Step | Hyp | Ref
| Expression |
1 | | sticksstones18.3 |
. . . . . . . . . . . . . 14
⊢ 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)} |
2 | 1 | eqimssi 3974 |
. . . . . . . . . . . . 13
⊢ 𝐴 ⊆ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)} |
3 | 2 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ⊆ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}) |
4 | 3 | sseld 3915 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑎 ∈ 𝐴 → 𝑎 ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)})) |
5 | 4 | imp 410 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}) |
6 | | vex 3425 |
. . . . . . . . . . 11
⊢ 𝑎 ∈ V |
7 | | feq1 6545 |
. . . . . . . . . . . 12
⊢ (𝑔 = 𝑎 → (𝑔:(1...𝐾)⟶ℕ0 ↔ 𝑎:(1...𝐾)⟶ℕ0)) |
8 | | simpl 486 |
. . . . . . . . . . . . . . 15
⊢ ((𝑔 = 𝑎 ∧ 𝑖 ∈ (1...𝐾)) → 𝑔 = 𝑎) |
9 | 8 | fveq1d 6738 |
. . . . . . . . . . . . . 14
⊢ ((𝑔 = 𝑎 ∧ 𝑖 ∈ (1...𝐾)) → (𝑔‘𝑖) = (𝑎‘𝑖)) |
10 | 9 | sumeq2dv 15292 |
. . . . . . . . . . . . 13
⊢ (𝑔 = 𝑎 → Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖)) |
11 | 10 | eqeq1d 2740 |
. . . . . . . . . . . 12
⊢ (𝑔 = 𝑎 → (Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖) = 𝑁)) |
12 | 7, 11 | anbi12d 634 |
. . . . . . . . . . 11
⊢ (𝑔 = 𝑎 → ((𝑔:(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁) ↔ (𝑎:(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖) = 𝑁))) |
13 | 6, 12 | elab 3600 |
. . . . . . . . . 10
⊢ (𝑎 ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)} ↔ (𝑎:(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖) = 𝑁)) |
14 | 5, 13 | sylib 221 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎:(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖) = 𝑁)) |
15 | 14 | simpld 498 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎:(1...𝐾)⟶ℕ0) |
16 | 15 | adantr 484 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝑆) → 𝑎:(1...𝐾)⟶ℕ0) |
17 | | sticksstones18.5 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑍:(1...𝐾)–1-1-onto→𝑆) |
18 | | f1ocnv 6692 |
. . . . . . . . . . 11
⊢ (𝑍:(1...𝐾)–1-1-onto→𝑆 → ◡𝑍:𝑆–1-1-onto→(1...𝐾)) |
19 | 17, 18 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ◡𝑍:𝑆–1-1-onto→(1...𝐾)) |
20 | | f1of 6680 |
. . . . . . . . . 10
⊢ (◡𝑍:𝑆–1-1-onto→(1...𝐾) → ◡𝑍:𝑆⟶(1...𝐾)) |
21 | 19, 20 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ◡𝑍:𝑆⟶(1...𝐾)) |
22 | 21 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ◡𝑍:𝑆⟶(1...𝐾)) |
23 | 22 | ffvelrnda 6923 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝑆) → (◡𝑍‘𝑥) ∈ (1...𝐾)) |
24 | 16, 23 | ffvelrnd 6924 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝑆) → (𝑎‘(◡𝑍‘𝑥)) ∈
ℕ0) |
25 | 24 | fmpttd 6951 |
. . . . 5
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))):𝑆⟶ℕ0) |
26 | | eqidd 2739 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) → (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))) |
27 | | simpr 488 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) ∧ 𝑥 = 𝑖) → 𝑥 = 𝑖) |
28 | 27 | fveq2d 6740 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) ∧ 𝑥 = 𝑖) → (◡𝑍‘𝑥) = (◡𝑍‘𝑖)) |
29 | 28 | fveq2d 6740 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) ∧ 𝑥 = 𝑖) → (𝑎‘(◡𝑍‘𝑥)) = (𝑎‘(◡𝑍‘𝑖))) |
30 | | simpr 488 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) → 𝑖 ∈ 𝑆) |
31 | | fvexd 6751 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) → (𝑎‘(◡𝑍‘𝑖)) ∈ V) |
32 | 26, 29, 30, 31 | fvmptd 6844 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = (𝑎‘(◡𝑍‘𝑖))) |
33 | 32 | sumeq2dv 15292 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = Σ𝑖 ∈ 𝑆 (𝑎‘(◡𝑍‘𝑖))) |
34 | | fveq2 6736 |
. . . . . . . . 9
⊢ (𝑛 = (◡𝑍‘𝑖) → (𝑎‘𝑛) = (𝑎‘(◡𝑍‘𝑖))) |
35 | | fzfid 13571 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (1...𝐾) ∈ Fin) |
36 | 17 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑍:(1...𝐾)–1-1-onto→𝑆) |
37 | | f1oenfi 8882 |
. . . . . . . . . . . 12
⊢
(((1...𝐾) ∈ Fin
∧ 𝑍:(1...𝐾)–1-1-onto→𝑆) → (1...𝐾) ≈ 𝑆) |
38 | 35, 36, 37 | syl2anc 587 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (1...𝐾) ≈ 𝑆) |
39 | 38 | ensymd 8702 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑆 ≈ (1...𝐾)) |
40 | | enfii 8888 |
. . . . . . . . . 10
⊢
(((1...𝐾) ∈ Fin
∧ 𝑆 ≈ (1...𝐾)) → 𝑆 ∈ Fin) |
41 | 35, 39, 40 | syl2anc 587 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑆 ∈ Fin) |
42 | 19 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ◡𝑍:𝑆–1-1-onto→(1...𝐾)) |
43 | | eqidd 2739 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) → (◡𝑍‘𝑖) = (◡𝑍‘𝑖)) |
44 | | nn0sscn 12120 |
. . . . . . . . . . . 12
⊢
ℕ0 ⊆ ℂ |
45 | 44 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ℕ0 ⊆
ℂ) |
46 | | fss 6581 |
. . . . . . . . . . 11
⊢ ((𝑎:(1...𝐾)⟶ℕ0 ∧
ℕ0 ⊆ ℂ) → 𝑎:(1...𝐾)⟶ℂ) |
47 | 15, 45, 46 | syl2anc 587 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎:(1...𝐾)⟶ℂ) |
48 | 47 | ffvelrnda 6923 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑛 ∈ (1...𝐾)) → (𝑎‘𝑛) ∈ ℂ) |
49 | 34, 41, 42, 43, 48 | fsumf1o 15312 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑛 ∈ (1...𝐾)(𝑎‘𝑛) = Σ𝑖 ∈ 𝑆 (𝑎‘(◡𝑍‘𝑖))) |
50 | 49 | eqcomd 2744 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ 𝑆 (𝑎‘(◡𝑍‘𝑖)) = Σ𝑛 ∈ (1...𝐾)(𝑎‘𝑛)) |
51 | | fveq2 6736 |
. . . . . . . . . 10
⊢ (𝑛 = 𝑖 → (𝑎‘𝑛) = (𝑎‘𝑖)) |
52 | 51 | cbvsumv 15285 |
. . . . . . . . 9
⊢
Σ𝑛 ∈
(1...𝐾)(𝑎‘𝑛) = Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖) |
53 | 52 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑛 ∈ (1...𝐾)(𝑎‘𝑛) = Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖)) |
54 | 14 | simprd 499 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖) = 𝑁) |
55 | 53, 54 | eqtrd 2778 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑛 ∈ (1...𝐾)(𝑎‘𝑛) = 𝑁) |
56 | 50, 55 | eqtrd 2778 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ 𝑆 (𝑎‘(◡𝑍‘𝑖)) = 𝑁) |
57 | 33, 56 | eqtrd 2778 |
. . . . 5
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = 𝑁) |
58 | 25, 57 | jca 515 |
. . . 4
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))):𝑆⟶ℕ0 ∧
Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = 𝑁)) |
59 | | fzfid 13571 |
. . . . . . . 8
⊢ (𝜑 → (1...𝐾) ∈ Fin) |
60 | 59 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (1...𝐾) ∈ Fin) |
61 | 59, 17, 37 | syl2anc 587 |
. . . . . . . . 9
⊢ (𝜑 → (1...𝐾) ≈ 𝑆) |
62 | 61 | ensymd 8702 |
. . . . . . . 8
⊢ (𝜑 → 𝑆 ≈ (1...𝐾)) |
63 | 62 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑆 ≈ (1...𝐾)) |
64 | 60, 63, 40 | syl2anc 587 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑆 ∈ Fin) |
65 | 64 | mptexd 7059 |
. . . . 5
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∈ V) |
66 | | feq1 6545 |
. . . . . . 7
⊢ (ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) → (ℎ:𝑆⟶ℕ0 ↔ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))):𝑆⟶ℕ0)) |
67 | | simpl 486 |
. . . . . . . . . 10
⊢ ((ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∧ 𝑖 ∈ 𝑆) → ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))) |
68 | 67 | fveq1d 6738 |
. . . . . . . . 9
⊢ ((ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∧ 𝑖 ∈ 𝑆) → (ℎ‘𝑖) = ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖)) |
69 | 68 | sumeq2dv 15292 |
. . . . . . . 8
⊢ (ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) → Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖)) |
70 | 69 | eqeq1d 2740 |
. . . . . . 7
⊢ (ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) → (Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = 𝑁)) |
71 | 66, 70 | anbi12d 634 |
. . . . . 6
⊢ (ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) → ((ℎ:𝑆⟶ℕ0 ∧
Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁) ↔ ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))):𝑆⟶ℕ0 ∧
Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = 𝑁))) |
72 | 71 | elabg 3598 |
. . . . 5
⊢ ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∈ V → ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧
Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)} ↔ ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))):𝑆⟶ℕ0 ∧
Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = 𝑁))) |
73 | 65, 72 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧
Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)} ↔ ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))):𝑆⟶ℕ0 ∧
Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = 𝑁))) |
74 | 58, 73 | mpbird 260 |
. . 3
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧
Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}) |
75 | | sticksstones18.4 |
. . . 4
⊢ 𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧
Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)} |
76 | 75 | a1i 11 |
. . 3
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧
Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}) |
77 | 74, 76 | eleqtrrd 2842 |
. 2
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∈ 𝐵) |
78 | | sticksstones18.6 |
. 2
⊢ 𝐹 = (𝑎 ∈ 𝐴 ↦ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))) |
79 | 77, 78 | fmptd 6950 |
1
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |