Step | Hyp | Ref
| Expression |
1 | | dpjidcl.3 |
. . . 4
⊢ (𝜑 → 𝐴 ∈ (𝐺 DProd 𝑆)) |
2 | | dpjfval.2 |
. . . . 5
⊢ (𝜑 → dom 𝑆 = 𝐼) |
3 | | dpjidcl.0 |
. . . . . 6
⊢ 0 =
(0g‘𝐺) |
4 | | dpjidcl.w |
. . . . . 6
⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } |
5 | 3, 4 | eldprd 19245 |
. . . . 5
⊢ (dom
𝑆 = 𝐼 → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)))) |
6 | 2, 5 | syl 17 |
. . . 4
⊢ (𝜑 → (𝐴 ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)))) |
7 | 1, 6 | mpbid 235 |
. . 3
⊢ (𝜑 → (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓))) |
8 | 7 | simprd 499 |
. 2
⊢ (𝜑 → ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)) |
9 | | dpjfval.1 |
. . . . 5
⊢ (𝜑 → 𝐺dom DProd 𝑆) |
10 | 9 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝐺dom DProd 𝑆) |
11 | 2 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → dom 𝑆 = 𝐼) |
12 | 9 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐺dom DProd 𝑆) |
13 | 2 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → dom 𝑆 = 𝐼) |
14 | | dpjfval.p |
. . . . . 6
⊢ 𝑃 = (𝐺dProj𝑆) |
15 | | simpr 488 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼) |
16 | 12, 13, 14, 15 | dpjf 19298 |
. . . . 5
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑃‘𝑥):(𝐺 DProd 𝑆)⟶(𝑆‘𝑥)) |
17 | 1 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ (𝐺 DProd 𝑆)) |
18 | 16, 17 | ffvelrnd 6862 |
. . . 4
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)‘𝐴) ∈ (𝑆‘𝑥)) |
19 | 9, 2 | dprddomcld 19242 |
. . . . . . 7
⊢ (𝜑 → 𝐼 ∈ V) |
20 | 19 | mptexd 6997 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ V) |
21 | 20 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ V) |
22 | | funmpt 6377 |
. . . . . 6
⊢ Fun
(𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) |
23 | 22 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → Fun (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴))) |
24 | | simprl 771 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝑓 ∈ 𝑊) |
25 | 4, 10, 11, 24 | dprdffsupp 19255 |
. . . . 5
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝑓 finSupp 0 ) |
26 | | eldifi 4017 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 )) → 𝑥 ∈ 𝐼) |
27 | | eqid 2738 |
. . . . . . . . . 10
⊢
(proj1‘𝐺) = (proj1‘𝐺) |
28 | 12, 13, 14, 27, 15 | dpjval 19297 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑃‘𝑥) = ((𝑆‘𝑥)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))) |
29 | 28 | fveq1d 6676 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)‘𝐴) = (((𝑆‘𝑥)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))‘𝐴)) |
30 | 26, 29 | sylan2 596 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → ((𝑃‘𝑥)‘𝐴) = (((𝑆‘𝑥)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))‘𝐴)) |
31 | | simplrr 778 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → 𝐴 = (𝐺 Σg 𝑓)) |
32 | | eqid 2738 |
. . . . . . . . . . 11
⊢
(Base‘𝐺) =
(Base‘𝐺) |
33 | | eqid 2738 |
. . . . . . . . . . 11
⊢
(Cntz‘𝐺) =
(Cntz‘𝐺) |
34 | | dprdgrp 19246 |
. . . . . . . . . . . . 13
⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) |
35 | | grpmnd 18226 |
. . . . . . . . . . . . 13
⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) |
36 | 10, 34, 35 | 3syl 18 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝐺 ∈ Mnd) |
37 | 36 | adantr 484 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → 𝐺 ∈ Mnd) |
38 | 19 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → 𝐼 ∈ V) |
39 | 4, 10, 11, 24, 32 | dprdff 19253 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝑓:𝐼⟶(Base‘𝐺)) |
40 | 39 | adantr 484 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → 𝑓:𝐼⟶(Base‘𝐺)) |
41 | 24 | adantr 484 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝑓 ∈ 𝑊) |
42 | 4, 12, 13, 41, 33 | dprdfcntz 19256 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ran 𝑓 ⊆ ((Cntz‘𝐺)‘ran 𝑓)) |
43 | 26, 42 | sylan2 596 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → ran 𝑓 ⊆ ((Cntz‘𝐺)‘ran 𝑓)) |
44 | | snssi 4696 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 )) → {𝑥} ⊆ (𝐼 ∖ (𝑓 supp 0 ))) |
45 | 44 | adantl 485 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → {𝑥} ⊆ (𝐼 ∖ (𝑓 supp 0 ))) |
46 | 45 | difss2d 4025 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → {𝑥} ⊆ 𝐼) |
47 | | suppssdm 7872 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 supp 0 ) ⊆ dom 𝑓 |
48 | 47, 39 | fssdm 6524 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → (𝑓 supp 0 ) ⊆ 𝐼) |
49 | 48 | adantr 484 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → (𝑓 supp 0 ) ⊆ 𝐼) |
50 | | ssconb 4028 |
. . . . . . . . . . . . 13
⊢ (({𝑥} ⊆ 𝐼 ∧ (𝑓 supp 0 ) ⊆ 𝐼) → ({𝑥} ⊆ (𝐼 ∖ (𝑓 supp 0 )) ↔ (𝑓 supp 0 ) ⊆ (𝐼 ∖ {𝑥}))) |
51 | 46, 49, 50 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → ({𝑥} ⊆ (𝐼 ∖ (𝑓 supp 0 )) ↔ (𝑓 supp 0 ) ⊆ (𝐼 ∖ {𝑥}))) |
52 | 45, 51 | mpbid 235 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → (𝑓 supp 0 ) ⊆ (𝐼 ∖ {𝑥})) |
53 | 25 | adantr 484 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → 𝑓 finSupp 0 ) |
54 | 32, 3, 33, 37, 38, 40, 43, 52, 53 | gsumzres 19148 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → (𝐺 Σg
(𝑓 ↾ (𝐼 ∖ {𝑥}))) = (𝐺 Σg 𝑓)) |
55 | 31, 54 | eqtr4d 2776 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → 𝐴 = (𝐺 Σg (𝑓 ↾ (𝐼 ∖ {𝑥})))) |
56 | | eqid 2738 |
. . . . . . . . . . 11
⊢ {ℎ ∈ X𝑖 ∈
(𝐼 ∖ {𝑥})((𝑆 ↾ (𝐼 ∖ {𝑥}))‘𝑖) ∣ ℎ finSupp 0 } = {ℎ ∈ X𝑖 ∈ (𝐼 ∖ {𝑥})((𝑆 ↾ (𝐼 ∖ {𝑥}))‘𝑖) ∣ ℎ finSupp 0 } |
57 | | difss 4022 |
. . . . . . . . . . . . . 14
⊢ (𝐼 ∖ {𝑥}) ⊆ 𝐼 |
58 | 57 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐼 ∖ {𝑥}) ⊆ 𝐼) |
59 | 12, 13, 58 | dprdres 19269 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑥})) ∧ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))) ⊆ (𝐺 DProd 𝑆))) |
60 | 59 | simpld 498 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))) |
61 | 12, 13 | dprdf2 19248 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
62 | | fssres 6544 |
. . . . . . . . . . . . 13
⊢ ((𝑆:𝐼⟶(SubGrp‘𝐺) ∧ (𝐼 ∖ {𝑥}) ⊆ 𝐼) → (𝑆 ↾ (𝐼 ∖ {𝑥})):(𝐼 ∖ {𝑥})⟶(SubGrp‘𝐺)) |
63 | 61, 57, 62 | sylancl 589 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑆 ↾ (𝐼 ∖ {𝑥})):(𝐼 ∖ {𝑥})⟶(SubGrp‘𝐺)) |
64 | 63 | fdmd 6515 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → dom (𝑆 ↾ (𝐼 ∖ {𝑥})) = (𝐼 ∖ {𝑥})) |
65 | 39 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝑓:𝐼⟶(Base‘𝐺)) |
66 | 65 | feqmptd 6737 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝑓 = (𝑘 ∈ 𝐼 ↦ (𝑓‘𝑘))) |
67 | 66 | reseq1d 5824 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑓 ↾ (𝐼 ∖ {𝑥})) = ((𝑘 ∈ 𝐼 ↦ (𝑓‘𝑘)) ↾ (𝐼 ∖ {𝑥}))) |
68 | | resmpt 5879 |
. . . . . . . . . . . . . 14
⊢ ((𝐼 ∖ {𝑥}) ⊆ 𝐼 → ((𝑘 ∈ 𝐼 ↦ (𝑓‘𝑘)) ↾ (𝐼 ∖ {𝑥})) = (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘))) |
69 | 57, 68 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ 𝐼 ↦ (𝑓‘𝑘)) ↾ (𝐼 ∖ {𝑥})) = (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) |
70 | 67, 69 | eqtrdi 2789 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑓 ↾ (𝐼 ∖ {𝑥})) = (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘))) |
71 | | eldifi 4017 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (𝐼 ∖ {𝑥}) → 𝑘 ∈ 𝐼) |
72 | 4, 12, 13, 41 | dprdfcl 19254 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑘) ∈ (𝑆‘𝑘)) |
73 | 71, 72 | sylan2 596 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) ∧ 𝑘 ∈ (𝐼 ∖ {𝑥})) → (𝑓‘𝑘) ∈ (𝑆‘𝑘)) |
74 | | fvres 6693 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (𝐼 ∖ {𝑥}) → ((𝑆 ↾ (𝐼 ∖ {𝑥}))‘𝑘) = (𝑆‘𝑘)) |
75 | 74 | adantl 485 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) ∧ 𝑘 ∈ (𝐼 ∖ {𝑥})) → ((𝑆 ↾ (𝐼 ∖ {𝑥}))‘𝑘) = (𝑆‘𝑘)) |
76 | 73, 75 | eleqtrrd 2836 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) ∧ 𝑘 ∈ (𝐼 ∖ {𝑥})) → (𝑓‘𝑘) ∈ ((𝑆 ↾ (𝐼 ∖ {𝑥}))‘𝑘)) |
77 | 19 | difexd 5197 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐼 ∖ {𝑥}) ∈ V) |
78 | 77 | mptexd 6997 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) ∈ V) |
79 | 78 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) ∈ V) |
80 | | funmpt 6377 |
. . . . . . . . . . . . . . 15
⊢ Fun
(𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) |
81 | 80 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → Fun (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘))) |
82 | 25 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝑓 finSupp 0 ) |
83 | | ssdif 4030 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐼 ∖ {𝑥}) ⊆ 𝐼 → ((𝐼 ∖ {𝑥}) ∖ (𝑓 supp 0 )) ⊆ (𝐼 ∖ (𝑓 supp 0 ))) |
84 | 57, 83 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐼 ∖ {𝑥}) ∖ (𝑓 supp 0 )) ⊆ (𝐼 ∖ (𝑓 supp 0 )) |
85 | 84 | sseli 3873 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ ((𝐼 ∖ {𝑥}) ∖ (𝑓 supp 0 )) → 𝑘 ∈ (𝐼 ∖ (𝑓 supp 0 ))) |
86 | | ssidd 3900 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑓 supp 0 ) ⊆ (𝑓 supp 0 )) |
87 | 19 | ad2antrr 726 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ V) |
88 | 3 | fvexi 6688 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 ∈
V |
89 | 88 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 0 ∈ V) |
90 | 65, 86, 87, 89 | suppssr 7891 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) ∧ 𝑘 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → (𝑓‘𝑘) = 0 ) |
91 | 85, 90 | sylan2 596 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) ∧ 𝑘 ∈ ((𝐼 ∖ {𝑥}) ∖ (𝑓 supp 0 ))) → (𝑓‘𝑘) = 0 ) |
92 | 77 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐼 ∖ {𝑥}) ∈ V) |
93 | 91, 92 | suppss2 7895 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ((𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) supp 0 ) ⊆ (𝑓 supp 0 )) |
94 | | fsuppsssupp 8922 |
. . . . . . . . . . . . . 14
⊢ ((((𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) ∈ V ∧ Fun (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘))) ∧ (𝑓 finSupp 0 ∧ ((𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) supp 0 ) ⊆ (𝑓 supp 0 ))) → (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) finSupp 0 ) |
95 | 79, 81, 82, 93, 94 | syl22anc 838 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) finSupp 0 ) |
96 | 56, 60, 64, 76, 95 | dprdwd 19252 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) ∈ {ℎ ∈ X𝑖 ∈ (𝐼 ∖ {𝑥})((𝑆 ↾ (𝐼 ∖ {𝑥}))‘𝑖) ∣ ℎ finSupp 0 }) |
97 | 70, 96 | eqeltrd 2833 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑓 ↾ (𝐼 ∖ {𝑥})) ∈ {ℎ ∈ X𝑖 ∈ (𝐼 ∖ {𝑥})((𝑆 ↾ (𝐼 ∖ {𝑥}))‘𝑖) ∣ ℎ finSupp 0 }) |
98 | 3, 56, 60, 64, 97 | eldprdi 19259 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg (𝑓 ↾ (𝐼 ∖ {𝑥}))) ∈ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))) |
99 | 26, 98 | sylan2 596 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → (𝐺 Σg
(𝑓 ↾ (𝐼 ∖ {𝑥}))) ∈ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))) |
100 | 55, 99 | eqeltrd 2833 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → 𝐴 ∈ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))) |
101 | | eqid 2738 |
. . . . . . . . . 10
⊢
(+g‘𝐺) = (+g‘𝐺) |
102 | | eqid 2738 |
. . . . . . . . . 10
⊢
(LSSum‘𝐺) =
(LSSum‘𝐺) |
103 | 61, 15 | ffvelrnd 6862 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ∈ (SubGrp‘𝐺)) |
104 | | dprdsubg 19265 |
. . . . . . . . . . 11
⊢ (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑥})) → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺)) |
105 | 60, 104 | syl 17 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺)) |
106 | 12, 13, 15, 3 | dpjdisj 19294 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ((𝑆‘𝑥) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))) = { 0 }) |
107 | 12, 13, 15, 33 | dpjcntz 19293 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))) |
108 | 101, 102,
3, 33, 103, 105, 106, 107, 27 | pj1rid 18946 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) ∧ 𝐴 ∈ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))) → (((𝑆‘𝑥)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))‘𝐴) = 0 ) |
109 | 26, 108 | sylanl2 681 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) ∧ 𝐴 ∈ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))) → (((𝑆‘𝑥)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))‘𝐴) = 0 ) |
110 | 100, 109 | mpdan 687 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → (((𝑆‘𝑥)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))‘𝐴) = 0 ) |
111 | 30, 110 | eqtrd 2773 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → ((𝑃‘𝑥)‘𝐴) = 0 ) |
112 | 19 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝐼 ∈ V) |
113 | 111, 112 | suppss2 7895 |
. . . . 5
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → ((𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) supp 0 ) ⊆ (𝑓 supp 0 )) |
114 | | fsuppsssupp 8922 |
. . . . 5
⊢ ((((𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ V ∧ Fun (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴))) ∧ (𝑓 finSupp 0 ∧ ((𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) supp 0 ) ⊆ (𝑓 supp 0 ))) → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) finSupp 0 ) |
115 | 21, 23, 25, 113, 114 | syl22anc 838 |
. . . 4
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) finSupp 0 ) |
116 | 4, 10, 11, 18, 115 | dprdwd 19252 |
. . 3
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ 𝑊) |
117 | | simprr 773 |
. . . 4
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝐴 = (𝐺 Σg 𝑓)) |
118 | 39 | feqmptd 6737 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝑓 = (𝑥 ∈ 𝐼 ↦ (𝑓‘𝑥))) |
119 | | simplrr 778 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐴 = (𝐺 Σg 𝑓)) |
120 | 12, 34, 35 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Mnd) |
121 | 4, 12, 13, 41 | dprdffsupp 19255 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝑓 finSupp 0 ) |
122 | | disjdif 4361 |
. . . . . . . . . . . . 13
⊢ ({𝑥} ∩ (𝐼 ∖ {𝑥})) = ∅ |
123 | 122 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ({𝑥} ∩ (𝐼 ∖ {𝑥})) = ∅) |
124 | | undif2 4366 |
. . . . . . . . . . . . 13
⊢ ({𝑥} ∪ (𝐼 ∖ {𝑥})) = ({𝑥} ∪ 𝐼) |
125 | 15 | snssd 4697 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → {𝑥} ⊆ 𝐼) |
126 | | ssequn1 4070 |
. . . . . . . . . . . . . 14
⊢ ({𝑥} ⊆ 𝐼 ↔ ({𝑥} ∪ 𝐼) = 𝐼) |
127 | 125, 126 | sylib 221 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ({𝑥} ∪ 𝐼) = 𝐼) |
128 | 124, 127 | eqtr2id 2786 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐼 = ({𝑥} ∪ (𝐼 ∖ {𝑥}))) |
129 | 32, 3, 101, 33, 120, 87, 65, 42, 121, 123, 128 | gsumzsplit 19166 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg 𝑓) = ((𝐺 Σg (𝑓 ↾ {𝑥}))(+g‘𝐺)(𝐺 Σg (𝑓 ↾ (𝐼 ∖ {𝑥}))))) |
130 | 65, 125 | feqresmpt 6738 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑓 ↾ {𝑥}) = (𝑘 ∈ {𝑥} ↦ (𝑓‘𝑘))) |
131 | 130 | oveq2d 7186 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg (𝑓 ↾ {𝑥})) = (𝐺 Σg (𝑘 ∈ {𝑥} ↦ (𝑓‘𝑘)))) |
132 | 65, 15 | ffvelrnd 6862 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ (Base‘𝐺)) |
133 | | fveq2 6674 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑥 → (𝑓‘𝑘) = (𝑓‘𝑥)) |
134 | 32, 133 | gsumsn 19193 |
. . . . . . . . . . . . . 14
⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐼 ∧ (𝑓‘𝑥) ∈ (Base‘𝐺)) → (𝐺 Σg (𝑘 ∈ {𝑥} ↦ (𝑓‘𝑘))) = (𝑓‘𝑥)) |
135 | 120, 15, 132, 134 | syl3anc 1372 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg (𝑘 ∈ {𝑥} ↦ (𝑓‘𝑘))) = (𝑓‘𝑥)) |
136 | 131, 135 | eqtrd 2773 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg (𝑓 ↾ {𝑥})) = (𝑓‘𝑥)) |
137 | 136 | oveq1d 7185 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ((𝐺 Σg (𝑓 ↾ {𝑥}))(+g‘𝐺)(𝐺 Σg (𝑓 ↾ (𝐼 ∖ {𝑥})))) = ((𝑓‘𝑥)(+g‘𝐺)(𝐺 Σg (𝑓 ↾ (𝐼 ∖ {𝑥}))))) |
138 | 119, 129,
137 | 3eqtrd 2777 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐴 = ((𝑓‘𝑥)(+g‘𝐺)(𝐺 Σg (𝑓 ↾ (𝐼 ∖ {𝑥}))))) |
139 | 12, 13, 15, 102 | dpjlsm 19295 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺 DProd 𝑆) = ((𝑆‘𝑥)(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))) |
140 | 17, 139 | eleqtrd 2835 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ ((𝑆‘𝑥)(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))) |
141 | 4, 10, 11, 24 | dprdfcl 19254 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ (𝑆‘𝑥)) |
142 | 101, 102,
3, 33, 103, 105, 106, 107, 27, 140, 141, 98 | pj1eq 18944 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐴 = ((𝑓‘𝑥)(+g‘𝐺)(𝐺 Σg (𝑓 ↾ (𝐼 ∖ {𝑥})))) ↔ ((((𝑆‘𝑥)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))‘𝐴) = (𝑓‘𝑥) ∧ (((𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))(proj1‘𝐺)(𝑆‘𝑥))‘𝐴) = (𝐺 Σg (𝑓 ↾ (𝐼 ∖ {𝑥})))))) |
143 | 138, 142 | mpbid 235 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ((((𝑆‘𝑥)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))‘𝐴) = (𝑓‘𝑥) ∧ (((𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))(proj1‘𝐺)(𝑆‘𝑥))‘𝐴) = (𝐺 Σg (𝑓 ↾ (𝐼 ∖ {𝑥}))))) |
144 | 143 | simpld 498 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (((𝑆‘𝑥)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))‘𝐴) = (𝑓‘𝑥)) |
145 | 29, 144 | eqtrd 2773 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)‘𝐴) = (𝑓‘𝑥)) |
146 | 145 | mpteq2dva 5125 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) = (𝑥 ∈ 𝐼 ↦ (𝑓‘𝑥))) |
147 | 118, 146 | eqtr4d 2776 |
. . . . 5
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝑓 = (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴))) |
148 | 147 | oveq2d 7186 |
. . . 4
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → (𝐺 Σg 𝑓) = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)))) |
149 | 117, 148 | eqtrd 2773 |
. . 3
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)))) |
150 | 116, 149 | jca 515 |
. 2
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → ((𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴))))) |
151 | 8, 150 | rexlimddv 3201 |
1
⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴))))) |