| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | prodeq1 15944 | . . . 4
⊢ (𝑤 = ∅ → ∏𝑘 ∈ 𝑤 𝐶 = ∏𝑘 ∈ ∅ 𝐶) | 
| 2 | 1 | mpteq2dv 5243 | . . 3
⊢ (𝑤 = ∅ → (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ ∅ 𝐶)) | 
| 3 | 2 | eleq1d 2825 | . 2
⊢ (𝑤 = ∅ → ((𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑤 𝐶) ∈ (𝐴–cn→ℂ) ↔ (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ ∅ 𝐶) ∈ (𝐴–cn→ℂ))) | 
| 4 |  | prodeq1 15944 | . . . 4
⊢ (𝑤 = 𝑧 → ∏𝑘 ∈ 𝑤 𝐶 = ∏𝑘 ∈ 𝑧 𝐶) | 
| 5 | 4 | mpteq2dv 5243 | . . 3
⊢ (𝑤 = 𝑧 → (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 𝐶)) | 
| 6 | 5 | eleq1d 2825 | . 2
⊢ (𝑤 = 𝑧 → ((𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑤 𝐶) ∈ (𝐴–cn→ℂ) ↔ (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 𝐶) ∈ (𝐴–cn→ℂ))) | 
| 7 |  | prodeq1 15944 | . . . 4
⊢ (𝑤 = (𝑧 ∪ {𝑦}) → ∏𝑘 ∈ 𝑤 𝐶 = ∏𝑘 ∈ (𝑧 ∪ {𝑦})𝐶) | 
| 8 | 7 | mpteq2dv 5243 | . . 3
⊢ (𝑤 = (𝑧 ∪ {𝑦}) → (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑦})𝐶)) | 
| 9 | 8 | eleq1d 2825 | . 2
⊢ (𝑤 = (𝑧 ∪ {𝑦}) → ((𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑤 𝐶) ∈ (𝐴–cn→ℂ) ↔ (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑦})𝐶) ∈ (𝐴–cn→ℂ))) | 
| 10 |  | prodeq1 15944 | . . . 4
⊢ (𝑤 = 𝐵 → ∏𝑘 ∈ 𝑤 𝐶 = ∏𝑘 ∈ 𝐵 𝐶) | 
| 11 | 10 | mpteq2dv 5243 | . . 3
⊢ (𝑤 = 𝐵 → (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑤 𝐶) = (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝐵 𝐶)) | 
| 12 | 11 | eleq1d 2825 | . 2
⊢ (𝑤 = 𝐵 → ((𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑤 𝐶) ∈ (𝐴–cn→ℂ) ↔ (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝐵 𝐶) ∈ (𝐴–cn→ℂ))) | 
| 13 |  | prod0 15980 | . . . . 5
⊢
∏𝑘 ∈
∅ 𝐶 =
1 | 
| 14 | 13 | a1i 11 | . . . 4
⊢ (𝜑 → ∏𝑘 ∈ ∅ 𝐶 = 1) | 
| 15 | 14 | mpteq2dv 5243 | . . 3
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ ∅ 𝐶) = (𝑥 ∈ 𝐴 ↦ 1)) | 
| 16 |  | fprodcncf.a | . . . 4
⊢ (𝜑 → 𝐴 ⊆ ℂ) | 
| 17 |  | 1cnd 11257 | . . . 4
⊢ (𝜑 → 1 ∈
ℂ) | 
| 18 |  | ssidd 4006 | . . . 4
⊢ (𝜑 → ℂ ⊆
ℂ) | 
| 19 | 16, 17, 18 | constcncfg 45892 | . . 3
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 1) ∈ (𝐴–cn→ℂ)) | 
| 20 | 15, 19 | eqeltrd 2840 | . 2
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ ∅ 𝐶) ∈ (𝐴–cn→ℂ)) | 
| 21 |  | nfcv 2904 | . . . . . 6
⊢
Ⅎ𝑢∏𝑘 ∈ (𝑧 ∪ {𝑦})𝐶 | 
| 22 |  | nfcv 2904 | . . . . . . 7
⊢
Ⅎ𝑥(𝑧 ∪ {𝑦}) | 
| 23 |  | nfcsb1v 3922 | . . . . . . 7
⊢
Ⅎ𝑥⦋𝑢 / 𝑥⦌𝐶 | 
| 24 | 22, 23 | nfcprod 15946 | . . . . . 6
⊢
Ⅎ𝑥∏𝑘 ∈ (𝑧 ∪ {𝑦})⦋𝑢 / 𝑥⦌𝐶 | 
| 25 |  | csbeq1a 3912 | . . . . . . . 8
⊢ (𝑥 = 𝑢 → 𝐶 = ⦋𝑢 / 𝑥⦌𝐶) | 
| 26 | 25 | adantr 480 | . . . . . . 7
⊢ ((𝑥 = 𝑢 ∧ 𝑘 ∈ (𝑧 ∪ {𝑦})) → 𝐶 = ⦋𝑢 / 𝑥⦌𝐶) | 
| 27 | 26 | prodeq2dv 15959 | . . . . . 6
⊢ (𝑥 = 𝑢 → ∏𝑘 ∈ (𝑧 ∪ {𝑦})𝐶 = ∏𝑘 ∈ (𝑧 ∪ {𝑦})⦋𝑢 / 𝑥⦌𝐶) | 
| 28 | 21, 24, 27 | cbvmpt 5252 | . . . . 5
⊢ (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑦})𝐶) = (𝑢 ∈ 𝐴 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑦})⦋𝑢 / 𝑥⦌𝐶) | 
| 29 | 28 | a1i 11 | . . . 4
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 𝐶) ∈ (𝐴–cn→ℂ)) → (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑦})𝐶) = (𝑢 ∈ 𝐴 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑦})⦋𝑢 / 𝑥⦌𝐶)) | 
| 30 |  | nfv 1913 | . . . . . . . 8
⊢
Ⅎ𝑘((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ 𝑢 ∈ 𝐴) | 
| 31 |  | nfcsb1v 3922 | . . . . . . . 8
⊢
Ⅎ𝑘⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶 | 
| 32 |  | fprodcncf.b | . . . . . . . . . . . 12
⊢ (𝜑 → 𝐵 ∈ Fin) | 
| 33 | 32 | adantr 480 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ⊆ 𝐵) → 𝐵 ∈ Fin) | 
| 34 |  | simpr 484 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑧 ⊆ 𝐵) → 𝑧 ⊆ 𝐵) | 
| 35 |  | ssfi 9214 | . . . . . . . . . . 11
⊢ ((𝐵 ∈ Fin ∧ 𝑧 ⊆ 𝐵) → 𝑧 ∈ Fin) | 
| 36 | 33, 34, 35 | syl2anc 584 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ⊆ 𝐵) → 𝑧 ∈ Fin) | 
| 37 | 36 | adantrr 717 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) → 𝑧 ∈ Fin) | 
| 38 | 37 | adantr 480 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ 𝑢 ∈ 𝐴) → 𝑧 ∈ Fin) | 
| 39 |  | vex 3483 | . . . . . . . . 9
⊢ 𝑦 ∈ V | 
| 40 | 39 | a1i 11 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ 𝑢 ∈ 𝐴) → 𝑦 ∈ V) | 
| 41 |  | eldifn 4131 | . . . . . . . . . 10
⊢ (𝑦 ∈ (𝐵 ∖ 𝑧) → ¬ 𝑦 ∈ 𝑧) | 
| 42 | 41 | ad2antll 729 | . . . . . . . . 9
⊢ ((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) → ¬ 𝑦 ∈ 𝑧) | 
| 43 | 42 | adantr 480 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ 𝑢 ∈ 𝐴) → ¬ 𝑦 ∈ 𝑧) | 
| 44 |  | simplll 774 | . . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ 𝑢 ∈ 𝐴) ∧ 𝑘 ∈ 𝑧) → 𝜑) | 
| 45 |  | simplr 768 | . . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ 𝑢 ∈ 𝐴) ∧ 𝑘 ∈ 𝑧) → 𝑢 ∈ 𝐴) | 
| 46 | 34 | adantrr 717 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) → 𝑧 ⊆ 𝐵) | 
| 47 | 46 | ad2antrr 726 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ 𝑢 ∈ 𝐴) ∧ 𝑘 ∈ 𝑧) → 𝑧 ⊆ 𝐵) | 
| 48 |  | simpr 484 | . . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ 𝑢 ∈ 𝐴) ∧ 𝑘 ∈ 𝑧) → 𝑘 ∈ 𝑧) | 
| 49 | 47, 48 | sseldd 3983 | . . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ 𝑢 ∈ 𝐴) ∧ 𝑘 ∈ 𝑧) → 𝑘 ∈ 𝐵) | 
| 50 |  | nfv 1913 | . . . . . . . . . . 11
⊢
Ⅎ𝑥(𝜑 ∧ 𝑢 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) | 
| 51 | 23 | nfel1 2921 | . . . . . . . . . . 11
⊢
Ⅎ𝑥⦋𝑢 / 𝑥⦌𝐶 ∈ ℂ | 
| 52 | 50, 51 | nfim 1895 | . . . . . . . . . 10
⊢
Ⅎ𝑥((𝜑 ∧ 𝑢 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → ⦋𝑢 / 𝑥⦌𝐶 ∈ ℂ) | 
| 53 |  | eleq1w 2823 | . . . . . . . . . . . 12
⊢ (𝑥 = 𝑢 → (𝑥 ∈ 𝐴 ↔ 𝑢 ∈ 𝐴)) | 
| 54 | 53 | 3anbi2d 1442 | . . . . . . . . . . 11
⊢ (𝑥 = 𝑢 → ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) ↔ (𝜑 ∧ 𝑢 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵))) | 
| 55 | 25 | eleq1d 2825 | . . . . . . . . . . 11
⊢ (𝑥 = 𝑢 → (𝐶 ∈ ℂ ↔ ⦋𝑢 / 𝑥⦌𝐶 ∈ ℂ)) | 
| 56 | 54, 55 | imbi12d 344 | . . . . . . . . . 10
⊢ (𝑥 = 𝑢 → (((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ) ↔ ((𝜑 ∧ 𝑢 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → ⦋𝑢 / 𝑥⦌𝐶 ∈ ℂ))) | 
| 57 |  | fprodcncf.c | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ) | 
| 58 | 52, 56, 57 | chvarfv 2239 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → ⦋𝑢 / 𝑥⦌𝐶 ∈ ℂ) | 
| 59 | 44, 45, 49, 58 | syl3anc 1372 | . . . . . . . 8
⊢ ((((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ 𝑢 ∈ 𝐴) ∧ 𝑘 ∈ 𝑧) → ⦋𝑢 / 𝑥⦌𝐶 ∈ ℂ) | 
| 60 |  | csbeq1a 3912 | . . . . . . . 8
⊢ (𝑘 = 𝑦 → ⦋𝑢 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶) | 
| 61 |  | simpll 766 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ 𝑢 ∈ 𝐴) → 𝜑) | 
| 62 |  | eldifi 4130 | . . . . . . . . . . 11
⊢ (𝑦 ∈ (𝐵 ∖ 𝑧) → 𝑦 ∈ 𝐵) | 
| 63 | 62 | ad2antll 729 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) → 𝑦 ∈ 𝐵) | 
| 64 | 63 | adantr 480 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ 𝑢 ∈ 𝐴) → 𝑦 ∈ 𝐵) | 
| 65 |  | simpr 484 | . . . . . . . . 9
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝐴) | 
| 66 |  | simpll 766 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ 𝐴) → 𝜑) | 
| 67 |  | simpr 484 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝐴) | 
| 68 |  | simplr 768 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ 𝐴) → 𝑦 ∈ 𝐵) | 
| 69 |  | nfv 1913 | . . . . . . . . . . . 12
⊢
Ⅎ𝑘(𝜑 ∧ 𝑢 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) | 
| 70 |  | nfcv 2904 | . . . . . . . . . . . . 13
⊢
Ⅎ𝑘ℂ | 
| 71 | 31, 70 | nfel 2919 | . . . . . . . . . . . 12
⊢
Ⅎ𝑘⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶 ∈ ℂ | 
| 72 | 69, 71 | nfim 1895 | . . . . . . . . . . 11
⊢
Ⅎ𝑘((𝜑 ∧ 𝑢 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶 ∈ ℂ) | 
| 73 |  | eleq1w 2823 | . . . . . . . . . . . . 13
⊢ (𝑘 = 𝑦 → (𝑘 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵)) | 
| 74 | 73 | 3anbi3d 1443 | . . . . . . . . . . . 12
⊢ (𝑘 = 𝑦 → ((𝜑 ∧ 𝑢 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) ↔ (𝜑 ∧ 𝑢 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) | 
| 75 | 60 | eleq1d 2825 | . . . . . . . . . . . 12
⊢ (𝑘 = 𝑦 → (⦋𝑢 / 𝑥⦌𝐶 ∈ ℂ ↔ ⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶 ∈ ℂ)) | 
| 76 | 74, 75 | imbi12d 344 | . . . . . . . . . . 11
⊢ (𝑘 = 𝑦 → (((𝜑 ∧ 𝑢 ∈ 𝐴 ∧ 𝑘 ∈ 𝐵) → ⦋𝑢 / 𝑥⦌𝐶 ∈ ℂ) ↔ ((𝜑 ∧ 𝑢 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶 ∈ ℂ))) | 
| 77 | 72, 76, 58 | chvarfv 2239 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶 ∈ ℂ) | 
| 78 | 66, 67, 68, 77 | syl3anc 1372 | . . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑢 ∈ 𝐴) → ⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶 ∈ ℂ) | 
| 79 | 61, 64, 65, 78 | syl21anc 837 | . . . . . . . 8
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ 𝑢 ∈ 𝐴) → ⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶 ∈ ℂ) | 
| 80 | 30, 31, 38, 40, 43, 59, 60, 79 | fprodsplitsn 16026 | . . . . . . 7
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ 𝑢 ∈ 𝐴) → ∏𝑘 ∈ (𝑧 ∪ {𝑦})⦋𝑢 / 𝑥⦌𝐶 = (∏𝑘 ∈ 𝑧 ⦋𝑢 / 𝑥⦌𝐶 · ⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶)) | 
| 81 | 80 | mpteq2dva 5241 | . . . . . 6
⊢ ((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) → (𝑢 ∈ 𝐴 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑦})⦋𝑢 / 𝑥⦌𝐶) = (𝑢 ∈ 𝐴 ↦ (∏𝑘 ∈ 𝑧 ⦋𝑢 / 𝑥⦌𝐶 · ⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶))) | 
| 82 | 81 | adantr 480 | . . . . 5
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 𝐶) ∈ (𝐴–cn→ℂ)) → (𝑢 ∈ 𝐴 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑦})⦋𝑢 / 𝑥⦌𝐶) = (𝑢 ∈ 𝐴 ↦ (∏𝑘 ∈ 𝑧 ⦋𝑢 / 𝑥⦌𝐶 · ⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶))) | 
| 83 |  | nfcv 2904 | . . . . . . . . . . 11
⊢
Ⅎ𝑢∏𝑘 ∈ 𝑧 𝐶 | 
| 84 |  | nfcv 2904 | . . . . . . . . . . . 12
⊢
Ⅎ𝑥𝑧 | 
| 85 | 84, 23 | nfcprod 15946 | . . . . . . . . . . 11
⊢
Ⅎ𝑥∏𝑘 ∈ 𝑧 ⦋𝑢 / 𝑥⦌𝐶 | 
| 86 | 25 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝑥 = 𝑢 ∧ 𝑘 ∈ 𝑧) → 𝐶 = ⦋𝑢 / 𝑥⦌𝐶) | 
| 87 | 86 | prodeq2dv 15959 | . . . . . . . . . . 11
⊢ (𝑥 = 𝑢 → ∏𝑘 ∈ 𝑧 𝐶 = ∏𝑘 ∈ 𝑧 ⦋𝑢 / 𝑥⦌𝐶) | 
| 88 | 83, 85, 87 | cbvmpt 5252 | . . . . . . . . . 10
⊢ (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 𝐶) = (𝑢 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 ⦋𝑢 / 𝑥⦌𝐶) | 
| 89 | 88 | eqcomi 2745 | . . . . . . . . 9
⊢ (𝑢 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 ⦋𝑢 / 𝑥⦌𝐶) = (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 𝐶) | 
| 90 | 89 | a1i 11 | . . . . . . . 8
⊢ ((𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 𝐶) ∈ (𝐴–cn→ℂ) → (𝑢 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 ⦋𝑢 / 𝑥⦌𝐶) = (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 𝐶)) | 
| 91 |  | id 22 | . . . . . . . 8
⊢ ((𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 𝐶) ∈ (𝐴–cn→ℂ) → (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 𝐶) ∈ (𝐴–cn→ℂ)) | 
| 92 | 90, 91 | eqeltrd 2840 | . . . . . . 7
⊢ ((𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 𝐶) ∈ (𝐴–cn→ℂ) → (𝑢 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 ⦋𝑢 / 𝑥⦌𝐶) ∈ (𝐴–cn→ℂ)) | 
| 93 | 92 | adantl 481 | . . . . . 6
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 𝐶) ∈ (𝐴–cn→ℂ)) → (𝑢 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 ⦋𝑢 / 𝑥⦌𝐶) ∈ (𝐴–cn→ℂ)) | 
| 94 |  | nfv 1913 | . . . . . . . . . 10
⊢
Ⅎ𝑘(𝜑 ∧ 𝑦 ∈ 𝐵) | 
| 95 |  | nfcv 2904 | . . . . . . . . . . . 12
⊢
Ⅎ𝑘𝐴 | 
| 96 | 95, 31 | nfmpt 5248 | . . . . . . . . . . 11
⊢
Ⅎ𝑘(𝑢 ∈ 𝐴 ↦ ⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶) | 
| 97 |  | nfcv 2904 | . . . . . . . . . . 11
⊢
Ⅎ𝑘(𝐴–cn→ℂ) | 
| 98 | 96, 97 | nfel 2919 | . . . . . . . . . 10
⊢
Ⅎ𝑘(𝑢 ∈ 𝐴 ↦ ⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶) ∈ (𝐴–cn→ℂ) | 
| 99 | 94, 98 | nfim 1895 | . . . . . . . . 9
⊢
Ⅎ𝑘((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑢 ∈ 𝐴 ↦ ⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶) ∈ (𝐴–cn→ℂ)) | 
| 100 | 73 | anbi2d 630 | . . . . . . . . . 10
⊢ (𝑘 = 𝑦 → ((𝜑 ∧ 𝑘 ∈ 𝐵) ↔ (𝜑 ∧ 𝑦 ∈ 𝐵))) | 
| 101 | 60 | adantr 480 | . . . . . . . . . . . 12
⊢ ((𝑘 = 𝑦 ∧ 𝑢 ∈ 𝐴) → ⦋𝑢 / 𝑥⦌𝐶 = ⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶) | 
| 102 | 101 | mpteq2dva 5241 | . . . . . . . . . . 11
⊢ (𝑘 = 𝑦 → (𝑢 ∈ 𝐴 ↦ ⦋𝑢 / 𝑥⦌𝐶) = (𝑢 ∈ 𝐴 ↦ ⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶)) | 
| 103 | 102 | eleq1d 2825 | . . . . . . . . . 10
⊢ (𝑘 = 𝑦 → ((𝑢 ∈ 𝐴 ↦ ⦋𝑢 / 𝑥⦌𝐶) ∈ (𝐴–cn→ℂ) ↔ (𝑢 ∈ 𝐴 ↦ ⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶) ∈ (𝐴–cn→ℂ))) | 
| 104 | 100, 103 | imbi12d 344 | . . . . . . . . 9
⊢ (𝑘 = 𝑦 → (((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑢 ∈ 𝐴 ↦ ⦋𝑢 / 𝑥⦌𝐶) ∈ (𝐴–cn→ℂ)) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑢 ∈ 𝐴 ↦ ⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶) ∈ (𝐴–cn→ℂ)))) | 
| 105 |  | nfcv 2904 | . . . . . . . . . . 11
⊢
Ⅎ𝑢𝐶 | 
| 106 | 105, 23, 25 | cbvmpt 5252 | . . . . . . . . . 10
⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑢 ∈ 𝐴 ↦ ⦋𝑢 / 𝑥⦌𝐶) | 
| 107 |  | fprodcncf.cn | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ (𝐴–cn→ℂ)) | 
| 108 | 106, 107 | eqeltrrid 2845 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑢 ∈ 𝐴 ↦ ⦋𝑢 / 𝑥⦌𝐶) ∈ (𝐴–cn→ℂ)) | 
| 109 | 99, 104, 108 | chvarfv 2239 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝑢 ∈ 𝐴 ↦ ⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶) ∈ (𝐴–cn→ℂ)) | 
| 110 | 63, 109 | syldan 591 | . . . . . . 7
⊢ ((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) → (𝑢 ∈ 𝐴 ↦ ⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶) ∈ (𝐴–cn→ℂ)) | 
| 111 | 110 | adantr 480 | . . . . . 6
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 𝐶) ∈ (𝐴–cn→ℂ)) → (𝑢 ∈ 𝐴 ↦ ⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶) ∈ (𝐴–cn→ℂ)) | 
| 112 | 93, 111 | mulcncf 25481 | . . . . 5
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 𝐶) ∈ (𝐴–cn→ℂ)) → (𝑢 ∈ 𝐴 ↦ (∏𝑘 ∈ 𝑧 ⦋𝑢 / 𝑥⦌𝐶 · ⦋𝑦 / 𝑘⦌⦋𝑢 / 𝑥⦌𝐶)) ∈ (𝐴–cn→ℂ)) | 
| 113 | 82, 112 | eqeltrd 2840 | . . . 4
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 𝐶) ∈ (𝐴–cn→ℂ)) → (𝑢 ∈ 𝐴 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑦})⦋𝑢 / 𝑥⦌𝐶) ∈ (𝐴–cn→ℂ)) | 
| 114 | 29, 113 | eqeltrd 2840 | . . 3
⊢ (((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) ∧ (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 𝐶) ∈ (𝐴–cn→ℂ)) → (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑦})𝐶) ∈ (𝐴–cn→ℂ)) | 
| 115 | 114 | ex 412 | . 2
⊢ ((𝜑 ∧ (𝑧 ⊆ 𝐵 ∧ 𝑦 ∈ (𝐵 ∖ 𝑧))) → ((𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝑧 𝐶) ∈ (𝐴–cn→ℂ) → (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ (𝑧 ∪ {𝑦})𝐶) ∈ (𝐴–cn→ℂ))) | 
| 116 | 3, 6, 9, 12, 20, 115, 32 | findcard2d 9207 | 1
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∏𝑘 ∈ 𝐵 𝐶) ∈ (𝐴–cn→ℂ)) |