| Step | Hyp | Ref
| Expression |
| 1 | | sticksstones17.4 |
. . . . . . . . . . . . . . . 16
⊢ 𝐵 = {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧
Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)} |
| 2 | 1 | eqimssi 4044 |
. . . . . . . . . . . . . . 15
⊢ 𝐵 ⊆ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧
Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)} |
| 3 | 2 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐵 ⊆ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧
Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}) |
| 4 | 3 | sseld 3982 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑏 ∈ 𝐵 → 𝑏 ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧
Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)})) |
| 5 | 4 | imp 406 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧
Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)}) |
| 6 | | vex 3484 |
. . . . . . . . . . . . 13
⊢ 𝑏 ∈ V |
| 7 | | feq1 6716 |
. . . . . . . . . . . . . 14
⊢ (ℎ = 𝑏 → (ℎ:𝑆⟶ℕ0 ↔ 𝑏:𝑆⟶ℕ0)) |
| 8 | | simpl 482 |
. . . . . . . . . . . . . . . . 17
⊢ ((ℎ = 𝑏 ∧ 𝑖 ∈ 𝑆) → ℎ = 𝑏) |
| 9 | 8 | fveq1d 6908 |
. . . . . . . . . . . . . . . 16
⊢ ((ℎ = 𝑏 ∧ 𝑖 ∈ 𝑆) → (ℎ‘𝑖) = (𝑏‘𝑖)) |
| 10 | 9 | sumeq2dv 15738 |
. . . . . . . . . . . . . . 15
⊢ (ℎ = 𝑏 → Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = Σ𝑖 ∈ 𝑆 (𝑏‘𝑖)) |
| 11 | 10 | eqeq1d 2739 |
. . . . . . . . . . . . . 14
⊢ (ℎ = 𝑏 → (Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) = 𝑁)) |
| 12 | 7, 11 | anbi12d 632 |
. . . . . . . . . . . . 13
⊢ (ℎ = 𝑏 → ((ℎ:𝑆⟶ℕ0 ∧
Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁) ↔ (𝑏:𝑆⟶ℕ0 ∧
Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) = 𝑁))) |
| 13 | 6, 12 | elab 3679 |
. . . . . . . . . . . 12
⊢ (𝑏 ∈ {ℎ ∣ (ℎ:𝑆⟶ℕ0 ∧
Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = 𝑁)} ↔ (𝑏:𝑆⟶ℕ0 ∧
Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) = 𝑁)) |
| 14 | 5, 13 | sylib 218 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑏:𝑆⟶ℕ0 ∧
Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) = 𝑁)) |
| 15 | 14 | simpld 494 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏:𝑆⟶ℕ0) |
| 16 | 15 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (1...𝐾)) → 𝑏:𝑆⟶ℕ0) |
| 17 | 16 | 3impa 1110 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑦 ∈ (1...𝐾)) → 𝑏:𝑆⟶ℕ0) |
| 18 | | sticksstones17.5 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑍:(1...𝐾)–1-1-onto→𝑆) |
| 19 | | f1of 6848 |
. . . . . . . . . . . . 13
⊢ (𝑍:(1...𝐾)–1-1-onto→𝑆 → 𝑍:(1...𝐾)⟶𝑆) |
| 20 | 18, 19 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑍:(1...𝐾)⟶𝑆) |
| 21 | 20 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑍:(1...𝐾)⟶𝑆) |
| 22 | 21 | adantr 480 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (1...𝐾)) → 𝑍:(1...𝐾)⟶𝑆) |
| 23 | 22 | 3impa 1110 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑦 ∈ (1...𝐾)) → 𝑍:(1...𝐾)⟶𝑆) |
| 24 | | simp3 1139 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑦 ∈ (1...𝐾)) → 𝑦 ∈ (1...𝐾)) |
| 25 | 23, 24 | ffvelcdmd 7105 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑦 ∈ (1...𝐾)) → (𝑍‘𝑦) ∈ 𝑆) |
| 26 | 17, 25 | ffvelcdmd 7105 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵 ∧ 𝑦 ∈ (1...𝐾)) → (𝑏‘(𝑍‘𝑦)) ∈
ℕ0) |
| 27 | 26 | 3expa 1119 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑦 ∈ (1...𝐾)) → (𝑏‘(𝑍‘𝑦)) ∈
ℕ0) |
| 28 | 27 | fmpttd 7135 |
. . . . 5
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))):(1...𝐾)⟶ℕ0) |
| 29 | | eqidd 2738 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))) |
| 30 | | simpr 484 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) ∧ 𝑦 = 𝑖) → 𝑦 = 𝑖) |
| 31 | 30 | fveq2d 6910 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) ∧ 𝑦 = 𝑖) → (𝑍‘𝑦) = (𝑍‘𝑖)) |
| 32 | 31 | fveq2d 6910 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) ∧ 𝑦 = 𝑖) → (𝑏‘(𝑍‘𝑦)) = (𝑏‘(𝑍‘𝑖))) |
| 33 | | simpr 484 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → 𝑖 ∈ (1...𝐾)) |
| 34 | | fvexd 6921 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → (𝑏‘(𝑍‘𝑖)) ∈ V) |
| 35 | 29, 32, 33, 34 | fvmptd 7023 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = (𝑏‘(𝑍‘𝑖))) |
| 36 | 35 | sumeq2dv 15738 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = Σ𝑖 ∈ (1...𝐾)(𝑏‘(𝑍‘𝑖))) |
| 37 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑠 = (𝑍‘𝑖) → (𝑏‘𝑠) = (𝑏‘(𝑍‘𝑖))) |
| 38 | | fzfi 14013 |
. . . . . . . . . 10
⊢
(1...𝐾) ∈
Fin |
| 39 | 38 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (1...𝐾) ∈ Fin) |
| 40 | 18 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑍:(1...𝐾)–1-1-onto→𝑆) |
| 41 | | eqidd 2738 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑖 ∈ (1...𝐾)) → (𝑍‘𝑖) = (𝑍‘𝑖)) |
| 42 | | nn0sscn 12531 |
. . . . . . . . . . . 12
⊢
ℕ0 ⊆ ℂ |
| 43 | 42 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ℕ0 ⊆
ℂ) |
| 44 | | fss 6752 |
. . . . . . . . . . 11
⊢ ((𝑏:𝑆⟶ℕ0 ∧
ℕ0 ⊆ ℂ) → 𝑏:𝑆⟶ℂ) |
| 45 | 15, 43, 44 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝑏:𝑆⟶ℂ) |
| 46 | 45 | ffvelcdmda 7104 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑏 ∈ 𝐵) ∧ 𝑠 ∈ 𝑆) → (𝑏‘𝑠) ∈ ℂ) |
| 47 | 37, 39, 40, 41, 46 | fsumf1o 15759 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑠 ∈ 𝑆 (𝑏‘𝑠) = Σ𝑖 ∈ (1...𝐾)(𝑏‘(𝑍‘𝑖))) |
| 48 | 47 | eqcomd 2743 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ (1...𝐾)(𝑏‘(𝑍‘𝑖)) = Σ𝑠 ∈ 𝑆 (𝑏‘𝑠)) |
| 49 | | fveq2 6906 |
. . . . . . . . . 10
⊢ (𝑠 = 𝑖 → (𝑏‘𝑠) = (𝑏‘𝑖)) |
| 50 | 49 | cbvsumv 15732 |
. . . . . . . . 9
⊢
Σ𝑠 ∈
𝑆 (𝑏‘𝑠) = Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) |
| 51 | 50 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑠 ∈ 𝑆 (𝑏‘𝑠) = Σ𝑖 ∈ 𝑆 (𝑏‘𝑖)) |
| 52 | 14 | simprd 495 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ 𝑆 (𝑏‘𝑖) = 𝑁) |
| 53 | 51, 52 | eqtrd 2777 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑠 ∈ 𝑆 (𝑏‘𝑠) = 𝑁) |
| 54 | 48, 53 | eqtrd 2777 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ (1...𝐾)(𝑏‘(𝑍‘𝑖)) = 𝑁) |
| 55 | 36, 54 | eqtrd 2777 |
. . . . 5
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = 𝑁) |
| 56 | 28, 55 | jca 511 |
. . . 4
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))):(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = 𝑁)) |
| 57 | | fzfid 14014 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (1...𝐾) ∈ Fin) |
| 58 | 57 | mptexd 7244 |
. . . . 5
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∈ V) |
| 59 | | feq1 6716 |
. . . . . . 7
⊢ (𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) → (𝑔:(1...𝐾)⟶ℕ0 ↔ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))):(1...𝐾)⟶ℕ0)) |
| 60 | | simpl 482 |
. . . . . . . . . 10
⊢ ((𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∧ 𝑖 ∈ (1...𝐾)) → 𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))) |
| 61 | 60 | fveq1d 6908 |
. . . . . . . . 9
⊢ ((𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∧ 𝑖 ∈ (1...𝐾)) → (𝑔‘𝑖) = ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖)) |
| 62 | 61 | sumeq2dv 15738 |
. . . . . . . 8
⊢ (𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) → Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖)) |
| 63 | 62 | eqeq1d 2739 |
. . . . . . 7
⊢ (𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) → (Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = 𝑁)) |
| 64 | 59, 63 | anbi12d 632 |
. . . . . 6
⊢ (𝑔 = (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) → ((𝑔:(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁) ↔ ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))):(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = 𝑁))) |
| 65 | 64 | elabg 3676 |
. . . . 5
⊢ ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∈ V → ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)} ↔ ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))):(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = 𝑁))) |
| 66 | 58, 65 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)} ↔ ((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))):(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)((𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))‘𝑖) = 𝑁))) |
| 67 | 56, 66 | mpbird 257 |
. . 3
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}) |
| 68 | | sticksstones17.3 |
. . . 4
⊢ 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)} |
| 69 | 68 | a1i 11 |
. . 3
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}) |
| 70 | 67, 69 | eleqtrrd 2844 |
. 2
⊢ ((𝜑 ∧ 𝑏 ∈ 𝐵) → (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦))) ∈ 𝐴) |
| 71 | | sticksstones17.6 |
. 2
⊢ 𝐺 = (𝑏 ∈ 𝐵 ↦ (𝑦 ∈ (1...𝐾) ↦ (𝑏‘(𝑍‘𝑦)))) |
| 72 | 70, 71 | fmptd 7134 |
1
⊢ (𝜑 → 𝐺:𝐵⟶𝐴) |