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 18757 |
. . . . 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 224 |
. . 3
⊢ (𝜑 → (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓))) |
8 | 7 | simprd 491 |
. 2
⊢ (𝜑 → ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)) |
9 | | dpjfval.1 |
. . . . 5
⊢ (𝜑 → 𝐺dom DProd 𝑆) |
10 | 9 | adantr 474 |
. . . 4
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝐺dom DProd 𝑆) |
11 | 2 | adantr 474 |
. . . 4
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → dom 𝑆 = 𝐼) |
12 | 9 | ad2antrr 719 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐺dom DProd 𝑆) |
13 | 2 | ad2antrr 719 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → dom 𝑆 = 𝐼) |
14 | | dpjfval.p |
. . . . . 6
⊢ 𝑃 = (𝐺dProj𝑆) |
15 | | simpr 479 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼) |
16 | 12, 13, 14, 15 | dpjf 18810 |
. . . . 5
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑃‘𝑥):(𝐺 DProd 𝑆)⟶(𝑆‘𝑥)) |
17 | 1 | ad2antrr 719 |
. . . . 5
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ (𝐺 DProd 𝑆)) |
18 | 16, 17 | ffvelrnd 6609 |
. . . 4
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)‘𝐴) ∈ (𝑆‘𝑥)) |
19 | 9, 2 | dprddomcld 18754 |
. . . . . . 7
⊢ (𝜑 → 𝐼 ∈ V) |
20 | | mptexg 6740 |
. . . . . . 7
⊢ (𝐼 ∈ V → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ V) |
21 | 19, 20 | syl 17 |
. . . . . 6
⊢ (𝜑 → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ V) |
22 | 21 | adantr 474 |
. . . . 5
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ V) |
23 | | funmpt 6161 |
. . . . . 6
⊢ Fun
(𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) |
24 | 23 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → Fun (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴))) |
25 | | simprl 789 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝑓 ∈ 𝑊) |
26 | 4, 10, 11, 25 | dprdffsupp 18767 |
. . . . 5
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝑓 finSupp 0 ) |
27 | | eldifi 3959 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 )) → 𝑥 ∈ 𝐼) |
28 | | eqid 2825 |
. . . . . . . . . 10
⊢
(proj1‘𝐺) = (proj1‘𝐺) |
29 | 12, 13, 14, 28, 15 | dpjval 18809 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑃‘𝑥) = ((𝑆‘𝑥)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))) |
30 | 29 | fveq1d 6435 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)‘𝐴) = (((𝑆‘𝑥)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))‘𝐴)) |
31 | 27, 30 | sylan2 588 |
. . . . . . 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)) |
32 | | simplrr 798 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → 𝐴 = (𝐺 Σg 𝑓)) |
33 | | eqid 2825 |
. . . . . . . . . . 11
⊢
(Base‘𝐺) =
(Base‘𝐺) |
34 | | eqid 2825 |
. . . . . . . . . . 11
⊢
(Cntz‘𝐺) =
(Cntz‘𝐺) |
35 | | dprdgrp 18758 |
. . . . . . . . . . . . 13
⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) |
36 | | grpmnd 17783 |
. . . . . . . . . . . . 13
⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) |
37 | 10, 35, 36 | 3syl 18 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝐺 ∈ Mnd) |
38 | 37 | adantr 474 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → 𝐺 ∈ Mnd) |
39 | 19 | ad2antrr 719 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → 𝐼 ∈ V) |
40 | 4, 10, 11, 25, 33 | dprdff 18765 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝑓:𝐼⟶(Base‘𝐺)) |
41 | 40 | adantr 474 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → 𝑓:𝐼⟶(Base‘𝐺)) |
42 | 25 | adantr 474 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝑓 ∈ 𝑊) |
43 | 4, 12, 13, 42, 34 | dprdfcntz 18768 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ran 𝑓 ⊆ ((Cntz‘𝐺)‘ran 𝑓)) |
44 | 27, 43 | sylan2 588 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → ran 𝑓 ⊆ ((Cntz‘𝐺)‘ran 𝑓)) |
45 | | snssi 4557 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 )) → {𝑥} ⊆ (𝐼 ∖ (𝑓 supp 0 ))) |
46 | 45 | adantl 475 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → {𝑥} ⊆ (𝐼 ∖ (𝑓 supp 0 ))) |
47 | 46 | difss2d 3967 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → {𝑥} ⊆ 𝐼) |
48 | | suppssdm 7572 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 supp 0 ) ⊆ dom 𝑓 |
49 | 48, 40 | fssdm 6294 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → (𝑓 supp 0 ) ⊆ 𝐼) |
50 | 49 | adantr 474 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → (𝑓 supp 0 ) ⊆ 𝐼) |
51 | | ssconb 3970 |
. . . . . . . . . . . . 13
⊢ (({𝑥} ⊆ 𝐼 ∧ (𝑓 supp 0 ) ⊆ 𝐼) → ({𝑥} ⊆ (𝐼 ∖ (𝑓 supp 0 )) ↔ (𝑓 supp 0 ) ⊆ (𝐼 ∖ {𝑥}))) |
52 | 47, 50, 51 | syl2anc 581 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → ({𝑥} ⊆ (𝐼 ∖ (𝑓 supp 0 )) ↔ (𝑓 supp 0 ) ⊆ (𝐼 ∖ {𝑥}))) |
53 | 46, 52 | mpbid 224 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → (𝑓 supp 0 ) ⊆ (𝐼 ∖ {𝑥})) |
54 | 26 | adantr 474 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → 𝑓 finSupp 0 ) |
55 | 33, 3, 34, 38, 39, 41, 44, 53, 54 | gsumzres 18663 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → (𝐺 Σg
(𝑓 ↾ (𝐼 ∖ {𝑥}))) = (𝐺 Σg 𝑓)) |
56 | 32, 55 | eqtr4d 2864 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → 𝐴 = (𝐺 Σg (𝑓 ↾ (𝐼 ∖ {𝑥})))) |
57 | | eqid 2825 |
. . . . . . . . . . 11
⊢ {ℎ ∈ X𝑖 ∈
(𝐼 ∖ {𝑥})((𝑆 ↾ (𝐼 ∖ {𝑥}))‘𝑖) ∣ ℎ finSupp 0 } = {ℎ ∈ X𝑖 ∈ (𝐼 ∖ {𝑥})((𝑆 ↾ (𝐼 ∖ {𝑥}))‘𝑖) ∣ ℎ finSupp 0 } |
58 | | difss 3964 |
. . . . . . . . . . . . . 14
⊢ (𝐼 ∖ {𝑥}) ⊆ 𝐼 |
59 | 58 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐼 ∖ {𝑥}) ⊆ 𝐼) |
60 | 12, 13, 59 | dprdres 18781 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑥})) ∧ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))) ⊆ (𝐺 DProd 𝑆))) |
61 | 60 | simpld 490 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))) |
62 | 12, 13 | dprdf2 18760 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
63 | | fssres 6307 |
. . . . . . . . . . . . 13
⊢ ((𝑆:𝐼⟶(SubGrp‘𝐺) ∧ (𝐼 ∖ {𝑥}) ⊆ 𝐼) → (𝑆 ↾ (𝐼 ∖ {𝑥})):(𝐼 ∖ {𝑥})⟶(SubGrp‘𝐺)) |
64 | 62, 58, 63 | sylancl 582 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑆 ↾ (𝐼 ∖ {𝑥})):(𝐼 ∖ {𝑥})⟶(SubGrp‘𝐺)) |
65 | 64 | fdmd 6287 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → dom (𝑆 ↾ (𝐼 ∖ {𝑥})) = (𝐼 ∖ {𝑥})) |
66 | 40 | adantr 474 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝑓:𝐼⟶(Base‘𝐺)) |
67 | 66 | feqmptd 6496 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝑓 = (𝑘 ∈ 𝐼 ↦ (𝑓‘𝑘))) |
68 | 67 | reseq1d 5628 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑓 ↾ (𝐼 ∖ {𝑥})) = ((𝑘 ∈ 𝐼 ↦ (𝑓‘𝑘)) ↾ (𝐼 ∖ {𝑥}))) |
69 | | resmpt 5686 |
. . . . . . . . . . . . . 14
⊢ ((𝐼 ∖ {𝑥}) ⊆ 𝐼 → ((𝑘 ∈ 𝐼 ↦ (𝑓‘𝑘)) ↾ (𝐼 ∖ {𝑥})) = (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘))) |
70 | 58, 69 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ 𝐼 ↦ (𝑓‘𝑘)) ↾ (𝐼 ∖ {𝑥})) = (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) |
71 | 68, 70 | syl6eq 2877 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑓 ↾ (𝐼 ∖ {𝑥})) = (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘))) |
72 | | eldifi 3959 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (𝐼 ∖ {𝑥}) → 𝑘 ∈ 𝐼) |
73 | 4, 12, 13, 42 | dprdfcl 18766 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑘) ∈ (𝑆‘𝑘)) |
74 | 72, 73 | sylan2 588 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) ∧ 𝑘 ∈ (𝐼 ∖ {𝑥})) → (𝑓‘𝑘) ∈ (𝑆‘𝑘)) |
75 | | fvres 6452 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (𝐼 ∖ {𝑥}) → ((𝑆 ↾ (𝐼 ∖ {𝑥}))‘𝑘) = (𝑆‘𝑘)) |
76 | 75 | adantl 475 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) ∧ 𝑘 ∈ (𝐼 ∖ {𝑥})) → ((𝑆 ↾ (𝐼 ∖ {𝑥}))‘𝑘) = (𝑆‘𝑘)) |
77 | 74, 76 | eleqtrrd 2909 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) ∧ 𝑘 ∈ (𝐼 ∖ {𝑥})) → (𝑓‘𝑘) ∈ ((𝑆 ↾ (𝐼 ∖ {𝑥}))‘𝑘)) |
78 | | difexg 5033 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐼 ∈ V → (𝐼 ∖ {𝑥}) ∈ V) |
79 | 19, 78 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐼 ∖ {𝑥}) ∈ V) |
80 | | mptexg 6740 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐼 ∖ {𝑥}) ∈ V → (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) ∈ V) |
81 | 79, 80 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) ∈ V) |
82 | 81 | ad2antrr 719 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) ∈ V) |
83 | | funmpt 6161 |
. . . . . . . . . . . . . . 15
⊢ Fun
(𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) |
84 | 83 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → Fun (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘))) |
85 | 26 | adantr 474 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝑓 finSupp 0 ) |
86 | | ssdif 3972 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐼 ∖ {𝑥}) ⊆ 𝐼 → ((𝐼 ∖ {𝑥}) ∖ (𝑓 supp 0 )) ⊆ (𝐼 ∖ (𝑓 supp 0 ))) |
87 | 58, 86 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐼 ∖ {𝑥}) ∖ (𝑓 supp 0 )) ⊆ (𝐼 ∖ (𝑓 supp 0 )) |
88 | 87 | sseli 3823 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ ((𝐼 ∖ {𝑥}) ∖ (𝑓 supp 0 )) → 𝑘 ∈ (𝐼 ∖ (𝑓 supp 0 ))) |
89 | | ssidd 3849 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑓 supp 0 ) ⊆ (𝑓 supp 0 )) |
90 | 19 | ad2antrr 719 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ V) |
91 | 3 | fvexi 6447 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 ∈
V |
92 | 91 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 0 ∈ V) |
93 | 66, 89, 90, 92 | suppssr 7591 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) ∧ 𝑘 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → (𝑓‘𝑘) = 0 ) |
94 | 88, 93 | sylan2 588 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) ∧ 𝑘 ∈ ((𝐼 ∖ {𝑥}) ∖ (𝑓 supp 0 ))) → (𝑓‘𝑘) = 0 ) |
95 | 79 | ad2antrr 719 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐼 ∖ {𝑥}) ∈ V) |
96 | 94, 95 | suppss2 7594 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ((𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) supp 0 ) ⊆ (𝑓 supp 0 )) |
97 | | fsuppsssupp 8560 |
. . . . . . . . . . . . . 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 ) |
98 | 82, 84, 85, 96, 97 | syl22anc 874 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) finSupp 0 ) |
99 | 57, 61, 65, 77, 98 | dprdwd 18764 |
. . . . . . . . . . . 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 }) |
100 | 71, 99 | eqeltrd 2906 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑓 ↾ (𝐼 ∖ {𝑥})) ∈ {ℎ ∈ X𝑖 ∈ (𝐼 ∖ {𝑥})((𝑆 ↾ (𝐼 ∖ {𝑥}))‘𝑖) ∣ ℎ finSupp 0 }) |
101 | 3, 57, 61, 65, 100 | eldprdi 18771 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg (𝑓 ↾ (𝐼 ∖ {𝑥}))) ∈ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))) |
102 | 27, 101 | sylan2 588 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → (𝐺 Σg
(𝑓 ↾ (𝐼 ∖ {𝑥}))) ∈ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))) |
103 | 56, 102 | eqeltrd 2906 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → 𝐴 ∈ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))) |
104 | | eqid 2825 |
. . . . . . . . . 10
⊢
(+g‘𝐺) = (+g‘𝐺) |
105 | | eqid 2825 |
. . . . . . . . . 10
⊢
(LSSum‘𝐺) =
(LSSum‘𝐺) |
106 | 62, 15 | ffvelrnd 6609 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ∈ (SubGrp‘𝐺)) |
107 | | dprdsubg 18777 |
. . . . . . . . . . 11
⊢ (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑥})) → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺)) |
108 | 61, 107 | syl 17 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺)) |
109 | 12, 13, 15, 3 | dpjdisj 18806 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ((𝑆‘𝑥) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))) = { 0 }) |
110 | 12, 13, 15, 34 | dpjcntz 18805 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))) |
111 | 104, 105,
3, 34, 106, 108, 109, 110, 28 | pj1rid 18466 |
. . . . . . . . 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 ) |
112 | 27, 111 | sylanl2 673 |
. . . . . . . 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 ) |
113 | 103, 112 | mpdan 680 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → (((𝑆‘𝑥)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))‘𝐴) = 0 ) |
114 | 31, 113 | eqtrd 2861 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → ((𝑃‘𝑥)‘𝐴) = 0 ) |
115 | 19 | adantr 474 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝐼 ∈ V) |
116 | 114, 115 | suppss2 7594 |
. . . . 5
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → ((𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) supp 0 ) ⊆ (𝑓 supp 0 )) |
117 | | fsuppsssupp 8560 |
. . . . 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 ) |
118 | 22, 24, 26, 116, 117 | syl22anc 874 |
. . . 4
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) finSupp 0 ) |
119 | 4, 10, 11, 18, 118 | dprdwd 18764 |
. . 3
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ 𝑊) |
120 | | simprr 791 |
. . . 4
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝐴 = (𝐺 Σg 𝑓)) |
121 | 40 | feqmptd 6496 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝑓 = (𝑥 ∈ 𝐼 ↦ (𝑓‘𝑥))) |
122 | | simplrr 798 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐴 = (𝐺 Σg 𝑓)) |
123 | 12, 35, 36 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Mnd) |
124 | 4, 12, 13, 42 | dprdffsupp 18767 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝑓 finSupp 0 ) |
125 | | disjdif 4263 |
. . . . . . . . . . . . 13
⊢ ({𝑥} ∩ (𝐼 ∖ {𝑥})) = ∅ |
126 | 125 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ({𝑥} ∩ (𝐼 ∖ {𝑥})) = ∅) |
127 | | undif2 4267 |
. . . . . . . . . . . . 13
⊢ ({𝑥} ∪ (𝐼 ∖ {𝑥})) = ({𝑥} ∪ 𝐼) |
128 | 15 | snssd 4558 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → {𝑥} ⊆ 𝐼) |
129 | | ssequn1 4010 |
. . . . . . . . . . . . . 14
⊢ ({𝑥} ⊆ 𝐼 ↔ ({𝑥} ∪ 𝐼) = 𝐼) |
130 | 128, 129 | sylib 210 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ({𝑥} ∪ 𝐼) = 𝐼) |
131 | 127, 130 | syl5req 2874 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐼 = ({𝑥} ∪ (𝐼 ∖ {𝑥}))) |
132 | 33, 3, 104, 34, 123, 90, 66, 43, 124, 126, 131 | gsumzsplit 18680 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg 𝑓) = ((𝐺 Σg (𝑓 ↾ {𝑥}))(+g‘𝐺)(𝐺 Σg (𝑓 ↾ (𝐼 ∖ {𝑥}))))) |
133 | 66, 128 | feqresmpt 6497 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑓 ↾ {𝑥}) = (𝑘 ∈ {𝑥} ↦ (𝑓‘𝑘))) |
134 | 133 | oveq2d 6921 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg (𝑓 ↾ {𝑥})) = (𝐺 Σg (𝑘 ∈ {𝑥} ↦ (𝑓‘𝑘)))) |
135 | 66, 15 | ffvelrnd 6609 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ (Base‘𝐺)) |
136 | | fveq2 6433 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑥 → (𝑓‘𝑘) = (𝑓‘𝑥)) |
137 | 33, 136 | gsumsn 18707 |
. . . . . . . . . . . . . 14
⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐼 ∧ (𝑓‘𝑥) ∈ (Base‘𝐺)) → (𝐺 Σg (𝑘 ∈ {𝑥} ↦ (𝑓‘𝑘))) = (𝑓‘𝑥)) |
138 | 123, 15, 135, 137 | syl3anc 1496 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg (𝑘 ∈ {𝑥} ↦ (𝑓‘𝑘))) = (𝑓‘𝑥)) |
139 | 134, 138 | eqtrd 2861 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg (𝑓 ↾ {𝑥})) = (𝑓‘𝑥)) |
140 | 139 | oveq1d 6920 |
. . . . . . . . . . 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}))))) |
141 | 122, 132,
140 | 3eqtrd 2865 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐴 = ((𝑓‘𝑥)(+g‘𝐺)(𝐺 Σg (𝑓 ↾ (𝐼 ∖ {𝑥}))))) |
142 | 12, 13, 15, 105 | dpjlsm 18807 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺 DProd 𝑆) = ((𝑆‘𝑥)(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))) |
143 | 17, 142 | eleqtrd 2908 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ ((𝑆‘𝑥)(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))) |
144 | 4, 10, 11, 25 | dprdfcl 18766 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ (𝑆‘𝑥)) |
145 | 104, 105,
3, 34, 106, 108, 109, 110, 28, 143, 144, 101 | pj1eq 18464 |
. . . . . . . . . 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})))))) |
146 | 141, 145 | mpbid 224 |
. . . . . . . . 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}))))) |
147 | 146 | simpld 490 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (((𝑆‘𝑥)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))‘𝐴) = (𝑓‘𝑥)) |
148 | 30, 147 | eqtrd 2861 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)‘𝐴) = (𝑓‘𝑥)) |
149 | 148 | mpteq2dva 4967 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) = (𝑥 ∈ 𝐼 ↦ (𝑓‘𝑥))) |
150 | 121, 149 | eqtr4d 2864 |
. . . . 5
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝑓 = (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴))) |
151 | 150 | oveq2d 6921 |
. . . 4
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → (𝐺 Σg 𝑓) = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)))) |
152 | 120, 151 | eqtrd 2861 |
. . 3
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)))) |
153 | 119, 152 | jca 509 |
. 2
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → ((𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴))))) |
154 | 8, 153 | rexlimddv 3245 |
1
⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴))))) |