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 18226 |
. . . . 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 221 |
. . 3
⊢ (𝜑 → (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓))) |
8 | 7 | simprd 478 |
. 2
⊢ (𝜑 → ∃𝑓 ∈ 𝑊 𝐴 = (𝐺 Σg 𝑓)) |
9 | | dpjfval.1 |
. . . . 5
⊢ (𝜑 → 𝐺dom DProd 𝑆) |
10 | 9 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝐺dom DProd 𝑆) |
11 | 2 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → dom 𝑆 = 𝐼) |
12 | 9 | ad2antrr 758 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐺dom DProd 𝑆) |
13 | 2 | ad2antrr 758 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → dom 𝑆 = 𝐼) |
14 | | dpjfval.p |
. . . . . 6
⊢ 𝑃 = (𝐺dProj𝑆) |
15 | | simpr 476 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝑥 ∈ 𝐼) |
16 | 12, 13, 14, 15 | dpjf 18279 |
. . . . 5
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑃‘𝑥):(𝐺 DProd 𝑆)⟶(𝑆‘𝑥)) |
17 | 1 | ad2antrr 758 |
. . . . 5
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ (𝐺 DProd 𝑆)) |
18 | 16, 17 | ffvelrnd 6268 |
. . . 4
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)‘𝐴) ∈ (𝑆‘𝑥)) |
19 | 9, 2 | dprddomcld 18223 |
. . . . . . 7
⊢ (𝜑 → 𝐼 ∈ V) |
20 | | mptexg 6389 |
. . . . . . 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 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ V) |
23 | | funmpt 5840 |
. . . . . 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 790 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝑓 ∈ 𝑊) |
26 | 4, 10, 11, 25 | dprdffsupp 18236 |
. . . . 5
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝑓 finSupp 0 ) |
27 | | eldifi 3694 |
. . . . . . . 8
⊢ (𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 )) → 𝑥 ∈ 𝐼) |
28 | | eqid 2610 |
. . . . . . . . . 10
⊢
(proj1‘𝐺) = (proj1‘𝐺) |
29 | 12, 13, 14, 28, 15 | dpjval 18278 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑃‘𝑥) = ((𝑆‘𝑥)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))) |
30 | 29 | fveq1d 6105 |
. . . . . . . 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 490 |
. . . . . . 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 797 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → 𝐴 = (𝐺 Σg 𝑓)) |
33 | | eqid 2610 |
. . . . . . . . . . 11
⊢
(Base‘𝐺) =
(Base‘𝐺) |
34 | | eqid 2610 |
. . . . . . . . . . 11
⊢
(Cntz‘𝐺) =
(Cntz‘𝐺) |
35 | | dprdgrp 18227 |
. . . . . . . . . . . . 13
⊢ (𝐺dom DProd 𝑆 → 𝐺 ∈ Grp) |
36 | | grpmnd 17252 |
. . . . . . . . . . . . 13
⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) |
37 | 10, 35, 36 | 3syl 18 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝐺 ∈ Mnd) |
38 | 37 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → 𝐺 ∈ Mnd) |
39 | 19 | ad2antrr 758 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → 𝐼 ∈ V) |
40 | 4, 10, 11, 25, 33 | dprdff 18234 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝑓:𝐼⟶(Base‘𝐺)) |
41 | 40 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → 𝑓:𝐼⟶(Base‘𝐺)) |
42 | 25 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝑓 ∈ 𝑊) |
43 | 4, 12, 13, 42, 34 | dprdfcntz 18237 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ran 𝑓 ⊆ ((Cntz‘𝐺)‘ran 𝑓)) |
44 | 27, 43 | sylan2 490 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → ran 𝑓 ⊆ ((Cntz‘𝐺)‘ran 𝑓)) |
45 | | snssi 4280 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 )) → {𝑥} ⊆ (𝐼 ∖ (𝑓 supp 0 ))) |
46 | 45 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → {𝑥} ⊆ (𝐼 ∖ (𝑓 supp 0 ))) |
47 | 46 | difss2d 3702 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → {𝑥} ⊆ 𝐼) |
48 | | suppssdm 7195 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 supp 0 ) ⊆ dom 𝑓 |
49 | | fdm 5964 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓:𝐼⟶(Base‘𝐺) → dom 𝑓 = 𝐼) |
50 | 40, 49 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → dom 𝑓 = 𝐼) |
51 | 48, 50 | syl5sseq 3616 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → (𝑓 supp 0 ) ⊆ 𝐼) |
52 | 51 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → (𝑓 supp 0 ) ⊆ 𝐼) |
53 | | ssconb 3705 |
. . . . . . . . . . . . 13
⊢ (({𝑥} ⊆ 𝐼 ∧ (𝑓 supp 0 ) ⊆ 𝐼) → ({𝑥} ⊆ (𝐼 ∖ (𝑓 supp 0 )) ↔ (𝑓 supp 0 ) ⊆ (𝐼 ∖ {𝑥}))) |
54 | 47, 52, 53 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → ({𝑥} ⊆ (𝐼 ∖ (𝑓 supp 0 )) ↔ (𝑓 supp 0 ) ⊆ (𝐼 ∖ {𝑥}))) |
55 | 46, 54 | mpbid 221 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → (𝑓 supp 0 ) ⊆ (𝐼 ∖ {𝑥})) |
56 | 26 | adantr 480 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → 𝑓 finSupp 0 ) |
57 | 33, 3, 34, 38, 39, 41, 44, 55, 56 | gsumzres 18133 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → (𝐺 Σg
(𝑓 ↾ (𝐼 ∖ {𝑥}))) = (𝐺 Σg 𝑓)) |
58 | 32, 57 | eqtr4d 2647 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → 𝐴 = (𝐺 Σg (𝑓 ↾ (𝐼 ∖ {𝑥})))) |
59 | | eqid 2610 |
. . . . . . . . . . 11
⊢ {ℎ ∈ X𝑖 ∈
(𝐼 ∖ {𝑥})((𝑆 ↾ (𝐼 ∖ {𝑥}))‘𝑖) ∣ ℎ finSupp 0 } = {ℎ ∈ X𝑖 ∈ (𝐼 ∖ {𝑥})((𝑆 ↾ (𝐼 ∖ {𝑥}))‘𝑖) ∣ ℎ finSupp 0 } |
60 | | difss 3699 |
. . . . . . . . . . . . . 14
⊢ (𝐼 ∖ {𝑥}) ⊆ 𝐼 |
61 | 60 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐼 ∖ {𝑥}) ⊆ 𝐼) |
62 | 12, 13, 61 | dprdres 18250 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑥})) ∧ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))) ⊆ (𝐺 DProd 𝑆))) |
63 | 62 | simpld 474 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))) |
64 | 12, 13 | dprdf2 18229 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝑆:𝐼⟶(SubGrp‘𝐺)) |
65 | | fssres 5983 |
. . . . . . . . . . . . 13
⊢ ((𝑆:𝐼⟶(SubGrp‘𝐺) ∧ (𝐼 ∖ {𝑥}) ⊆ 𝐼) → (𝑆 ↾ (𝐼 ∖ {𝑥})):(𝐼 ∖ {𝑥})⟶(SubGrp‘𝐺)) |
66 | 64, 60, 65 | sylancl 693 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑆 ↾ (𝐼 ∖ {𝑥})):(𝐼 ∖ {𝑥})⟶(SubGrp‘𝐺)) |
67 | | fdm 5964 |
. . . . . . . . . . . 12
⊢ ((𝑆 ↾ (𝐼 ∖ {𝑥})):(𝐼 ∖ {𝑥})⟶(SubGrp‘𝐺) → dom (𝑆 ↾ (𝐼 ∖ {𝑥})) = (𝐼 ∖ {𝑥})) |
68 | 66, 67 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → dom (𝑆 ↾ (𝐼 ∖ {𝑥})) = (𝐼 ∖ {𝑥})) |
69 | 40 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝑓:𝐼⟶(Base‘𝐺)) |
70 | 69 | feqmptd 6159 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝑓 = (𝑘 ∈ 𝐼 ↦ (𝑓‘𝑘))) |
71 | 70 | reseq1d 5316 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑓 ↾ (𝐼 ∖ {𝑥})) = ((𝑘 ∈ 𝐼 ↦ (𝑓‘𝑘)) ↾ (𝐼 ∖ {𝑥}))) |
72 | | resmpt 5369 |
. . . . . . . . . . . . . 14
⊢ ((𝐼 ∖ {𝑥}) ⊆ 𝐼 → ((𝑘 ∈ 𝐼 ↦ (𝑓‘𝑘)) ↾ (𝐼 ∖ {𝑥})) = (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘))) |
73 | 60, 72 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ 𝐼 ↦ (𝑓‘𝑘)) ↾ (𝐼 ∖ {𝑥})) = (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) |
74 | 71, 73 | syl6eq 2660 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑓 ↾ (𝐼 ∖ {𝑥})) = (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘))) |
75 | | eldifi 3694 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (𝐼 ∖ {𝑥}) → 𝑘 ∈ 𝐼) |
76 | 4, 12, 13, 42 | dprdfcl 18235 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) ∧ 𝑘 ∈ 𝐼) → (𝑓‘𝑘) ∈ (𝑆‘𝑘)) |
77 | 75, 76 | sylan2 490 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) ∧ 𝑘 ∈ (𝐼 ∖ {𝑥})) → (𝑓‘𝑘) ∈ (𝑆‘𝑘)) |
78 | | fvres 6117 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (𝐼 ∖ {𝑥}) → ((𝑆 ↾ (𝐼 ∖ {𝑥}))‘𝑘) = (𝑆‘𝑘)) |
79 | 78 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) ∧ 𝑘 ∈ (𝐼 ∖ {𝑥})) → ((𝑆 ↾ (𝐼 ∖ {𝑥}))‘𝑘) = (𝑆‘𝑘)) |
80 | 77, 79 | eleqtrrd 2691 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) ∧ 𝑘 ∈ (𝐼 ∖ {𝑥})) → (𝑓‘𝑘) ∈ ((𝑆 ↾ (𝐼 ∖ {𝑥}))‘𝑘)) |
81 | | difexg 4735 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐼 ∈ V → (𝐼 ∖ {𝑥}) ∈ V) |
82 | 19, 81 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐼 ∖ {𝑥}) ∈ V) |
83 | | mptexg 6389 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐼 ∖ {𝑥}) ∈ V → (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) ∈ V) |
84 | 82, 83 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) ∈ V) |
85 | 84 | ad2antrr 758 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) ∈ V) |
86 | | funmpt 5840 |
. . . . . . . . . . . . . . 15
⊢ Fun
(𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) |
87 | 86 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → Fun (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘))) |
88 | 26 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝑓 finSupp 0 ) |
89 | | ssdif 3707 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐼 ∖ {𝑥}) ⊆ 𝐼 → ((𝐼 ∖ {𝑥}) ∖ (𝑓 supp 0 )) ⊆ (𝐼 ∖ (𝑓 supp 0 ))) |
90 | 60, 89 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐼 ∖ {𝑥}) ∖ (𝑓 supp 0 )) ⊆ (𝐼 ∖ (𝑓 supp 0 )) |
91 | 90 | sseli 3564 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ ((𝐼 ∖ {𝑥}) ∖ (𝑓 supp 0 )) → 𝑘 ∈ (𝐼 ∖ (𝑓 supp 0 ))) |
92 | | ssid 3587 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓 supp 0 ) ⊆ (𝑓 supp 0 ) |
93 | 92 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑓 supp 0 ) ⊆ (𝑓 supp 0 )) |
94 | 19 | ad2antrr 758 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐼 ∈ V) |
95 | | fvex 6113 |
. . . . . . . . . . . . . . . . . . 19
⊢
(0g‘𝐺) ∈ V |
96 | 3, 95 | eqeltri 2684 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 ∈
V |
97 | 96 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 0 ∈ V) |
98 | 69, 93, 94, 97 | suppssr 7213 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) ∧ 𝑘 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → (𝑓‘𝑘) = 0 ) |
99 | 91, 98 | sylan2 490 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) ∧ 𝑘 ∈ ((𝐼 ∖ {𝑥}) ∖ (𝑓 supp 0 ))) → (𝑓‘𝑘) = 0 ) |
100 | 82 | ad2antrr 758 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐼 ∖ {𝑥}) ∈ V) |
101 | 99, 100 | suppss2 7216 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ((𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) supp 0 ) ⊆ (𝑓 supp 0 )) |
102 | | fsuppsssupp 8174 |
. . . . . . . . . . . . . 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 ) |
103 | 85, 87, 88, 101, 102 | syl22anc 1319 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑘 ∈ (𝐼 ∖ {𝑥}) ↦ (𝑓‘𝑘)) finSupp 0 ) |
104 | 59, 63, 68, 80, 103 | dprdwd 18233 |
. . . . . . . . . . . 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 }) |
105 | 74, 104 | eqeltrd 2688 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑓 ↾ (𝐼 ∖ {𝑥})) ∈ {ℎ ∈ X𝑖 ∈ (𝐼 ∖ {𝑥})((𝑆 ↾ (𝐼 ∖ {𝑥}))‘𝑖) ∣ ℎ finSupp 0 }) |
106 | 3, 59, 63, 68, 105 | eldprdi 18240 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg (𝑓 ↾ (𝐼 ∖ {𝑥}))) ∈ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))) |
107 | 27, 106 | sylan2 490 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → (𝐺 Σg
(𝑓 ↾ (𝐼 ∖ {𝑥}))) ∈ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))) |
108 | 58, 107 | eqeltrd 2688 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → 𝐴 ∈ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))) |
109 | | eqid 2610 |
. . . . . . . . . 10
⊢
(+g‘𝐺) = (+g‘𝐺) |
110 | | eqid 2610 |
. . . . . . . . . 10
⊢
(LSSum‘𝐺) =
(LSSum‘𝐺) |
111 | 64, 15 | ffvelrnd 6268 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ∈ (SubGrp‘𝐺)) |
112 | | dprdsubg 18246 |
. . . . . . . . . . 11
⊢ (𝐺dom DProd (𝑆 ↾ (𝐼 ∖ {𝑥})) → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺)) |
113 | 63, 112 | syl 17 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))) ∈ (SubGrp‘𝐺)) |
114 | 12, 13, 15, 3 | dpjdisj 18275 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ((𝑆‘𝑥) ∩ (𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥})))) = { 0 }) |
115 | 12, 13, 15, 34 | dpjcntz 18274 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑆‘𝑥) ⊆ ((Cntz‘𝐺)‘(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))) |
116 | 109, 110,
3, 34, 111, 113, 114, 115, 28 | pj1rid 17938 |
. . . . . . . . 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 ) |
117 | 27, 116 | 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 ) |
118 | 108, 117 | mpdan 699 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → (((𝑆‘𝑥)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))‘𝐴) = 0 ) |
119 | 31, 118 | eqtrd 2644 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ (𝐼 ∖ (𝑓 supp 0 ))) → ((𝑃‘𝑥)‘𝐴) = 0 ) |
120 | 19 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝐼 ∈ V) |
121 | 119, 120 | suppss2 7216 |
. . . . 5
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → ((𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) supp 0 ) ⊆ (𝑓 supp 0 )) |
122 | | fsuppsssupp 8174 |
. . . . 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 ) |
123 | 22, 24, 26, 121, 122 | syl22anc 1319 |
. . . 4
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) finSupp 0 ) |
124 | 4, 10, 11, 18, 123 | dprdwd 18233 |
. . 3
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ 𝑊) |
125 | | simprr 792 |
. . . 4
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝐴 = (𝐺 Σg 𝑓)) |
126 | 40 | feqmptd 6159 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝑓 = (𝑥 ∈ 𝐼 ↦ (𝑓‘𝑥))) |
127 | | simplrr 797 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐴 = (𝐺 Σg 𝑓)) |
128 | 12, 35, 36 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐺 ∈ Mnd) |
129 | 4, 12, 13, 42 | dprdffsupp 18236 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝑓 finSupp 0 ) |
130 | | disjdif 3992 |
. . . . . . . . . . . . 13
⊢ ({𝑥} ∩ (𝐼 ∖ {𝑥})) = ∅ |
131 | 130 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ({𝑥} ∩ (𝐼 ∖ {𝑥})) = ∅) |
132 | | undif2 3996 |
. . . . . . . . . . . . 13
⊢ ({𝑥} ∪ (𝐼 ∖ {𝑥})) = ({𝑥} ∪ 𝐼) |
133 | 15 | snssd 4281 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → {𝑥} ⊆ 𝐼) |
134 | | ssequn1 3745 |
. . . . . . . . . . . . . 14
⊢ ({𝑥} ⊆ 𝐼 ↔ ({𝑥} ∪ 𝐼) = 𝐼) |
135 | 133, 134 | sylib 207 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ({𝑥} ∪ 𝐼) = 𝐼) |
136 | 132, 135 | syl5req 2657 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐼 = ({𝑥} ∪ (𝐼 ∖ {𝑥}))) |
137 | 33, 3, 109, 34, 128, 94, 69, 43, 129, 131, 136 | gsumzsplit 18150 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg 𝑓) = ((𝐺 Σg (𝑓 ↾ {𝑥}))(+g‘𝐺)(𝐺 Σg (𝑓 ↾ (𝐼 ∖ {𝑥}))))) |
138 | 69, 133 | feqresmpt 6160 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑓 ↾ {𝑥}) = (𝑘 ∈ {𝑥} ↦ (𝑓‘𝑘))) |
139 | 138 | oveq2d 6565 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg (𝑓 ↾ {𝑥})) = (𝐺 Σg (𝑘 ∈ {𝑥} ↦ (𝑓‘𝑘)))) |
140 | 69, 15 | ffvelrnd 6268 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ (Base‘𝐺)) |
141 | | fveq2 6103 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝑥 → (𝑓‘𝑘) = (𝑓‘𝑥)) |
142 | 33, 141 | gsumsn 18177 |
. . . . . . . . . . . . . 14
⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐼 ∧ (𝑓‘𝑥) ∈ (Base‘𝐺)) → (𝐺 Σg (𝑘 ∈ {𝑥} ↦ (𝑓‘𝑘))) = (𝑓‘𝑥)) |
143 | 128, 15, 140, 142 | syl3anc 1318 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg (𝑘 ∈ {𝑥} ↦ (𝑓‘𝑘))) = (𝑓‘𝑥)) |
144 | 139, 143 | eqtrd 2644 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺 Σg (𝑓 ↾ {𝑥})) = (𝑓‘𝑥)) |
145 | 144 | oveq1d 6564 |
. . . . . . . . . . 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}))))) |
146 | 127, 137,
145 | 3eqtrd 2648 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐴 = ((𝑓‘𝑥)(+g‘𝐺)(𝐺 Σg (𝑓 ↾ (𝐼 ∖ {𝑥}))))) |
147 | 12, 13, 15, 110 | dpjlsm 18276 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝐺 DProd 𝑆) = ((𝑆‘𝑥)(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))) |
148 | 17, 147 | eleqtrd 2690 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → 𝐴 ∈ ((𝑆‘𝑥)(LSSum‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))) |
149 | 4, 10, 11, 25 | dprdfcl 18235 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (𝑓‘𝑥) ∈ (𝑆‘𝑥)) |
150 | 109, 110,
3, 34, 111, 113, 114, 115, 28, 148, 149, 106 | pj1eq 17936 |
. . . . . . . . . 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})))))) |
151 | 146, 150 | mpbid 221 |
. . . . . . . . 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}))))) |
152 | 151 | simpld 474 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → (((𝑆‘𝑥)(proj1‘𝐺)(𝐺 DProd (𝑆 ↾ (𝐼 ∖ {𝑥}))))‘𝐴) = (𝑓‘𝑥)) |
153 | 30, 152 | eqtrd 2644 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) ∧ 𝑥 ∈ 𝐼) → ((𝑃‘𝑥)‘𝐴) = (𝑓‘𝑥)) |
154 | 153 | mpteq2dva 4672 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) = (𝑥 ∈ 𝐼 ↦ (𝑓‘𝑥))) |
155 | 126, 154 | eqtr4d 2647 |
. . . . 5
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝑓 = (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴))) |
156 | 155 | oveq2d 6565 |
. . . 4
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → (𝐺 Σg 𝑓) = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)))) |
157 | 125, 156 | eqtrd 2644 |
. . 3
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → 𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)))) |
158 | 124, 157 | jca 553 |
. 2
⊢ ((𝜑 ∧ (𝑓 ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg 𝑓))) → ((𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴))))) |
159 | 8, 158 | rexlimddv 3017 |
1
⊢ (𝜑 → ((𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴)) ∈ 𝑊 ∧ 𝐴 = (𝐺 Σg (𝑥 ∈ 𝐼 ↦ ((𝑃‘𝑥)‘𝐴))))) |