| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | sticksstones18.3 | . . . . . . . . . . . . . 14
⊢ 𝐴 = {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)} | 
| 2 | 1 | eqimssi 4043 | . . . . . . . . . . . . 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 3981 | . . . . . . . . . . 11
⊢ (𝜑 → (𝑎 ∈ 𝐴 → 𝑎 ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)})) | 
| 5 | 4 | imp 406 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎 ∈ {𝑔 ∣ (𝑔:(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁)}) | 
| 6 |  | vex 3483 | . . . . . . . . . . 11
⊢ 𝑎 ∈ V | 
| 7 |  | feq1 6715 | . . . . . . . . . . . 12
⊢ (𝑔 = 𝑎 → (𝑔:(1...𝐾)⟶ℕ0 ↔ 𝑎:(1...𝐾)⟶ℕ0)) | 
| 8 |  | simpl 482 | . . . . . . . . . . . . . . 15
⊢ ((𝑔 = 𝑎 ∧ 𝑖 ∈ (1...𝐾)) → 𝑔 = 𝑎) | 
| 9 | 8 | fveq1d 6907 | . . . . . . . . . . . . . 14
⊢ ((𝑔 = 𝑎 ∧ 𝑖 ∈ (1...𝐾)) → (𝑔‘𝑖) = (𝑎‘𝑖)) | 
| 10 | 9 | sumeq2dv 15739 | . . . . . . . . . . . . 13
⊢ (𝑔 = 𝑎 → Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖)) | 
| 11 | 10 | eqeq1d 2738 | . . . . . . . . . . . 12
⊢ (𝑔 = 𝑎 → (Σ𝑖 ∈ (1...𝐾)(𝑔‘𝑖) = 𝑁 ↔ Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖) = 𝑁)) | 
| 12 | 7, 11 | anbi12d 632 | . . . . . . . . . . 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 3678 | . . . . . . . . . 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 218 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑎:(1...𝐾)⟶ℕ0 ∧
Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖) = 𝑁)) | 
| 15 | 14 | simpld 494 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎:(1...𝐾)⟶ℕ0) | 
| 16 | 15 | adantr 480 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝑆) → 𝑎:(1...𝐾)⟶ℕ0) | 
| 17 |  | sticksstones18.5 | . . . . . . . . . . 11
⊢ (𝜑 → 𝑍:(1...𝐾)–1-1-onto→𝑆) | 
| 18 |  | f1ocnv 6859 | . . . . . . . . . . 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 6847 | . . . . . . . . . 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 480 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ◡𝑍:𝑆⟶(1...𝐾)) | 
| 23 | 22 | ffvelcdmda 7103 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝑆) → (◡𝑍‘𝑥) ∈ (1...𝐾)) | 
| 24 | 16, 23 | ffvelcdmd 7104 | . . . . . 6
⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑥 ∈ 𝑆) → (𝑎‘(◡𝑍‘𝑥)) ∈
ℕ0) | 
| 25 | 24 | fmpttd 7134 | . . . . 5
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))):𝑆⟶ℕ0) | 
| 26 |  | eqidd 2737 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) → (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))) | 
| 27 |  | simpr 484 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) ∧ 𝑥 = 𝑖) → 𝑥 = 𝑖) | 
| 28 | 27 | fveq2d 6909 | . . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) ∧ 𝑥 = 𝑖) → (◡𝑍‘𝑥) = (◡𝑍‘𝑖)) | 
| 29 | 28 | fveq2d 6909 | . . . . . . . 8
⊢ ((((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) ∧ 𝑥 = 𝑖) → (𝑎‘(◡𝑍‘𝑥)) = (𝑎‘(◡𝑍‘𝑖))) | 
| 30 |  | simpr 484 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) → 𝑖 ∈ 𝑆) | 
| 31 |  | fvexd 6920 | . . . . . . . 8
⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) → (𝑎‘(◡𝑍‘𝑖)) ∈ V) | 
| 32 | 26, 29, 30, 31 | fvmptd 7022 | . . . . . . 7
⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) → ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = (𝑎‘(◡𝑍‘𝑖))) | 
| 33 | 32 | sumeq2dv 15739 | . . . . . 6
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = Σ𝑖 ∈ 𝑆 (𝑎‘(◡𝑍‘𝑖))) | 
| 34 |  | fveq2 6905 | . . . . . . . . 9
⊢ (𝑛 = (◡𝑍‘𝑖) → (𝑎‘𝑛) = (𝑎‘(◡𝑍‘𝑖))) | 
| 35 |  | fzfid 14015 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (1...𝐾) ∈ Fin) | 
| 36 | 17 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑍:(1...𝐾)–1-1-onto→𝑆) | 
| 37 |  | f1oenfi 9220 | . . . . . . . . . . . 12
⊢
(((1...𝐾) ∈ Fin
∧ 𝑍:(1...𝐾)–1-1-onto→𝑆) → (1...𝐾) ≈ 𝑆) | 
| 38 | 35, 36, 37 | syl2anc 584 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (1...𝐾) ≈ 𝑆) | 
| 39 | 38 | ensymd 9046 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑆 ≈ (1...𝐾)) | 
| 40 |  | enfii 9227 | . . . . . . . . . 10
⊢
(((1...𝐾) ∈ Fin
∧ 𝑆 ≈ (1...𝐾)) → 𝑆 ∈ Fin) | 
| 41 | 35, 39, 40 | syl2anc 584 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑆 ∈ Fin) | 
| 42 | 19 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ◡𝑍:𝑆–1-1-onto→(1...𝐾)) | 
| 43 |  | eqidd 2737 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑖 ∈ 𝑆) → (◡𝑍‘𝑖) = (◡𝑍‘𝑖)) | 
| 44 |  | nn0sscn 12533 | . . . . . . . . . . . 12
⊢
ℕ0 ⊆ ℂ | 
| 45 | 44 | a1i 11 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ℕ0 ⊆
ℂ) | 
| 46 |  | fss 6751 | . . . . . . . . . . 11
⊢ ((𝑎:(1...𝐾)⟶ℕ0 ∧
ℕ0 ⊆ ℂ) → 𝑎:(1...𝐾)⟶ℂ) | 
| 47 | 15, 45, 46 | syl2anc 584 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑎:(1...𝐾)⟶ℂ) | 
| 48 | 47 | ffvelcdmda 7103 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑎 ∈ 𝐴) ∧ 𝑛 ∈ (1...𝐾)) → (𝑎‘𝑛) ∈ ℂ) | 
| 49 | 34, 41, 42, 43, 48 | fsumf1o 15760 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑛 ∈ (1...𝐾)(𝑎‘𝑛) = Σ𝑖 ∈ 𝑆 (𝑎‘(◡𝑍‘𝑖))) | 
| 50 | 49 | eqcomd 2742 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ 𝑆 (𝑎‘(◡𝑍‘𝑖)) = Σ𝑛 ∈ (1...𝐾)(𝑎‘𝑛)) | 
| 51 |  | fveq2 6905 | . . . . . . . . . 10
⊢ (𝑛 = 𝑖 → (𝑎‘𝑛) = (𝑎‘𝑖)) | 
| 52 | 51 | cbvsumv 15733 | . . . . . . . . 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 495 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ (1...𝐾)(𝑎‘𝑖) = 𝑁) | 
| 55 | 53, 54 | eqtrd 2776 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑛 ∈ (1...𝐾)(𝑎‘𝑛) = 𝑁) | 
| 56 | 50, 55 | eqtrd 2776 | . . . . . 6
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ 𝑆 (𝑎‘(◡𝑍‘𝑖)) = 𝑁) | 
| 57 | 33, 56 | eqtrd 2776 | . . . . 5
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = 𝑁) | 
| 58 | 25, 57 | jca 511 | . . . 4
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))):𝑆⟶ℕ0 ∧
Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖) = 𝑁)) | 
| 59 |  | fzfid 14015 | . . . . . . . 8
⊢ (𝜑 → (1...𝐾) ∈ Fin) | 
| 60 | 59 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (1...𝐾) ∈ Fin) | 
| 61 | 59, 17, 37 | syl2anc 584 | . . . . . . . . 9
⊢ (𝜑 → (1...𝐾) ≈ 𝑆) | 
| 62 | 61 | ensymd 9046 | . . . . . . . 8
⊢ (𝜑 → 𝑆 ≈ (1...𝐾)) | 
| 63 | 62 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑆 ≈ (1...𝐾)) | 
| 64 | 60, 63, 40 | syl2anc 584 | . . . . . 6
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → 𝑆 ∈ Fin) | 
| 65 | 64 | mptexd 7245 | . . . . 5
⊢ ((𝜑 ∧ 𝑎 ∈ 𝐴) → (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∈ V) | 
| 66 |  | feq1 6715 | . . . . . . 7
⊢ (ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) → (ℎ:𝑆⟶ℕ0 ↔ (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))):𝑆⟶ℕ0)) | 
| 67 |  | simpl 482 | . . . . . . . . . 10
⊢ ((ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∧ 𝑖 ∈ 𝑆) → ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))) | 
| 68 | 67 | fveq1d 6907 | . . . . . . . . 9
⊢ ((ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) ∧ 𝑖 ∈ 𝑆) → (ℎ‘𝑖) = ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖)) | 
| 69 | 68 | sumeq2dv 15739 | . . . . . . . 8
⊢ (ℎ = (𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥))) → Σ𝑖 ∈ 𝑆 (ℎ‘𝑖) = Σ𝑖 ∈ 𝑆 ((𝑥 ∈ 𝑆 ↦ (𝑎‘(◡𝑍‘𝑥)))‘𝑖)) | 
| 70 | 69 | eqeq1d 2738 | . . . . . . 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 632 | . . . . . 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 3675 | . . . . 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 257 | . . 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 2843 | . 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 7133 | 1
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |