Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvmptfprod Structured version   Visualization version   GIF version

Theorem dvmptfprod 43161
Description: Function-builder for derivative, finite product rule. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Hypotheses
Ref Expression
dvmptfprod.iph 𝑖𝜑
dvmptfprod.jph 𝑗𝜑
dvmptfprod.j 𝐽 = (𝐾t 𝑆)
dvmptfprod.k 𝐾 = (TopOpen‘ℂfld)
dvmptfprod.s (𝜑𝑆 ∈ {ℝ, ℂ})
dvmptfprod.x (𝜑𝑋𝐽)
dvmptfprod.i (𝜑𝐼 ∈ Fin)
dvmptfprod.a ((𝜑𝑖𝐼𝑥𝑋) → 𝐴 ∈ ℂ)
dvmptfprod.b ((𝜑𝑖𝐼𝑥𝑋) → 𝐵 ∈ ℂ)
dvmptfprod.d ((𝜑𝑖𝐼) → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))
dvmptfprod.bc (𝑖 = 𝑗𝐵 = 𝐶)
Assertion
Ref Expression
dvmptfprod (𝜑 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝐼 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)))
Distinct variable groups:   𝐴,𝑗   𝐶,𝑖   𝑖,𝐼,𝑗,𝑥   𝑆,𝑖,𝑗,𝑥   𝑖,𝑋,𝑗,𝑥   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑖,𝑗)   𝐴(𝑥,𝑖)   𝐵(𝑥,𝑖,𝑗)   𝐶(𝑥,𝑗)   𝐽(𝑥,𝑖,𝑗)   𝐾(𝑥,𝑖,𝑗)

Proof of Theorem dvmptfprod
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvmptfprod.i . 2 (𝜑𝐼 ∈ Fin)
2 ssid 3923 . . 3 𝐼𝐼
32jctr 528 . 2 (𝜑 → (𝜑𝐼𝐼))
4 sseq1 3926 . . . . 5 (𝑎 = ∅ → (𝑎𝐼 ↔ ∅ ⊆ 𝐼))
54anbi2d 632 . . . 4 (𝑎 = ∅ → ((𝜑𝑎𝐼) ↔ (𝜑 ∧ ∅ ⊆ 𝐼)))
6 prodeq1 15471 . . . . . . 7 (𝑎 = ∅ → ∏𝑖𝑎 𝐴 = ∏𝑖 ∈ ∅ 𝐴)
76mpteq2dv 5151 . . . . . 6 (𝑎 = ∅ → (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴) = (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴))
87oveq2d 7229 . . . . 5 (𝑎 = ∅ → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)))
9 sumeq1 15252 . . . . . . 7 (𝑎 = ∅ → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
10 difeq1 4030 . . . . . . . . . 10 (𝑎 = ∅ → (𝑎 ∖ {𝑗}) = (∅ ∖ {𝑗}))
1110prodeq1d 15483 . . . . . . . . 9 (𝑎 = ∅ → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)
1211oveq2d 7229 . . . . . . . 8 (𝑎 = ∅ → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
1312sumeq2sdv 15268 . . . . . . 7 (𝑎 = ∅ → Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
149, 13eqtrd 2777 . . . . . 6 (𝑎 = ∅ → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
1514mpteq2dv 5151 . . . . 5 (𝑎 = ∅ → (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
168, 15eqeq12d 2753 . . . 4 (𝑎 = ∅ → ((𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) ↔ (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))))
175, 16imbi12d 348 . . 3 (𝑎 = ∅ → (((𝜑𝑎𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))) ↔ ((𝜑 ∧ ∅ ⊆ 𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))))
18 sseq1 3926 . . . . 5 (𝑎 = 𝑏 → (𝑎𝐼𝑏𝐼))
1918anbi2d 632 . . . 4 (𝑎 = 𝑏 → ((𝜑𝑎𝐼) ↔ (𝜑𝑏𝐼)))
20 prodeq1 15471 . . . . . . 7 (𝑎 = 𝑏 → ∏𝑖𝑎 𝐴 = ∏𝑖𝑏 𝐴)
2120mpteq2dv 5151 . . . . . 6 (𝑎 = 𝑏 → (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴) = (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴))
2221oveq2d 7229 . . . . 5 (𝑎 = 𝑏 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)))
23 sumeq1 15252 . . . . . . 7 (𝑎 = 𝑏 → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
24 difeq1 4030 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝑎 ∖ {𝑗}) = (𝑏 ∖ {𝑗}))
2524prodeq1d 15483 . . . . . . . . 9 (𝑎 = 𝑏 → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
2625oveq2d 7229 . . . . . . . 8 (𝑎 = 𝑏 → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
2726sumeq2sdv 15268 . . . . . . 7 (𝑎 = 𝑏 → Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
2823, 27eqtrd 2777 . . . . . 6 (𝑎 = 𝑏 → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
2928mpteq2dv 5151 . . . . 5 (𝑎 = 𝑏 → (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
3022, 29eqeq12d 2753 . . . 4 (𝑎 = 𝑏 → ((𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) ↔ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))))
3119, 30imbi12d 348 . . 3 (𝑎 = 𝑏 → (((𝜑𝑎𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))) ↔ ((𝜑𝑏𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))))
32 sseq1 3926 . . . . 5 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎𝐼 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝐼))
3332anbi2d 632 . . . 4 (𝑎 = (𝑏 ∪ {𝑐}) → ((𝜑𝑎𝐼) ↔ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)))
34 prodeq1 15471 . . . . . . 7 (𝑎 = (𝑏 ∪ {𝑐}) → ∏𝑖𝑎 𝐴 = ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)
3534mpteq2dv 5151 . . . . . 6 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴) = (𝑥𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴))
3635oveq2d 7229 . . . . 5 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)))
37 sumeq1 15252 . . . . . . 7 (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
38 difeq1 4030 . . . . . . . . . 10 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ∖ {𝑗}) = ((𝑏 ∪ {𝑐}) ∖ {𝑗}))
3938prodeq1d 15483 . . . . . . . . 9 (𝑎 = (𝑏 ∪ {𝑐}) → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)
4039oveq2d 7229 . . . . . . . 8 (𝑎 = (𝑏 ∪ {𝑐}) → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))
4140sumeq2sdv 15268 . . . . . . 7 (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))
4237, 41eqtrd 2777 . . . . . 6 (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))
4342mpteq2dv 5151 . . . . 5 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)))
4436, 43eqeq12d 2753 . . . 4 (𝑎 = (𝑏 ∪ {𝑐}) → ((𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) ↔ (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))))
4533, 44imbi12d 348 . . 3 (𝑎 = (𝑏 ∪ {𝑐}) → (((𝜑𝑎𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))) ↔ ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)))))
46 sseq1 3926 . . . . 5 (𝑎 = 𝐼 → (𝑎𝐼𝐼𝐼))
4746anbi2d 632 . . . 4 (𝑎 = 𝐼 → ((𝜑𝑎𝐼) ↔ (𝜑𝐼𝐼)))
48 prodeq1 15471 . . . . . . 7 (𝑎 = 𝐼 → ∏𝑖𝑎 𝐴 = ∏𝑖𝐼 𝐴)
4948mpteq2dv 5151 . . . . . 6 (𝑎 = 𝐼 → (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴) = (𝑥𝑋 ↦ ∏𝑖𝐼 𝐴))
5049oveq2d 7229 . . . . 5 (𝑎 = 𝐼 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝐼 𝐴)))
51 sumeq1 15252 . . . . . . 7 (𝑎 = 𝐼 → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
52 difeq1 4030 . . . . . . . . . . . 12 (𝑎 = 𝐼 → (𝑎 ∖ {𝑗}) = (𝐼 ∖ {𝑗}))
5352prodeq1d 15483 . . . . . . . . . . 11 (𝑎 = 𝐼 → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)
5453oveq2d 7229 . . . . . . . . . 10 (𝑎 = 𝐼 → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))
5554a1d 25 . . . . . . . . 9 (𝑎 = 𝐼 → (𝑗𝐼 → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)))
5655ralrimiv 3104 . . . . . . . 8 (𝑎 = 𝐼 → ∀𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))
5756sumeq2d 15266 . . . . . . 7 (𝑎 = 𝐼 → Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))
5851, 57eqtrd 2777 . . . . . 6 (𝑎 = 𝐼 → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))
5958mpteq2dv 5151 . . . . 5 (𝑎 = 𝐼 → (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)))
6050, 59eqeq12d 2753 . . . 4 (𝑎 = 𝐼 → ((𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) ↔ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝐼 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))))
6147, 60imbi12d 348 . . 3 (𝑎 = 𝐼 → (((𝜑𝑎𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))) ↔ ((𝜑𝐼𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝐼 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)))))
62 prod0 15505 . . . . . . . 8 𝑖 ∈ ∅ 𝐴 = 1
6362mpteq2i 5147 . . . . . . 7 (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴) = (𝑥𝑋 ↦ 1)
6463oveq2i 7224 . . . . . 6 (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑆 D (𝑥𝑋 ↦ 1))
6564a1i 11 . . . . 5 (𝜑 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑆 D (𝑥𝑋 ↦ 1)))
66 dvmptfprod.s . . . . . 6 (𝜑𝑆 ∈ {ℝ, ℂ})
67 dvmptfprod.x . . . . . . 7 (𝜑𝑋𝐽)
68 dvmptfprod.j . . . . . . . 8 𝐽 = (𝐾t 𝑆)
69 dvmptfprod.k . . . . . . . . 9 𝐾 = (TopOpen‘ℂfld)
7069oveq1i 7223 . . . . . . . 8 (𝐾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
7168, 70eqtri 2765 . . . . . . 7 𝐽 = ((TopOpen‘ℂfld) ↾t 𝑆)
7267, 71eleqtrdi 2848 . . . . . 6 (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
73 1cnd 10828 . . . . . 6 (𝜑 → 1 ∈ ℂ)
7466, 72, 73dvmptconst 43131 . . . . 5 (𝜑 → (𝑆 D (𝑥𝑋 ↦ 1)) = (𝑥𝑋 ↦ 0))
75 sum0 15285 . . . . . . . 8 Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴) = 0
7675eqcomi 2746 . . . . . . 7 0 = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)
7776mpteq2i 5147 . . . . . 6 (𝑥𝑋 ↦ 0) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
7877a1i 11 . . . . 5 (𝜑 → (𝑥𝑋 ↦ 0) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
7965, 74, 783eqtrd 2781 . . . 4 (𝜑 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
8079adantr 484 . . 3 ((𝜑 ∧ ∅ ⊆ 𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
81 simp3 1140 . . . . 5 (((𝑏 ∈ Fin ∧ ¬ 𝑐𝑏) ∧ ((𝜑𝑏𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼))
82 simp1r 1200 . . . . 5 (((𝑏 ∈ Fin ∧ ¬ 𝑐𝑏) ∧ ((𝜑𝑏𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → ¬ 𝑐𝑏)
83 simpl 486 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → 𝜑)
84 ssun1 4086 . . . . . . . . . . 11 𝑏 ⊆ (𝑏 ∪ {𝑐})
85 sstr2 3908 . . . . . . . . . . 11 (𝑏 ⊆ (𝑏 ∪ {𝑐}) → ((𝑏 ∪ {𝑐}) ⊆ 𝐼𝑏𝐼))
8684, 85ax-mp 5 . . . . . . . . . 10 ((𝑏 ∪ {𝑐}) ⊆ 𝐼𝑏𝐼)
8786adantl 485 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → 𝑏𝐼)
8883, 87jca 515 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝜑𝑏𝐼))
8988adantl 485 . . . . . . 7 ((((𝜑𝑏𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → (𝜑𝑏𝐼))
90 simpl 486 . . . . . . 7 ((((𝜑𝑏𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → ((𝜑𝑏𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))))
9189, 90mpd 15 . . . . . 6 ((((𝜑𝑏𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
92913adant1 1132 . . . . 5 (((𝑏 ∈ Fin ∧ ¬ 𝑐𝑏) ∧ ((𝜑𝑏𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
93 nfv 1922 . . . . . . 7 𝑥((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏)
94 nfcv 2904 . . . . . . . . 9 𝑥𝑆
95 nfcv 2904 . . . . . . . . 9 𝑥 D
96 nfmpt1 5153 . . . . . . . . 9 𝑥(𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)
9794, 95, 96nfov 7243 . . . . . . . 8 𝑥(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴))
98 nfmpt1 5153 . . . . . . . 8 𝑥(𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
9997, 98nfeq 2917 . . . . . . 7 𝑥(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
10093, 99nfan 1907 . . . . . 6 𝑥(((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
101 dvmptfprod.iph . . . . . . . . 9 𝑖𝜑
102 nfv 1922 . . . . . . . . 9 𝑖(𝑏 ∪ {𝑐}) ⊆ 𝐼
103101, 102nfan 1907 . . . . . . . 8 𝑖(𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)
104 nfv 1922 . . . . . . . 8 𝑖 ¬ 𝑐𝑏
105103, 104nfan 1907 . . . . . . 7 𝑖((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏)
106 nfcv 2904 . . . . . . . . 9 𝑖𝑆
107 nfcv 2904 . . . . . . . . 9 𝑖 D
108 nfcv 2904 . . . . . . . . . 10 𝑖𝑋
109 nfcv 2904 . . . . . . . . . . 11 𝑖𝑏
110109nfcprod1 15472 . . . . . . . . . 10 𝑖𝑖𝑏 𝐴
111108, 110nfmpt 5152 . . . . . . . . 9 𝑖(𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)
112106, 107, 111nfov 7243 . . . . . . . 8 𝑖(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴))
113 nfcv 2904 . . . . . . . . . . 11 𝑖𝐶
114 nfcv 2904 . . . . . . . . . . 11 𝑖 ·
115 nfcv 2904 . . . . . . . . . . . 12 𝑖(𝑏 ∖ {𝑗})
116115nfcprod1 15472 . . . . . . . . . . 11 𝑖𝑖 ∈ (𝑏 ∖ {𝑗})𝐴
117113, 114, 116nfov 7243 . . . . . . . . . 10 𝑖(𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
118109, 117nfsum 15254 . . . . . . . . 9 𝑖Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
119108, 118nfmpt 5152 . . . . . . . 8 𝑖(𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
120112, 119nfeq 2917 . . . . . . 7 𝑖(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
121105, 120nfan 1907 . . . . . 6 𝑖(((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
122 dvmptfprod.jph . . . . . . . . 9 𝑗𝜑
123 nfv 1922 . . . . . . . . 9 𝑗(𝑏 ∪ {𝑐}) ⊆ 𝐼
124122, 123nfan 1907 . . . . . . . 8 𝑗(𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)
125 nfv 1922 . . . . . . . 8 𝑗 ¬ 𝑐𝑏
126124, 125nfan 1907 . . . . . . 7 𝑗((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏)
127 nfcv 2904 . . . . . . . 8 𝑗(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴))
128 nfcv 2904 . . . . . . . . 9 𝑗𝑋
129 nfcv 2904 . . . . . . . . . 10 𝑗𝑏
130129nfsum1 15253 . . . . . . . . 9 𝑗Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
131128, 130nfmpt 5152 . . . . . . . 8 𝑗(𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
132127, 131nfeq 2917 . . . . . . 7 𝑗(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
133126, 132nfan 1907 . . . . . 6 𝑗(((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
134 nfcsb1v 3836 . . . . . 6 𝑖𝑐 / 𝑖𝐴
135 nfcsb1v 3836 . . . . . 6 𝑗𝑐 / 𝑗𝐶
13683ad2antrr 726 . . . . . . . 8 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝜑)
1371363ad2ant1 1135 . . . . . . 7 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑖𝐼𝑥𝑋) → 𝜑)
138 simp2 1139 . . . . . . 7 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑖𝐼𝑥𝑋) → 𝑖𝐼)
139 simp3 1140 . . . . . . 7 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑖𝐼𝑥𝑋) → 𝑥𝑋)
140 dvmptfprod.a . . . . . . 7 ((𝜑𝑖𝐼𝑥𝑋) → 𝐴 ∈ ℂ)
141137, 138, 139, 140syl3anc 1373 . . . . . 6 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑖𝐼𝑥𝑋) → 𝐴 ∈ ℂ)
142136, 1syl 17 . . . . . . 7 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝐼 ∈ Fin)
14387ad2antrr 726 . . . . . . 7 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑏𝐼)
144 ssfi 8851 . . . . . . 7 ((𝐼 ∈ Fin ∧ 𝑏𝐼) → 𝑏 ∈ Fin)
145142, 143, 144syl2anc 587 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑏 ∈ Fin)
146 vex 3412 . . . . . . 7 𝑐 ∈ V
147146a1i 11 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑐 ∈ V)
148 simplr 769 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → ¬ 𝑐𝑏)
149 simpllr 776 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → (𝑏 ∪ {𝑐}) ⊆ 𝐼)
15066ad3antrrr 730 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑆 ∈ {ℝ, ℂ})
151136ad2antrr 726 . . . . . . 7 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝜑)
152143ad2antrr 726 . . . . . . . 8 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝑏𝐼)
153 simpr 488 . . . . . . . 8 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝑗𝑏)
154152, 153sseldd 3902 . . . . . . 7 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝑗𝐼)
155 simplr 769 . . . . . . 7 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝑥𝑋)
156 nfv 1922 . . . . . . . . . 10 𝑖 𝑗𝐼
157 nfv 1922 . . . . . . . . . 10 𝑖 𝑥𝑋
158101, 156, 157nf3an 1909 . . . . . . . . 9 𝑖(𝜑𝑗𝐼𝑥𝑋)
159 nfv 1922 . . . . . . . . 9 𝑖 𝐶 ∈ ℂ
160158, 159nfim 1904 . . . . . . . 8 𝑖((𝜑𝑗𝐼𝑥𝑋) → 𝐶 ∈ ℂ)
161 eleq1w 2820 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑖𝐼𝑗𝐼))
1621613anbi2d 1443 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝜑𝑖𝐼𝑥𝑋) ↔ (𝜑𝑗𝐼𝑥𝑋)))
163 dvmptfprod.bc . . . . . . . . . 10 (𝑖 = 𝑗𝐵 = 𝐶)
164163eleq1d 2822 . . . . . . . . 9 (𝑖 = 𝑗 → (𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ))
165162, 164imbi12d 348 . . . . . . . 8 (𝑖 = 𝑗 → (((𝜑𝑖𝐼𝑥𝑋) → 𝐵 ∈ ℂ) ↔ ((𝜑𝑗𝐼𝑥𝑋) → 𝐶 ∈ ℂ)))
166 dvmptfprod.b . . . . . . . 8 ((𝜑𝑖𝐼𝑥𝑋) → 𝐵 ∈ ℂ)
167160, 165, 166chvarfv 2238 . . . . . . 7 ((𝜑𝑗𝐼𝑥𝑋) → 𝐶 ∈ ℂ)
168151, 154, 155, 167syl3anc 1373 . . . . . 6 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝐶 ∈ ℂ)
169 simpr 488 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
17083adantr 484 . . . . . . . . 9 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥𝑋) → 𝜑)
171 id 22 . . . . . . . . . . 11 ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑏 ∪ {𝑐}) ⊆ 𝐼)
172146snid 4577 . . . . . . . . . . . . 13 𝑐 ∈ {𝑐}
173 elun2 4091 . . . . . . . . . . . . 13 (𝑐 ∈ {𝑐} → 𝑐 ∈ (𝑏 ∪ {𝑐}))
174172, 173ax-mp 5 . . . . . . . . . . . 12 𝑐 ∈ (𝑏 ∪ {𝑐})
175174a1i 11 . . . . . . . . . . 11 ((𝑏 ∪ {𝑐}) ⊆ 𝐼𝑐 ∈ (𝑏 ∪ {𝑐}))
176171, 175sseldd 3902 . . . . . . . . . 10 ((𝑏 ∪ {𝑐}) ⊆ 𝐼𝑐𝐼)
177176ad2antlr 727 . . . . . . . . 9 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥𝑋) → 𝑐𝐼)
178 simpr 488 . . . . . . . . 9 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥𝑋) → 𝑥𝑋)
179 nfv 1922 . . . . . . . . . . . 12 𝑗 𝑐𝐼
180 nfv 1922 . . . . . . . . . . . 12 𝑗 𝑥𝑋
181122, 179, 180nf3an 1909 . . . . . . . . . . 11 𝑗(𝜑𝑐𝐼𝑥𝑋)
182 nfcv 2904 . . . . . . . . . . . 12 𝑗
183135, 182nfel 2918 . . . . . . . . . . 11 𝑗𝑐 / 𝑗𝐶 ∈ ℂ
184181, 183nfim 1904 . . . . . . . . . 10 𝑗((𝜑𝑐𝐼𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
185 eleq1w 2820 . . . . . . . . . . . 12 (𝑗 = 𝑐 → (𝑗𝐼𝑐𝐼))
1861853anbi2d 1443 . . . . . . . . . . 11 (𝑗 = 𝑐 → ((𝜑𝑗𝐼𝑥𝑋) ↔ (𝜑𝑐𝐼𝑥𝑋)))
187 csbeq1a 3825 . . . . . . . . . . . 12 (𝑗 = 𝑐𝐶 = 𝑐 / 𝑗𝐶)
188187eleq1d 2822 . . . . . . . . . . 11 (𝑗 = 𝑐 → (𝐶 ∈ ℂ ↔ 𝑐 / 𝑗𝐶 ∈ ℂ))
189186, 188imbi12d 348 . . . . . . . . . 10 (𝑗 = 𝑐 → (((𝜑𝑗𝐼𝑥𝑋) → 𝐶 ∈ ℂ) ↔ ((𝜑𝑐𝐼𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)))
190184, 189, 167chvarfv 2238 . . . . . . . . 9 ((𝜑𝑐𝐼𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
191170, 177, 178, 190syl3anc 1373 . . . . . . . 8 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
192191adantlr 715 . . . . . . 7 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ 𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
193192adantlr 715 . . . . . 6 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
194122, 179nfan 1907 . . . . . . . . . 10 𝑗(𝜑𝑐𝐼)
195 nfcv 2904 . . . . . . . . . . 11 𝑗(𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴))
196128, 135nfmpt 5152 . . . . . . . . . . 11 𝑗(𝑥𝑋𝑐 / 𝑗𝐶)
197195, 196nfeq 2917 . . . . . . . . . 10 𝑗(𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶)
198194, 197nfim 1904 . . . . . . . . 9 𝑗((𝜑𝑐𝐼) → (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶))
199185anbi2d 632 . . . . . . . . . 10 (𝑗 = 𝑐 → ((𝜑𝑗𝐼) ↔ (𝜑𝑐𝐼)))
200 csbeq1a 3825 . . . . . . . . . . . . . 14 (𝑗 = 𝑐𝑗 / 𝑖𝐴 = 𝑐 / 𝑗𝑗 / 𝑖𝐴)
201 csbcow 3826 . . . . . . . . . . . . . . 15 𝑐 / 𝑗𝑗 / 𝑖𝐴 = 𝑐 / 𝑖𝐴
202201a1i 11 . . . . . . . . . . . . . 14 (𝑗 = 𝑐𝑐 / 𝑗𝑗 / 𝑖𝐴 = 𝑐 / 𝑖𝐴)
203200, 202eqtrd 2777 . . . . . . . . . . . . 13 (𝑗 = 𝑐𝑗 / 𝑖𝐴 = 𝑐 / 𝑖𝐴)
204203mpteq2dv 5151 . . . . . . . . . . . 12 (𝑗 = 𝑐 → (𝑥𝑋𝑗 / 𝑖𝐴) = (𝑥𝑋𝑐 / 𝑖𝐴))
205204oveq2d 7229 . . . . . . . . . . 11 (𝑗 = 𝑐 → (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)))
206187mpteq2dv 5151 . . . . . . . . . . 11 (𝑗 = 𝑐 → (𝑥𝑋𝐶) = (𝑥𝑋𝑐 / 𝑗𝐶))
207205, 206eqeq12d 2753 . . . . . . . . . 10 (𝑗 = 𝑐 → ((𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶) ↔ (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶)))
208199, 207imbi12d 348 . . . . . . . . 9 (𝑗 = 𝑐 → (((𝜑𝑗𝐼) → (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶)) ↔ ((𝜑𝑐𝐼) → (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶))))
209101, 156nfan 1907 . . . . . . . . . . 11 𝑖(𝜑𝑗𝐼)
210 nfcsb1v 3836 . . . . . . . . . . . . . 14 𝑖𝑗 / 𝑖𝐴
211108, 210nfmpt 5152 . . . . . . . . . . . . 13 𝑖(𝑥𝑋𝑗 / 𝑖𝐴)
212106, 107, 211nfov 7243 . . . . . . . . . . . 12 𝑖(𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴))
213 nfcv 2904 . . . . . . . . . . . 12 𝑖(𝑥𝑋𝐶)
214212, 213nfeq 2917 . . . . . . . . . . 11 𝑖(𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶)
215209, 214nfim 1904 . . . . . . . . . 10 𝑖((𝜑𝑗𝐼) → (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶))
216161anbi2d 632 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝜑𝑖𝐼) ↔ (𝜑𝑗𝐼)))
217 csbeq1a 3825 . . . . . . . . . . . . . 14 (𝑖 = 𝑗𝐴 = 𝑗 / 𝑖𝐴)
218217mpteq2dv 5151 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑥𝑋𝐴) = (𝑥𝑋𝑗 / 𝑖𝐴))
219218oveq2d 7229 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑆 D (𝑥𝑋𝐴)) = (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)))
220163idi 1 . . . . . . . . . . . . 13 (𝑖 = 𝑗𝐵 = 𝐶)
221220mpteq2dv 5151 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑥𝑋𝐵) = (𝑥𝑋𝐶))
222219, 221eqeq12d 2753 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵) ↔ (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶)))
223216, 222imbi12d 348 . . . . . . . . . 10 (𝑖 = 𝑗 → (((𝜑𝑖𝐼) → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵)) ↔ ((𝜑𝑗𝐼) → (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶))))
224 dvmptfprod.d . . . . . . . . . 10 ((𝜑𝑖𝐼) → (𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵))
225215, 223, 224chvarfv 2238 . . . . . . . . 9 ((𝜑𝑗𝐼) → (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶))
226198, 208, 225chvarfv 2238 . . . . . . . 8 ((𝜑𝑐𝐼) → (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶))
227176, 226sylan2 596 . . . . . . 7 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶))
228227ad2antrr 726 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶))
229 csbeq1a 3825 . . . . . 6 (𝑖 = 𝑐𝐴 = 𝑐 / 𝑖𝐴)
230100, 121, 133, 134, 135, 141, 145, 147, 148, 149, 150, 168, 169, 193, 228, 229, 187dvmptfprodlem 43160 . . . . 5 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)))
23181, 82, 92, 230syl21anc 838 . . . 4 (((𝑏 ∈ Fin ∧ ¬ 𝑐𝑏) ∧ ((𝜑𝑏𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)))
2322313exp 1121 . . 3 ((𝑏 ∈ Fin ∧ ¬ 𝑐𝑏) → (((𝜑𝑏𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)))))
23317, 31, 45, 61, 80, 232findcard2s 8843 . 2 (𝐼 ∈ Fin → ((𝜑𝐼𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝐼 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))))
2341, 3, 233sylc 65 1 (𝜑 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝐼 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1089   = wceq 1543  wnf 1791  wcel 2110  Vcvv 3408  csb 3811  cdif 3863  cun 3864  wss 3866  c0 4237  {csn 4541  {cpr 4543  cmpt 5135  cfv 6380  (class class class)co 7213  Fincfn 8626  cc 10727  cr 10728  0cc0 10729  1c1 10730   · cmul 10734  Σcsu 15249  cprod 15467  t crest 16925  TopOpenctopn 16926  fldccnfld 20363   D cdv 24760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-inf2 9256  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807  ax-addf 10808  ax-mulf 10809
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-om 7645  df-1st 7761  df-2nd 7762  df-supp 7904  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-er 8391  df-map 8510  df-pm 8511  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fsupp 8986  df-fi 9027  df-sup 9058  df-inf 9059  df-oi 9126  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-z 12177  df-dec 12294  df-uz 12439  df-q 12545  df-rp 12587  df-xneg 12704  df-xadd 12705  df-xmul 12706  df-icc 12942  df-fz 13096  df-fzo 13239  df-seq 13575  df-exp 13636  df-hash 13897  df-cj 14662  df-re 14663  df-im 14664  df-sqrt 14798  df-abs 14799  df-clim 15049  df-sum 15250  df-prod 15468  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-mulr 16816  df-starv 16817  df-sca 16818  df-vsca 16819  df-ip 16820  df-tset 16821  df-ple 16822  df-ds 16824  df-unif 16825  df-hom 16826  df-cco 16827  df-rest 16927  df-topn 16928  df-0g 16946  df-gsum 16947  df-topgen 16948  df-pt 16949  df-prds 16952  df-xrs 17007  df-qtop 17012  df-imas 17013  df-xps 17015  df-mre 17089  df-mrc 17090  df-acs 17092  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-submnd 18219  df-mulg 18489  df-cntz 18711  df-cmn 19172  df-psmet 20355  df-xmet 20356  df-met 20357  df-bl 20358  df-mopn 20359  df-fbas 20360  df-fg 20361  df-cnfld 20364  df-top 21791  df-topon 21808  df-topsp 21830  df-bases 21843  df-cld 21916  df-ntr 21917  df-cls 21918  df-nei 21995  df-lp 22033  df-perf 22034  df-cn 22124  df-cnp 22125  df-haus 22212  df-tx 22459  df-hmeo 22652  df-fil 22743  df-fm 22835  df-flim 22836  df-flf 22837  df-xms 23218  df-ms 23219  df-tms 23220  df-cncf 23775  df-limc 24763  df-dv 24764
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator