Step | Hyp | Ref
| Expression |
1 | | gsumval2.b |
. . . 4
⊢ 𝐵 = (Base‘𝐺) |
2 | | eqid 2622 |
. . . 4
⊢
(0g‘𝐺) = (0g‘𝐺) |
3 | | gsumval2.p |
. . . 4
⊢ + =
(+g‘𝐺) |
4 | | gsumval2a.o |
. . . 4
⊢ 𝑂 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} |
5 | | eqidd 2623 |
. . . 4
⊢ (𝜑 → (◡𝐹 “ (V ∖ 𝑂)) = (◡𝐹 “ (V ∖ 𝑂))) |
6 | | gsumval2.g |
. . . 4
⊢ (𝜑 → 𝐺 ∈ 𝑉) |
7 | | ovexd 6680 |
. . . 4
⊢ (𝜑 → (𝑀...𝑁) ∈ V) |
8 | | gsumval2.f |
. . . 4
⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶𝐵) |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | gsumval 17271 |
. . 3
⊢ (𝜑 → (𝐺 Σg 𝐹) = if(ran 𝐹 ⊆ 𝑂, (0g‘𝐺), if((𝑀...𝑁) ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(#‘(◡𝐹 “ (V ∖ 𝑂))))–1-1-onto→(◡𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(#‘(◡𝐹 “ (V ∖ 𝑂))))))))) |
10 | | gsumval2a.f |
. . . . 5
⊢ (𝜑 → ¬ ran 𝐹 ⊆ 𝑂) |
11 | 10 | iffalsed 4097 |
. . . 4
⊢ (𝜑 → if(ran 𝐹 ⊆ 𝑂, (0g‘𝐺), if((𝑀...𝑁) ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(#‘(◡𝐹 “ (V ∖ 𝑂))))–1-1-onto→(◡𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(#‘(◡𝐹 “ (V ∖ 𝑂)))))))) = if((𝑀...𝑁) ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(#‘(◡𝐹 “ (V ∖ 𝑂))))–1-1-onto→(◡𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(#‘(◡𝐹 “ (V ∖ 𝑂)))))))) |
12 | | gsumval2.n |
. . . . . . 7
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
13 | | eluzel2 11692 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
14 | 12, 13 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑀 ∈ ℤ) |
15 | | eluzelz 11697 |
. . . . . . 7
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ ℤ) |
16 | 12, 15 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ ℤ) |
17 | | fzf 12330 |
. . . . . . . 8
⊢
...:(ℤ × ℤ)⟶𝒫 ℤ |
18 | | ffn 6045 |
. . . . . . . 8
⊢
(...:(ℤ × ℤ)⟶𝒫 ℤ → ... Fn
(ℤ × ℤ)) |
19 | 17, 18 | ax-mp 5 |
. . . . . . 7
⊢ ... Fn
(ℤ × ℤ) |
20 | | fnovrn 6809 |
. . . . . . 7
⊢ ((... Fn
(ℤ × ℤ) ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) ∈ ran ...) |
21 | 19, 20 | mp3an1 1411 |
. . . . . 6
⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) ∈ ran ...) |
22 | 14, 16, 21 | syl2anc 693 |
. . . . 5
⊢ (𝜑 → (𝑀...𝑁) ∈ ran ...) |
23 | 22 | iftrued 4094 |
. . . 4
⊢ (𝜑 → if((𝑀...𝑁) ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(#‘(◡𝐹 “ (V ∖ 𝑂))))–1-1-onto→(◡𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(#‘(◡𝐹 “ (V ∖ 𝑂))))))) = (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)))) |
24 | 11, 23 | eqtrd 2656 |
. . 3
⊢ (𝜑 → if(ran 𝐹 ⊆ 𝑂, (0g‘𝐺), if((𝑀...𝑁) ∈ ran ..., (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))), (℩𝑧∃𝑓(𝑓:(1...(#‘(◡𝐹 “ (V ∖ 𝑂))))–1-1-onto→(◡𝐹 “ (V ∖ 𝑂)) ∧ 𝑧 = (seq1( + , (𝐹 ∘ 𝑓))‘(#‘(◡𝐹 “ (V ∖ 𝑂)))))))) = (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)))) |
25 | 9, 24 | eqtrd 2656 |
. 2
⊢ (𝜑 → (𝐺 Σg 𝐹) = (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)))) |
26 | | fvex 6201 |
. . 3
⊢ (seq𝑀( + , 𝐹)‘𝑁) ∈ V |
27 | | fzopth 12378 |
. . . . . . . . . . 11
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀 = 𝑚 ∧ 𝑁 = 𝑛))) |
28 | 12, 27 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀 = 𝑚 ∧ 𝑁 = 𝑛))) |
29 | | simpl 473 |
. . . . . . . . . . . . . 14
⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → 𝑀 = 𝑚) |
30 | 29 | seqeq1d 12807 |
. . . . . . . . . . . . 13
⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → seq𝑀( + , 𝐹) = seq𝑚( + , 𝐹)) |
31 | | simpr 477 |
. . . . . . . . . . . . 13
⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → 𝑁 = 𝑛) |
32 | 30, 31 | fveq12d 6197 |
. . . . . . . . . . . 12
⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → (seq𝑀( + , 𝐹)‘𝑁) = (seq𝑚( + , 𝐹)‘𝑛)) |
33 | 32 | eqcomd 2628 |
. . . . . . . . . . 11
⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁)) |
34 | | eqeq1 2626 |
. . . . . . . . . . 11
⊢ (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → (𝑧 = (seq𝑀( + , 𝐹)‘𝑁) ↔ (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁))) |
35 | 33, 34 | syl5ibrcom 237 |
. . . . . . . . . 10
⊢ ((𝑀 = 𝑚 ∧ 𝑁 = 𝑛) → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁))) |
36 | 28, 35 | syl6bi 243 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑀...𝑁) = (𝑚...𝑛) → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)))) |
37 | 36 | impd 447 |
. . . . . . . 8
⊢ (𝜑 → (((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁))) |
38 | 37 | rexlimdvw 3034 |
. . . . . . 7
⊢ (𝜑 → (∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁))) |
39 | 38 | exlimdv 1861 |
. . . . . 6
⊢ (𝜑 → (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) → 𝑧 = (seq𝑀( + , 𝐹)‘𝑁))) |
40 | 14 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → 𝑀 ∈ ℤ) |
41 | | oveq2 6658 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑁 → (𝑀...𝑛) = (𝑀...𝑁)) |
42 | 41 | eqcomd 2628 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑁 → (𝑀...𝑁) = (𝑀...𝑛)) |
43 | 42 | biantrurd 529 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑁 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑛) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)))) |
44 | | fveq2 6191 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑁 → (seq𝑀( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑁)) |
45 | 44 | eqeq2d 2632 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑁 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁))) |
46 | 43, 45 | bitr3d 270 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑁 → (((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁))) |
47 | 46 | rspcev 3309 |
. . . . . . . . 9
⊢ ((𝑁 ∈
(ℤ≥‘𝑀) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑛 ∈ (ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛))) |
48 | 12, 47 | sylan 488 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑛 ∈ (ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛))) |
49 | | fveq2 6191 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑀 → (ℤ≥‘𝑚) =
(ℤ≥‘𝑀)) |
50 | | oveq1 6657 |
. . . . . . . . . . . 12
⊢ (𝑚 = 𝑀 → (𝑚...𝑛) = (𝑀...𝑛)) |
51 | 50 | eqeq2d 2632 |
. . . . . . . . . . 11
⊢ (𝑚 = 𝑀 → ((𝑀...𝑁) = (𝑚...𝑛) ↔ (𝑀...𝑁) = (𝑀...𝑛))) |
52 | | seqeq1 12804 |
. . . . . . . . . . . . 13
⊢ (𝑚 = 𝑀 → seq𝑚( + , 𝐹) = seq𝑀( + , 𝐹)) |
53 | 52 | fveq1d 6193 |
. . . . . . . . . . . 12
⊢ (𝑚 = 𝑀 → (seq𝑚( + , 𝐹)‘𝑛) = (seq𝑀( + , 𝐹)‘𝑛)) |
54 | 53 | eqeq2d 2632 |
. . . . . . . . . . 11
⊢ (𝑚 = 𝑀 → (𝑧 = (seq𝑚( + , 𝐹)‘𝑛) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛))) |
55 | 51, 54 | anbi12d 747 |
. . . . . . . . . 10
⊢ (𝑚 = 𝑀 → (((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)))) |
56 | 49, 55 | rexeqbidv 3153 |
. . . . . . . . 9
⊢ (𝑚 = 𝑀 → (∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ ∃𝑛 ∈ (ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)))) |
57 | 56 | spcegv 3294 |
. . . . . . . 8
⊢ (𝑀 ∈ ℤ →
(∃𝑛 ∈
(ℤ≥‘𝑀)((𝑀...𝑁) = (𝑀...𝑛) ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑛)) → ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)))) |
58 | 40, 48, 57 | sylc 65 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁)) → ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))) |
59 | 58 | ex 450 |
. . . . . 6
⊢ (𝜑 → (𝑧 = (seq𝑀( + , 𝐹)‘𝑁) → ∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)))) |
60 | 39, 59 | impbid 202 |
. . . . 5
⊢ (𝜑 → (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁))) |
61 | 60 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ (seq𝑀( + , 𝐹)‘𝑁) ∈ V) → (∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛)) ↔ 𝑧 = (seq𝑀( + , 𝐹)‘𝑁))) |
62 | 61 | iota5 5871 |
. . 3
⊢ ((𝜑 ∧ (seq𝑀( + , 𝐹)‘𝑁) ∈ V) → (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))) = (seq𝑀( + , 𝐹)‘𝑁)) |
63 | 26, 62 | mpan2 707 |
. 2
⊢ (𝜑 → (℩𝑧∃𝑚∃𝑛 ∈ (ℤ≥‘𝑚)((𝑀...𝑁) = (𝑚...𝑛) ∧ 𝑧 = (seq𝑚( + , 𝐹)‘𝑛))) = (seq𝑀( + , 𝐹)‘𝑁)) |
64 | 25, 63 | eqtrd 2656 |
1
⊢ (𝜑 → (𝐺 Σg 𝐹) = (seq𝑀( + , 𝐹)‘𝑁)) |