45, 54 | eqeq12d 2753 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (((𝑠 ∩ 𝑈) ⊕ 𝑇) = ((𝑇 ∩ 𝑈) ⊕ 𝑇) ↔ 𝑠 = 𝑇)) |
| 56 | 39, 55 | imbitrid 244 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) → 𝑠 = 𝑇)) |
| 57 | | oveq1 7438 |
. . . . . . 7
⊢ ((𝑠 ∩ 𝑈) = 𝑈 → ((𝑠 ∩ 𝑈) ⊕ 𝑇) = (𝑈 ⊕ 𝑇)) |
| 58 | | lmodabl 20907 |
. . . . . . . . . . 11
⊢ (𝑊 ∈ LMod → 𝑊 ∈ Abel) |
| 59 | 3, 58 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑊 ∈ Abel) |
| 60 | 47, 5 | sseldd 3984 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝑊)) |
| 61 | 10 | lsmcom 19876 |
. . . . . . . . . 10
⊢ ((𝑊 ∈ Abel ∧ 𝑈 ∈ (SubGrp‘𝑊) ∧ 𝑇 ∈ (SubGrp‘𝑊)) → (𝑈 ⊕ 𝑇) = (𝑇 ⊕ 𝑈)) |
| 62 | 59, 60, 49, 61 | syl3anc 1373 |
. . . . . . . . 9
⊢ (𝜑 → (𝑈 ⊕ 𝑇) = (𝑇 ⊕ 𝑈)) |
| 63 | 62 | 3ad2ant1 1134 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (𝑈 ⊕ 𝑇) = (𝑇 ⊕ 𝑈)) |
| 64 | 45, 63 | eqeq12d 2753 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (((𝑠 ∩ 𝑈) ⊕ 𝑇) = (𝑈 ⊕ 𝑇) ↔ 𝑠 = (𝑇 ⊕ 𝑈))) |
| 65 | 57, 64 | imbitrid 244 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → ((𝑠 ∩ 𝑈) = 𝑈 → 𝑠 = (𝑇 ⊕ 𝑈))) |
| 66 | 56, 65 | orim12d 967 |
. . . . 5
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (((𝑠 ∩ 𝑈) = (𝑇 ∩ 𝑈) ∨ (𝑠 ∩ 𝑈) = 𝑈) → (𝑠 = 𝑇 ∨ 𝑠 = (𝑇 ⊕ 𝑈)))) |
| 67 | 38, 66 | mpd 15 |
. . . 4
⊢ ((𝜑 ∧ 𝑠 ∈ 𝑆 ∧ (𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈))) → (𝑠 = 𝑇 ∨ 𝑠 = (𝑇 ⊕ 𝑈))) |
| 68 | 67 | 3exp 1120 |
. . 3
⊢ (𝜑 → (𝑠 ∈ 𝑆 → ((𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈)) → (𝑠 = 𝑇 ∨ 𝑠 = (𝑇 ⊕ 𝑈))))) |
| 69 | 68 | ralrimiv 3145 |
. 2
⊢ (𝜑 → ∀𝑠 ∈ 𝑆 ((𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈)) → (𝑠 = 𝑇 ∨ 𝑠 = (𝑇 ⊕ 𝑈)))) |
| 70 | 1, 10 | lsmcl 21082 |
. . . 4
⊢ ((𝑊 ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ⊕ 𝑈) ∈ 𝑆) |
| 71 | 3, 4, 5, 70 | syl3anc 1373 |
. . 3
⊢ (𝜑 → (𝑇 ⊕ 𝑈) ∈ 𝑆) |
| 72 | 1, 2, 3, 4, 71 | lcvbr3 39024 |
. 2
⊢ (𝜑 → (𝑇𝐶(𝑇 ⊕ 𝑈) ↔ (𝑇 ⊊ (𝑇 ⊕ 𝑈) ∧ ∀𝑠 ∈ 𝑆 ((𝑇 ⊆ 𝑠 ∧ 𝑠 ⊆ (𝑇 ⊕ 𝑈)) → (𝑠 = 𝑇 ∨ 𝑠 = (𝑇 ⊕ 𝑈)))))) |
| 73 | 12, 69, 72 | mpbir2and 713 |
1
⊢ (𝜑 → 𝑇𝐶(𝑇 ⊕ 𝑈)) |