| Step |
Hyp |
Ref |
Expression |
| 1 |
|
dvmptfprod.iph |
⊢ Ⅎ 𝑖 𝜑 |
| 2 |
|
dvmptfprod.jph |
⊢ Ⅎ 𝑗 𝜑 |
| 3 |
|
dvmptfprod.j |
⊢ 𝐽 = ( 𝐾 ↾t 𝑆 ) |
| 4 |
|
dvmptfprod.k |
⊢ 𝐾 = ( TopOpen ‘ ℂfld ) |
| 5 |
|
dvmptfprod.s |
⊢ ( 𝜑 → 𝑆 ∈ { ℝ , ℂ } ) |
| 6 |
|
dvmptfprod.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐽 ) |
| 7 |
|
dvmptfprod.i |
⊢ ( 𝜑 → 𝐼 ∈ Fin ) |
| 8 |
|
dvmptfprod.a |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋 ) → 𝐴 ∈ ℂ ) |
| 9 |
|
dvmptfprod.b |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋 ) → 𝐵 ∈ ℂ ) |
| 10 |
|
dvmptfprod.d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ 𝐵 ) ) |
| 11 |
|
dvmptfprod.bc |
⊢ ( 𝑖 = 𝑗 → 𝐵 = 𝐶 ) |
| 12 |
|
ssid |
⊢ 𝐼 ⊆ 𝐼 |
| 13 |
12
|
jctr |
⊢ ( 𝜑 → ( 𝜑 ∧ 𝐼 ⊆ 𝐼 ) ) |
| 14 |
|
sseq1 |
⊢ ( 𝑎 = ∅ → ( 𝑎 ⊆ 𝐼 ↔ ∅ ⊆ 𝐼 ) ) |
| 15 |
14
|
anbi2d |
⊢ ( 𝑎 = ∅ → ( ( 𝜑 ∧ 𝑎 ⊆ 𝐼 ) ↔ ( 𝜑 ∧ ∅ ⊆ 𝐼 ) ) ) |
| 16 |
|
prodeq1 |
⊢ ( 𝑎 = ∅ → ∏ 𝑖 ∈ 𝑎 𝐴 = ∏ 𝑖 ∈ ∅ 𝐴 ) |
| 17 |
16
|
mpteq2dv |
⊢ ( 𝑎 = ∅ → ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑎 𝐴 ) = ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ ∅ 𝐴 ) ) |
| 18 |
17
|
oveq2d |
⊢ ( 𝑎 = ∅ → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑎 𝐴 ) ) = ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ ∅ 𝐴 ) ) ) |
| 19 |
|
sumeq1 |
⊢ ( 𝑎 = ∅ → Σ 𝑗 ∈ 𝑎 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) = Σ 𝑗 ∈ ∅ ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) ) |
| 20 |
|
difeq1 |
⊢ ( 𝑎 = ∅ → ( 𝑎 ∖ { 𝑗 } ) = ( ∅ ∖ { 𝑗 } ) ) |
| 21 |
20
|
prodeq1d |
⊢ ( 𝑎 = ∅ → ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 = ∏ 𝑖 ∈ ( ∅ ∖ { 𝑗 } ) 𝐴 ) |
| 22 |
21
|
oveq2d |
⊢ ( 𝑎 = ∅ → ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) = ( 𝐶 · ∏ 𝑖 ∈ ( ∅ ∖ { 𝑗 } ) 𝐴 ) ) |
| 23 |
22
|
sumeq2sdv |
⊢ ( 𝑎 = ∅ → Σ 𝑗 ∈ ∅ ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) = Σ 𝑗 ∈ ∅ ( 𝐶 · ∏ 𝑖 ∈ ( ∅ ∖ { 𝑗 } ) 𝐴 ) ) |
| 24 |
19 23
|
eqtrd |
⊢ ( 𝑎 = ∅ → Σ 𝑗 ∈ 𝑎 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) = Σ 𝑗 ∈ ∅ ( 𝐶 · ∏ 𝑖 ∈ ( ∅ ∖ { 𝑗 } ) 𝐴 ) ) |
| 25 |
24
|
mpteq2dv |
⊢ ( 𝑎 = ∅ → ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑎 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ ∅ ( 𝐶 · ∏ 𝑖 ∈ ( ∅ ∖ { 𝑗 } ) 𝐴 ) ) ) |
| 26 |
18 25
|
eqeq12d |
⊢ ( 𝑎 = ∅ → ( ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑎 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑎 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) ) ↔ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ ∅ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ ∅ ( 𝐶 · ∏ 𝑖 ∈ ( ∅ ∖ { 𝑗 } ) 𝐴 ) ) ) ) |
| 27 |
15 26
|
imbi12d |
⊢ ( 𝑎 = ∅ → ( ( ( 𝜑 ∧ 𝑎 ⊆ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑎 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑎 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) ) ) ↔ ( ( 𝜑 ∧ ∅ ⊆ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ ∅ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ ∅ ( 𝐶 · ∏ 𝑖 ∈ ( ∅ ∖ { 𝑗 } ) 𝐴 ) ) ) ) ) |
| 28 |
|
sseq1 |
⊢ ( 𝑎 = 𝑏 → ( 𝑎 ⊆ 𝐼 ↔ 𝑏 ⊆ 𝐼 ) ) |
| 29 |
28
|
anbi2d |
⊢ ( 𝑎 = 𝑏 → ( ( 𝜑 ∧ 𝑎 ⊆ 𝐼 ) ↔ ( 𝜑 ∧ 𝑏 ⊆ 𝐼 ) ) ) |
| 30 |
|
prodeq1 |
⊢ ( 𝑎 = 𝑏 → ∏ 𝑖 ∈ 𝑎 𝐴 = ∏ 𝑖 ∈ 𝑏 𝐴 ) |
| 31 |
30
|
mpteq2dv |
⊢ ( 𝑎 = 𝑏 → ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑎 𝐴 ) = ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) |
| 32 |
31
|
oveq2d |
⊢ ( 𝑎 = 𝑏 → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑎 𝐴 ) ) = ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) ) |
| 33 |
|
sumeq1 |
⊢ ( 𝑎 = 𝑏 → Σ 𝑗 ∈ 𝑎 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) = Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) ) |
| 34 |
|
difeq1 |
⊢ ( 𝑎 = 𝑏 → ( 𝑎 ∖ { 𝑗 } ) = ( 𝑏 ∖ { 𝑗 } ) ) |
| 35 |
34
|
prodeq1d |
⊢ ( 𝑎 = 𝑏 → ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 = ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) |
| 36 |
35
|
oveq2d |
⊢ ( 𝑎 = 𝑏 → ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) = ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) |
| 37 |
36
|
sumeq2sdv |
⊢ ( 𝑎 = 𝑏 → Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) = Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) |
| 38 |
33 37
|
eqtrd |
⊢ ( 𝑎 = 𝑏 → Σ 𝑗 ∈ 𝑎 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) = Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) |
| 39 |
38
|
mpteq2dv |
⊢ ( 𝑎 = 𝑏 → ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑎 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) |
| 40 |
32 39
|
eqeq12d |
⊢ ( 𝑎 = 𝑏 → ( ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑎 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑎 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) ) ↔ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) ) |
| 41 |
29 40
|
imbi12d |
⊢ ( 𝑎 = 𝑏 → ( ( ( 𝜑 ∧ 𝑎 ⊆ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑎 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑎 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) ) ) ↔ ( ( 𝜑 ∧ 𝑏 ⊆ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) ) ) |
| 42 |
|
sseq1 |
⊢ ( 𝑎 = ( 𝑏 ∪ { 𝑐 } ) → ( 𝑎 ⊆ 𝐼 ↔ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ) |
| 43 |
42
|
anbi2d |
⊢ ( 𝑎 = ( 𝑏 ∪ { 𝑐 } ) → ( ( 𝜑 ∧ 𝑎 ⊆ 𝐼 ) ↔ ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ) ) |
| 44 |
|
prodeq1 |
⊢ ( 𝑎 = ( 𝑏 ∪ { 𝑐 } ) → ∏ 𝑖 ∈ 𝑎 𝐴 = ∏ 𝑖 ∈ ( 𝑏 ∪ { 𝑐 } ) 𝐴 ) |
| 45 |
44
|
mpteq2dv |
⊢ ( 𝑎 = ( 𝑏 ∪ { 𝑐 } ) → ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑎 𝐴 ) = ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ ( 𝑏 ∪ { 𝑐 } ) 𝐴 ) ) |
| 46 |
45
|
oveq2d |
⊢ ( 𝑎 = ( 𝑏 ∪ { 𝑐 } ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑎 𝐴 ) ) = ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ ( 𝑏 ∪ { 𝑐 } ) 𝐴 ) ) ) |
| 47 |
|
sumeq1 |
⊢ ( 𝑎 = ( 𝑏 ∪ { 𝑐 } ) → Σ 𝑗 ∈ 𝑎 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) = Σ 𝑗 ∈ ( 𝑏 ∪ { 𝑐 } ) ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) ) |
| 48 |
|
difeq1 |
⊢ ( 𝑎 = ( 𝑏 ∪ { 𝑐 } ) → ( 𝑎 ∖ { 𝑗 } ) = ( ( 𝑏 ∪ { 𝑐 } ) ∖ { 𝑗 } ) ) |
| 49 |
48
|
prodeq1d |
⊢ ( 𝑎 = ( 𝑏 ∪ { 𝑐 } ) → ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 = ∏ 𝑖 ∈ ( ( 𝑏 ∪ { 𝑐 } ) ∖ { 𝑗 } ) 𝐴 ) |
| 50 |
49
|
oveq2d |
⊢ ( 𝑎 = ( 𝑏 ∪ { 𝑐 } ) → ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) = ( 𝐶 · ∏ 𝑖 ∈ ( ( 𝑏 ∪ { 𝑐 } ) ∖ { 𝑗 } ) 𝐴 ) ) |
| 51 |
50
|
sumeq2sdv |
⊢ ( 𝑎 = ( 𝑏 ∪ { 𝑐 } ) → Σ 𝑗 ∈ ( 𝑏 ∪ { 𝑐 } ) ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) = Σ 𝑗 ∈ ( 𝑏 ∪ { 𝑐 } ) ( 𝐶 · ∏ 𝑖 ∈ ( ( 𝑏 ∪ { 𝑐 } ) ∖ { 𝑗 } ) 𝐴 ) ) |
| 52 |
47 51
|
eqtrd |
⊢ ( 𝑎 = ( 𝑏 ∪ { 𝑐 } ) → Σ 𝑗 ∈ 𝑎 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) = Σ 𝑗 ∈ ( 𝑏 ∪ { 𝑐 } ) ( 𝐶 · ∏ 𝑖 ∈ ( ( 𝑏 ∪ { 𝑐 } ) ∖ { 𝑗 } ) 𝐴 ) ) |
| 53 |
52
|
mpteq2dv |
⊢ ( 𝑎 = ( 𝑏 ∪ { 𝑐 } ) → ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑎 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ ( 𝑏 ∪ { 𝑐 } ) ( 𝐶 · ∏ 𝑖 ∈ ( ( 𝑏 ∪ { 𝑐 } ) ∖ { 𝑗 } ) 𝐴 ) ) ) |
| 54 |
46 53
|
eqeq12d |
⊢ ( 𝑎 = ( 𝑏 ∪ { 𝑐 } ) → ( ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑎 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑎 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) ) ↔ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ ( 𝑏 ∪ { 𝑐 } ) 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ ( 𝑏 ∪ { 𝑐 } ) ( 𝐶 · ∏ 𝑖 ∈ ( ( 𝑏 ∪ { 𝑐 } ) ∖ { 𝑗 } ) 𝐴 ) ) ) ) |
| 55 |
43 54
|
imbi12d |
⊢ ( 𝑎 = ( 𝑏 ∪ { 𝑐 } ) → ( ( ( 𝜑 ∧ 𝑎 ⊆ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑎 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑎 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) ) ) ↔ ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ ( 𝑏 ∪ { 𝑐 } ) 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ ( 𝑏 ∪ { 𝑐 } ) ( 𝐶 · ∏ 𝑖 ∈ ( ( 𝑏 ∪ { 𝑐 } ) ∖ { 𝑗 } ) 𝐴 ) ) ) ) ) |
| 56 |
|
sseq1 |
⊢ ( 𝑎 = 𝐼 → ( 𝑎 ⊆ 𝐼 ↔ 𝐼 ⊆ 𝐼 ) ) |
| 57 |
56
|
anbi2d |
⊢ ( 𝑎 = 𝐼 → ( ( 𝜑 ∧ 𝑎 ⊆ 𝐼 ) ↔ ( 𝜑 ∧ 𝐼 ⊆ 𝐼 ) ) ) |
| 58 |
|
prodeq1 |
⊢ ( 𝑎 = 𝐼 → ∏ 𝑖 ∈ 𝑎 𝐴 = ∏ 𝑖 ∈ 𝐼 𝐴 ) |
| 59 |
58
|
mpteq2dv |
⊢ ( 𝑎 = 𝐼 → ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑎 𝐴 ) = ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝐼 𝐴 ) ) |
| 60 |
59
|
oveq2d |
⊢ ( 𝑎 = 𝐼 → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑎 𝐴 ) ) = ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝐼 𝐴 ) ) ) |
| 61 |
|
sumeq1 |
⊢ ( 𝑎 = 𝐼 → Σ 𝑗 ∈ 𝑎 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) = Σ 𝑗 ∈ 𝐼 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) ) |
| 62 |
|
difeq1 |
⊢ ( 𝑎 = 𝐼 → ( 𝑎 ∖ { 𝑗 } ) = ( 𝐼 ∖ { 𝑗 } ) ) |
| 63 |
62
|
prodeq1d |
⊢ ( 𝑎 = 𝐼 → ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 = ∏ 𝑖 ∈ ( 𝐼 ∖ { 𝑗 } ) 𝐴 ) |
| 64 |
63
|
oveq2d |
⊢ ( 𝑎 = 𝐼 → ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) = ( 𝐶 · ∏ 𝑖 ∈ ( 𝐼 ∖ { 𝑗 } ) 𝐴 ) ) |
| 65 |
64
|
sumeq2sdv |
⊢ ( 𝑎 = 𝐼 → Σ 𝑗 ∈ 𝐼 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) = Σ 𝑗 ∈ 𝐼 ( 𝐶 · ∏ 𝑖 ∈ ( 𝐼 ∖ { 𝑗 } ) 𝐴 ) ) |
| 66 |
61 65
|
eqtrd |
⊢ ( 𝑎 = 𝐼 → Σ 𝑗 ∈ 𝑎 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) = Σ 𝑗 ∈ 𝐼 ( 𝐶 · ∏ 𝑖 ∈ ( 𝐼 ∖ { 𝑗 } ) 𝐴 ) ) |
| 67 |
66
|
mpteq2dv |
⊢ ( 𝑎 = 𝐼 → ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑎 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝐼 ( 𝐶 · ∏ 𝑖 ∈ ( 𝐼 ∖ { 𝑗 } ) 𝐴 ) ) ) |
| 68 |
60 67
|
eqeq12d |
⊢ ( 𝑎 = 𝐼 → ( ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑎 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑎 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) ) ↔ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝐼 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝐼 ( 𝐶 · ∏ 𝑖 ∈ ( 𝐼 ∖ { 𝑗 } ) 𝐴 ) ) ) ) |
| 69 |
57 68
|
imbi12d |
⊢ ( 𝑎 = 𝐼 → ( ( ( 𝜑 ∧ 𝑎 ⊆ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑎 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑎 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑎 ∖ { 𝑗 } ) 𝐴 ) ) ) ↔ ( ( 𝜑 ∧ 𝐼 ⊆ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝐼 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝐼 ( 𝐶 · ∏ 𝑖 ∈ ( 𝐼 ∖ { 𝑗 } ) 𝐴 ) ) ) ) ) |
| 70 |
|
prod0 |
⊢ ∏ 𝑖 ∈ ∅ 𝐴 = 1 |
| 71 |
70
|
mpteq2i |
⊢ ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ ∅ 𝐴 ) = ( 𝑥 ∈ 𝑋 ↦ 1 ) |
| 72 |
71
|
oveq2i |
⊢ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ ∅ 𝐴 ) ) = ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ 1 ) ) |
| 73 |
72
|
a1i |
⊢ ( 𝜑 → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ ∅ 𝐴 ) ) = ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ 1 ) ) ) |
| 74 |
4
|
oveq1i |
⊢ ( 𝐾 ↾t 𝑆 ) = ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 ) |
| 75 |
3 74
|
eqtri |
⊢ 𝐽 = ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 ) |
| 76 |
6 75
|
eleqtrdi |
⊢ ( 𝜑 → 𝑋 ∈ ( ( TopOpen ‘ ℂfld ) ↾t 𝑆 ) ) |
| 77 |
|
1cnd |
⊢ ( 𝜑 → 1 ∈ ℂ ) |
| 78 |
5 76 77
|
dvmptconst |
⊢ ( 𝜑 → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ 1 ) ) = ( 𝑥 ∈ 𝑋 ↦ 0 ) ) |
| 79 |
|
sum0 |
⊢ Σ 𝑗 ∈ ∅ ( 𝐶 · ∏ 𝑖 ∈ ( ∅ ∖ { 𝑗 } ) 𝐴 ) = 0 |
| 80 |
79
|
eqcomi |
⊢ 0 = Σ 𝑗 ∈ ∅ ( 𝐶 · ∏ 𝑖 ∈ ( ∅ ∖ { 𝑗 } ) 𝐴 ) |
| 81 |
80
|
mpteq2i |
⊢ ( 𝑥 ∈ 𝑋 ↦ 0 ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ ∅ ( 𝐶 · ∏ 𝑖 ∈ ( ∅ ∖ { 𝑗 } ) 𝐴 ) ) |
| 82 |
81
|
a1i |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 ↦ 0 ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ ∅ ( 𝐶 · ∏ 𝑖 ∈ ( ∅ ∖ { 𝑗 } ) 𝐴 ) ) ) |
| 83 |
73 78 82
|
3eqtrd |
⊢ ( 𝜑 → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ ∅ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ ∅ ( 𝐶 · ∏ 𝑖 ∈ ( ∅ ∖ { 𝑗 } ) 𝐴 ) ) ) |
| 84 |
83
|
adantr |
⊢ ( ( 𝜑 ∧ ∅ ⊆ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ ∅ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ ∅ ( 𝐶 · ∏ 𝑖 ∈ ( ∅ ∖ { 𝑗 } ) 𝐴 ) ) ) |
| 85 |
|
simp3 |
⊢ ( ( ( 𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( ( 𝜑 ∧ 𝑏 ⊆ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) ∧ ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ) → ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ) |
| 86 |
|
simp1r |
⊢ ( ( ( 𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( ( 𝜑 ∧ 𝑏 ⊆ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) ∧ ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ) → ¬ 𝑐 ∈ 𝑏 ) |
| 87 |
|
ssun1 |
⊢ 𝑏 ⊆ ( 𝑏 ∪ { 𝑐 } ) |
| 88 |
|
sstr2 |
⊢ ( 𝑏 ⊆ ( 𝑏 ∪ { 𝑐 } ) → ( ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 → 𝑏 ⊆ 𝐼 ) ) |
| 89 |
87 88
|
ax-mp |
⊢ ( ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 → 𝑏 ⊆ 𝐼 ) |
| 90 |
89
|
anim2i |
⊢ ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) → ( 𝜑 ∧ 𝑏 ⊆ 𝐼 ) ) |
| 91 |
90
|
adantl |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ⊆ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) ∧ ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ) → ( 𝜑 ∧ 𝑏 ⊆ 𝐼 ) ) |
| 92 |
|
simpl |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ⊆ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) ∧ ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ) → ( ( 𝜑 ∧ 𝑏 ⊆ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) ) |
| 93 |
91 92
|
mpd |
⊢ ( ( ( ( 𝜑 ∧ 𝑏 ⊆ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) ∧ ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) |
| 94 |
93
|
3adant1 |
⊢ ( ( ( 𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( ( 𝜑 ∧ 𝑏 ⊆ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) ∧ ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) |
| 95 |
|
nfv |
⊢ Ⅎ 𝑥 ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) |
| 96 |
|
nfcv |
⊢ Ⅎ 𝑥 𝑆 |
| 97 |
|
nfcv |
⊢ Ⅎ 𝑥 D |
| 98 |
|
nfmpt1 |
⊢ Ⅎ 𝑥 ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) |
| 99 |
96 97 98
|
nfov |
⊢ Ⅎ 𝑥 ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) |
| 100 |
|
nfmpt1 |
⊢ Ⅎ 𝑥 ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) |
| 101 |
99 100
|
nfeq |
⊢ Ⅎ 𝑥 ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) |
| 102 |
95 101
|
nfan |
⊢ Ⅎ 𝑥 ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) |
| 103 |
|
nfv |
⊢ Ⅎ 𝑖 ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 |
| 104 |
1 103
|
nfan |
⊢ Ⅎ 𝑖 ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) |
| 105 |
|
nfv |
⊢ Ⅎ 𝑖 ¬ 𝑐 ∈ 𝑏 |
| 106 |
104 105
|
nfan |
⊢ Ⅎ 𝑖 ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) |
| 107 |
|
nfcv |
⊢ Ⅎ 𝑖 𝑆 |
| 108 |
|
nfcv |
⊢ Ⅎ 𝑖 D |
| 109 |
|
nfcv |
⊢ Ⅎ 𝑖 𝑋 |
| 110 |
|
nfcv |
⊢ Ⅎ 𝑖 𝑏 |
| 111 |
110
|
nfcprod1 |
⊢ Ⅎ 𝑖 ∏ 𝑖 ∈ 𝑏 𝐴 |
| 112 |
109 111
|
nfmpt |
⊢ Ⅎ 𝑖 ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) |
| 113 |
107 108 112
|
nfov |
⊢ Ⅎ 𝑖 ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) |
| 114 |
|
nfcv |
⊢ Ⅎ 𝑖 𝐶 |
| 115 |
|
nfcv |
⊢ Ⅎ 𝑖 · |
| 116 |
|
nfcv |
⊢ Ⅎ 𝑖 ( 𝑏 ∖ { 𝑗 } ) |
| 117 |
116
|
nfcprod1 |
⊢ Ⅎ 𝑖 ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 |
| 118 |
114 115 117
|
nfov |
⊢ Ⅎ 𝑖 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) |
| 119 |
110 118
|
nfsum |
⊢ Ⅎ 𝑖 Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) |
| 120 |
109 119
|
nfmpt |
⊢ Ⅎ 𝑖 ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) |
| 121 |
113 120
|
nfeq |
⊢ Ⅎ 𝑖 ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) |
| 122 |
106 121
|
nfan |
⊢ Ⅎ 𝑖 ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) |
| 123 |
|
nfv |
⊢ Ⅎ 𝑗 ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 |
| 124 |
2 123
|
nfan |
⊢ Ⅎ 𝑗 ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) |
| 125 |
|
nfv |
⊢ Ⅎ 𝑗 ¬ 𝑐 ∈ 𝑏 |
| 126 |
124 125
|
nfan |
⊢ Ⅎ 𝑗 ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) |
| 127 |
|
nfcv |
⊢ Ⅎ 𝑗 ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) |
| 128 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑋 |
| 129 |
|
nfcv |
⊢ Ⅎ 𝑗 𝑏 |
| 130 |
129
|
nfsum1 |
⊢ Ⅎ 𝑗 Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) |
| 131 |
128 130
|
nfmpt |
⊢ Ⅎ 𝑗 ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) |
| 132 |
127 131
|
nfeq |
⊢ Ⅎ 𝑗 ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) |
| 133 |
126 132
|
nfan |
⊢ Ⅎ 𝑗 ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) |
| 134 |
|
nfcsb1v |
⊢ Ⅎ 𝑖 ⦋ 𝑐 / 𝑖 ⦌ 𝐴 |
| 135 |
|
nfcsb1v |
⊢ Ⅎ 𝑗 ⦋ 𝑐 / 𝑗 ⦌ 𝐶 |
| 136 |
|
simpl |
⊢ ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) → 𝜑 ) |
| 137 |
136
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) → 𝜑 ) |
| 138 |
137 8
|
syl3an1 |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋 ) → 𝐴 ∈ ℂ ) |
| 139 |
7
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) → 𝐼 ∈ Fin ) |
| 140 |
89
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) → 𝑏 ⊆ 𝐼 ) |
| 141 |
140
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) → 𝑏 ⊆ 𝐼 ) |
| 142 |
139 141
|
ssfid |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) → 𝑏 ∈ Fin ) |
| 143 |
|
vex |
⊢ 𝑐 ∈ V |
| 144 |
143
|
a1i |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) → 𝑐 ∈ V ) |
| 145 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) → ¬ 𝑐 ∈ 𝑏 ) |
| 146 |
|
simpllr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) → ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) |
| 147 |
5
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) → 𝑆 ∈ { ℝ , ℂ } ) |
| 148 |
137
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑗 ∈ 𝑏 ) → 𝜑 ) |
| 149 |
141
|
ad2antrr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑗 ∈ 𝑏 ) → 𝑏 ⊆ 𝐼 ) |
| 150 |
|
simpr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑗 ∈ 𝑏 ) → 𝑗 ∈ 𝑏 ) |
| 151 |
149 150
|
sseldd |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑗 ∈ 𝑏 ) → 𝑗 ∈ 𝐼 ) |
| 152 |
|
simplr |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑗 ∈ 𝑏 ) → 𝑥 ∈ 𝑋 ) |
| 153 |
|
nfv |
⊢ Ⅎ 𝑖 𝑗 ∈ 𝐼 |
| 154 |
|
nfv |
⊢ Ⅎ 𝑖 𝑥 ∈ 𝑋 |
| 155 |
1 153 154
|
nf3an |
⊢ Ⅎ 𝑖 ( 𝜑 ∧ 𝑗 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋 ) |
| 156 |
|
nfv |
⊢ Ⅎ 𝑖 𝐶 ∈ ℂ |
| 157 |
155 156
|
nfim |
⊢ Ⅎ 𝑖 ( ( 𝜑 ∧ 𝑗 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋 ) → 𝐶 ∈ ℂ ) |
| 158 |
|
eleq1w |
⊢ ( 𝑖 = 𝑗 → ( 𝑖 ∈ 𝐼 ↔ 𝑗 ∈ 𝐼 ) ) |
| 159 |
158
|
3anbi2d |
⊢ ( 𝑖 = 𝑗 → ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋 ) ↔ ( 𝜑 ∧ 𝑗 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋 ) ) ) |
| 160 |
11
|
eleq1d |
⊢ ( 𝑖 = 𝑗 → ( 𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ ) ) |
| 161 |
159 160
|
imbi12d |
⊢ ( 𝑖 = 𝑗 → ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋 ) → 𝐵 ∈ ℂ ) ↔ ( ( 𝜑 ∧ 𝑗 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋 ) → 𝐶 ∈ ℂ ) ) ) |
| 162 |
157 161 9
|
chvarfv |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋 ) → 𝐶 ∈ ℂ ) |
| 163 |
148 151 152 162
|
syl3anc |
⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) ∧ 𝑥 ∈ 𝑋 ) ∧ 𝑗 ∈ 𝑏 ) → 𝐶 ∈ ℂ ) |
| 164 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) |
| 165 |
136
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ 𝑥 ∈ 𝑋 ) → 𝜑 ) |
| 166 |
|
id |
⊢ ( ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 → ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) |
| 167 |
|
vsnid |
⊢ 𝑐 ∈ { 𝑐 } |
| 168 |
|
elun2 |
⊢ ( 𝑐 ∈ { 𝑐 } → 𝑐 ∈ ( 𝑏 ∪ { 𝑐 } ) ) |
| 169 |
167 168
|
mp1i |
⊢ ( ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 → 𝑐 ∈ ( 𝑏 ∪ { 𝑐 } ) ) |
| 170 |
166 169
|
sseldd |
⊢ ( ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 → 𝑐 ∈ 𝐼 ) |
| 171 |
170
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ 𝑥 ∈ 𝑋 ) → 𝑐 ∈ 𝐼 ) |
| 172 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ 𝑥 ∈ 𝑋 ) → 𝑥 ∈ 𝑋 ) |
| 173 |
|
nfv |
⊢ Ⅎ 𝑗 𝑐 ∈ 𝐼 |
| 174 |
|
nfv |
⊢ Ⅎ 𝑗 𝑥 ∈ 𝑋 |
| 175 |
2 173 174
|
nf3an |
⊢ Ⅎ 𝑗 ( 𝜑 ∧ 𝑐 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋 ) |
| 176 |
135
|
nfel1 |
⊢ Ⅎ 𝑗 ⦋ 𝑐 / 𝑗 ⦌ 𝐶 ∈ ℂ |
| 177 |
175 176
|
nfim |
⊢ Ⅎ 𝑗 ( ( 𝜑 ∧ 𝑐 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋 ) → ⦋ 𝑐 / 𝑗 ⦌ 𝐶 ∈ ℂ ) |
| 178 |
|
eleq1w |
⊢ ( 𝑗 = 𝑐 → ( 𝑗 ∈ 𝐼 ↔ 𝑐 ∈ 𝐼 ) ) |
| 179 |
178
|
3anbi2d |
⊢ ( 𝑗 = 𝑐 → ( ( 𝜑 ∧ 𝑗 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋 ) ↔ ( 𝜑 ∧ 𝑐 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋 ) ) ) |
| 180 |
|
csbeq1a |
⊢ ( 𝑗 = 𝑐 → 𝐶 = ⦋ 𝑐 / 𝑗 ⦌ 𝐶 ) |
| 181 |
180
|
eleq1d |
⊢ ( 𝑗 = 𝑐 → ( 𝐶 ∈ ℂ ↔ ⦋ 𝑐 / 𝑗 ⦌ 𝐶 ∈ ℂ ) ) |
| 182 |
179 181
|
imbi12d |
⊢ ( 𝑗 = 𝑐 → ( ( ( 𝜑 ∧ 𝑗 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋 ) → 𝐶 ∈ ℂ ) ↔ ( ( 𝜑 ∧ 𝑐 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋 ) → ⦋ 𝑐 / 𝑗 ⦌ 𝐶 ∈ ℂ ) ) ) |
| 183 |
177 182 162
|
chvarfv |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐼 ∧ 𝑥 ∈ 𝑋 ) → ⦋ 𝑐 / 𝑗 ⦌ 𝐶 ∈ ℂ ) |
| 184 |
165 171 172 183
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ 𝑥 ∈ 𝑋 ) → ⦋ 𝑐 / 𝑗 ⦌ 𝐶 ∈ ℂ ) |
| 185 |
184
|
ad4ant14 |
⊢ ( ( ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) ∧ 𝑥 ∈ 𝑋 ) → ⦋ 𝑐 / 𝑗 ⦌ 𝐶 ∈ ℂ ) |
| 186 |
2 173
|
nfan |
⊢ Ⅎ 𝑗 ( 𝜑 ∧ 𝑐 ∈ 𝐼 ) |
| 187 |
|
nfcv |
⊢ Ⅎ 𝑗 ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑐 / 𝑖 ⦌ 𝐴 ) ) |
| 188 |
128 135
|
nfmpt |
⊢ Ⅎ 𝑗 ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑐 / 𝑗 ⦌ 𝐶 ) |
| 189 |
187 188
|
nfeq |
⊢ Ⅎ 𝑗 ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑐 / 𝑖 ⦌ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑐 / 𝑗 ⦌ 𝐶 ) |
| 190 |
186 189
|
nfim |
⊢ Ⅎ 𝑗 ( ( 𝜑 ∧ 𝑐 ∈ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑐 / 𝑖 ⦌ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑐 / 𝑗 ⦌ 𝐶 ) ) |
| 191 |
178
|
anbi2d |
⊢ ( 𝑗 = 𝑐 → ( ( 𝜑 ∧ 𝑗 ∈ 𝐼 ) ↔ ( 𝜑 ∧ 𝑐 ∈ 𝐼 ) ) ) |
| 192 |
|
csbeq1 |
⊢ ( 𝑗 = 𝑐 → ⦋ 𝑗 / 𝑖 ⦌ 𝐴 = ⦋ 𝑐 / 𝑖 ⦌ 𝐴 ) |
| 193 |
192
|
mpteq2dv |
⊢ ( 𝑗 = 𝑐 → ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑗 / 𝑖 ⦌ 𝐴 ) = ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑐 / 𝑖 ⦌ 𝐴 ) ) |
| 194 |
193
|
oveq2d |
⊢ ( 𝑗 = 𝑐 → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑗 / 𝑖 ⦌ 𝐴 ) ) = ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑐 / 𝑖 ⦌ 𝐴 ) ) ) |
| 195 |
180
|
mpteq2dv |
⊢ ( 𝑗 = 𝑐 → ( 𝑥 ∈ 𝑋 ↦ 𝐶 ) = ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑐 / 𝑗 ⦌ 𝐶 ) ) |
| 196 |
194 195
|
eqeq12d |
⊢ ( 𝑗 = 𝑐 → ( ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑗 / 𝑖 ⦌ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ 𝐶 ) ↔ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑐 / 𝑖 ⦌ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑐 / 𝑗 ⦌ 𝐶 ) ) ) |
| 197 |
191 196
|
imbi12d |
⊢ ( 𝑗 = 𝑐 → ( ( ( 𝜑 ∧ 𝑗 ∈ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑗 / 𝑖 ⦌ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ 𝐶 ) ) ↔ ( ( 𝜑 ∧ 𝑐 ∈ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑐 / 𝑖 ⦌ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑐 / 𝑗 ⦌ 𝐶 ) ) ) ) |
| 198 |
1 153
|
nfan |
⊢ Ⅎ 𝑖 ( 𝜑 ∧ 𝑗 ∈ 𝐼 ) |
| 199 |
|
nfcsb1v |
⊢ Ⅎ 𝑖 ⦋ 𝑗 / 𝑖 ⦌ 𝐴 |
| 200 |
109 199
|
nfmpt |
⊢ Ⅎ 𝑖 ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑗 / 𝑖 ⦌ 𝐴 ) |
| 201 |
107 108 200
|
nfov |
⊢ Ⅎ 𝑖 ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑗 / 𝑖 ⦌ 𝐴 ) ) |
| 202 |
|
nfcv |
⊢ Ⅎ 𝑖 ( 𝑥 ∈ 𝑋 ↦ 𝐶 ) |
| 203 |
201 202
|
nfeq |
⊢ Ⅎ 𝑖 ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑗 / 𝑖 ⦌ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ 𝐶 ) |
| 204 |
198 203
|
nfim |
⊢ Ⅎ 𝑖 ( ( 𝜑 ∧ 𝑗 ∈ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑗 / 𝑖 ⦌ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ 𝐶 ) ) |
| 205 |
158
|
anbi2d |
⊢ ( 𝑖 = 𝑗 → ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) ↔ ( 𝜑 ∧ 𝑗 ∈ 𝐼 ) ) ) |
| 206 |
|
csbeq1a |
⊢ ( 𝑖 = 𝑗 → 𝐴 = ⦋ 𝑗 / 𝑖 ⦌ 𝐴 ) |
| 207 |
206
|
mpteq2dv |
⊢ ( 𝑖 = 𝑗 → ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) = ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑗 / 𝑖 ⦌ 𝐴 ) ) |
| 208 |
207
|
oveq2d |
⊢ ( 𝑖 = 𝑗 → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ) = ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑗 / 𝑖 ⦌ 𝐴 ) ) ) |
| 209 |
11
|
mpteq2dv |
⊢ ( 𝑖 = 𝑗 → ( 𝑥 ∈ 𝑋 ↦ 𝐵 ) = ( 𝑥 ∈ 𝑋 ↦ 𝐶 ) ) |
| 210 |
208 209
|
eqeq12d |
⊢ ( 𝑖 = 𝑗 → ( ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ 𝐵 ) ↔ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑗 / 𝑖 ⦌ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ 𝐶 ) ) ) |
| 211 |
205 210
|
imbi12d |
⊢ ( 𝑖 = 𝑗 → ( ( ( 𝜑 ∧ 𝑖 ∈ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ 𝐵 ) ) ↔ ( ( 𝜑 ∧ 𝑗 ∈ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑗 / 𝑖 ⦌ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ 𝐶 ) ) ) ) |
| 212 |
204 211 10
|
chvarfv |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑗 / 𝑖 ⦌ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ 𝐶 ) ) |
| 213 |
190 197 212
|
chvarfv |
⊢ ( ( 𝜑 ∧ 𝑐 ∈ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑐 / 𝑖 ⦌ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑐 / 𝑗 ⦌ 𝐶 ) ) |
| 214 |
170 213
|
sylan2 |
⊢ ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑐 / 𝑖 ⦌ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑐 / 𝑗 ⦌ 𝐶 ) ) |
| 215 |
214
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑐 / 𝑖 ⦌ 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ ⦋ 𝑐 / 𝑗 ⦌ 𝐶 ) ) |
| 216 |
|
csbeq1a |
⊢ ( 𝑖 = 𝑐 → 𝐴 = ⦋ 𝑐 / 𝑖 ⦌ 𝐴 ) |
| 217 |
102 122 133 134 135 138 142 144 145 146 147 163 164 185 215 216 180
|
dvmptfprodlem |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ ( 𝑏 ∪ { 𝑐 } ) 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ ( 𝑏 ∪ { 𝑐 } ) ( 𝐶 · ∏ 𝑖 ∈ ( ( 𝑏 ∪ { 𝑐 } ) ∖ { 𝑗 } ) 𝐴 ) ) ) |
| 218 |
85 86 94 217
|
syl21anc |
⊢ ( ( ( 𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏 ) ∧ ( ( 𝜑 ∧ 𝑏 ⊆ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) ∧ ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ ( 𝑏 ∪ { 𝑐 } ) 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ ( 𝑏 ∪ { 𝑐 } ) ( 𝐶 · ∏ 𝑖 ∈ ( ( 𝑏 ∪ { 𝑐 } ) ∖ { 𝑗 } ) 𝐴 ) ) ) |
| 219 |
218
|
3exp |
⊢ ( ( 𝑏 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑏 ) → ( ( ( 𝜑 ∧ 𝑏 ⊆ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝑏 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝑏 ( 𝐶 · ∏ 𝑖 ∈ ( 𝑏 ∖ { 𝑗 } ) 𝐴 ) ) ) → ( ( 𝜑 ∧ ( 𝑏 ∪ { 𝑐 } ) ⊆ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ ( 𝑏 ∪ { 𝑐 } ) 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ ( 𝑏 ∪ { 𝑐 } ) ( 𝐶 · ∏ 𝑖 ∈ ( ( 𝑏 ∪ { 𝑐 } ) ∖ { 𝑗 } ) 𝐴 ) ) ) ) ) |
| 220 |
27 41 55 69 84 219
|
findcard2s |
⊢ ( 𝐼 ∈ Fin → ( ( 𝜑 ∧ 𝐼 ⊆ 𝐼 ) → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝐼 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝐼 ( 𝐶 · ∏ 𝑖 ∈ ( 𝐼 ∖ { 𝑗 } ) 𝐴 ) ) ) ) |
| 221 |
7 13 220
|
sylc |
⊢ ( 𝜑 → ( 𝑆 D ( 𝑥 ∈ 𝑋 ↦ ∏ 𝑖 ∈ 𝐼 𝐴 ) ) = ( 𝑥 ∈ 𝑋 ↦ Σ 𝑗 ∈ 𝐼 ( 𝐶 · ∏ 𝑖 ∈ ( 𝐼 ∖ { 𝑗 } ) 𝐴 ) ) ) |