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Theorem dvmptfprod 40798
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 3783 . . 3 𝐼𝐼
32jctr 520 . 2 (𝜑 → (𝜑𝐼𝐼))
4 sseq1 3786 . . . . 5 (𝑎 = ∅ → (𝑎𝐼 ↔ ∅ ⊆ 𝐼))
54anbi2d 622 . . . 4 (𝑎 = ∅ → ((𝜑𝑎𝐼) ↔ (𝜑 ∧ ∅ ⊆ 𝐼)))
6 prodeq1 14922 . . . . . . 7 (𝑎 = ∅ → ∏𝑖𝑎 𝐴 = ∏𝑖 ∈ ∅ 𝐴)
76mpteq2dv 4904 . . . . . 6 (𝑎 = ∅ → (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴) = (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴))
87oveq2d 6858 . . . . 5 (𝑎 = ∅ → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)))
9 sumeq1 14704 . . . . . . 7 (𝑎 = ∅ → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
10 difeq1 3883 . . . . . . . . . 10 (𝑎 = ∅ → (𝑎 ∖ {𝑗}) = (∅ ∖ {𝑗}))
1110prodeq1d 14934 . . . . . . . . 9 (𝑎 = ∅ → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)
1211oveq2d 6858 . . . . . . . 8 (𝑎 = ∅ → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
1312sumeq2sdv 14720 . . . . . . 7 (𝑎 = ∅ → Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
149, 13eqtrd 2799 . . . . . 6 (𝑎 = ∅ → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
1514mpteq2dv 4904 . . . . 5 (𝑎 = ∅ → (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
168, 15eqeq12d 2780 . . . 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 335 . . 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 3786 . . . . 5 (𝑎 = 𝑏 → (𝑎𝐼𝑏𝐼))
1918anbi2d 622 . . . 4 (𝑎 = 𝑏 → ((𝜑𝑎𝐼) ↔ (𝜑𝑏𝐼)))
20 prodeq1 14922 . . . . . . 7 (𝑎 = 𝑏 → ∏𝑖𝑎 𝐴 = ∏𝑖𝑏 𝐴)
2120mpteq2dv 4904 . . . . . 6 (𝑎 = 𝑏 → (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴) = (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴))
2221oveq2d 6858 . . . . 5 (𝑎 = 𝑏 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)))
23 sumeq1 14704 . . . . . . 7 (𝑎 = 𝑏 → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
24 difeq1 3883 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝑎 ∖ {𝑗}) = (𝑏 ∖ {𝑗}))
2524prodeq1d 14934 . . . . . . . . 9 (𝑎 = 𝑏 → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
2625oveq2d 6858 . . . . . . . 8 (𝑎 = 𝑏 → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
2726sumeq2sdv 14720 . . . . . . 7 (𝑎 = 𝑏 → Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
2823, 27eqtrd 2799 . . . . . 6 (𝑎 = 𝑏 → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
2928mpteq2dv 4904 . . . . 5 (𝑎 = 𝑏 → (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
3022, 29eqeq12d 2780 . . . 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 335 . . 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 3786 . . . . 5 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎𝐼 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝐼))
3332anbi2d 622 . . . 4 (𝑎 = (𝑏 ∪ {𝑐}) → ((𝜑𝑎𝐼) ↔ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)))
34 prodeq1 14922 . . . . . . 7 (𝑎 = (𝑏 ∪ {𝑐}) → ∏𝑖𝑎 𝐴 = ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)
3534mpteq2dv 4904 . . . . . 6 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴) = (𝑥𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴))
3635oveq2d 6858 . . . . 5 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)))
37 sumeq1 14704 . . . . . . 7 (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
38 difeq1 3883 . . . . . . . . . 10 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ∖ {𝑗}) = ((𝑏 ∪ {𝑐}) ∖ {𝑗}))
3938prodeq1d 14934 . . . . . . . . 9 (𝑎 = (𝑏 ∪ {𝑐}) → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)
4039oveq2d 6858 . . . . . . . 8 (𝑎 = (𝑏 ∪ {𝑐}) → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))
4140sumeq2sdv 14720 . . . . . . 7 (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))
4237, 41eqtrd 2799 . . . . . 6 (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))
4342mpteq2dv 4904 . . . . 5 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)))
4436, 43eqeq12d 2780 . . . 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 335 . . 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 3786 . . . . 5 (𝑎 = 𝐼 → (𝑎𝐼𝐼𝐼))
4746anbi2d 622 . . . 4 (𝑎 = 𝐼 → ((𝜑𝑎𝐼) ↔ (𝜑𝐼𝐼)))
48 prodeq1 14922 . . . . . . 7 (𝑎 = 𝐼 → ∏𝑖𝑎 𝐴 = ∏𝑖𝐼 𝐴)
4948mpteq2dv 4904 . . . . . 6 (𝑎 = 𝐼 → (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴) = (𝑥𝑋 ↦ ∏𝑖𝐼 𝐴))
5049oveq2d 6858 . . . . 5 (𝑎 = 𝐼 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝐼 𝐴)))
51 sumeq1 14704 . . . . . . 7 (𝑎 = 𝐼 → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
52 difeq1 3883 . . . . . . . . . . . 12 (𝑎 = 𝐼 → (𝑎 ∖ {𝑗}) = (𝐼 ∖ {𝑗}))
5352prodeq1d 14934 . . . . . . . . . . 11 (𝑎 = 𝐼 → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)
5453oveq2d 6858 . . . . . . . . . 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 3112 . . . . . . . 8 (𝑎 = 𝐼 → ∀𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))
5756sumeq2d 14717 . . . . . . 7 (𝑎 = 𝐼 → Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))
5851, 57eqtrd 2799 . . . . . 6 (𝑎 = 𝐼 → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))
5958mpteq2dv 4904 . . . . 5 (𝑎 = 𝐼 → (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)))
6050, 59eqeq12d 2780 . . . 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 335 . . 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 14956 . . . . . . . 8 𝑖 ∈ ∅ 𝐴 = 1
6362mpteq2i 4900 . . . . . . 7 (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴) = (𝑥𝑋 ↦ 1)
6463oveq2i 6853 . . . . . 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 6852 . . . . . . . 8 (𝐾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
7168, 70eqtri 2787 . . . . . . 7 𝐽 = ((TopOpen‘ℂfld) ↾t 𝑆)
7267, 71syl6eleq 2854 . . . . . 6 (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
73 1cnd 10288 . . . . . 6 (𝜑 → 1 ∈ ℂ)
7466, 72, 73dvmptconst 40767 . . . . 5 (𝜑 → (𝑆 D (𝑥𝑋 ↦ 1)) = (𝑥𝑋 ↦ 0))
75 sum0 14737 . . . . . . . 8 Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴) = 0
7675eqcomi 2774 . . . . . . 7 0 = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)
7776mpteq2i 4900 . . . . . 6 (𝑥𝑋 ↦ 0) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
7877a1i 11 . . . . 5 (𝜑 → (𝑥𝑋 ↦ 0) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
7965, 74, 783eqtrd 2803 . . . 4 (𝜑 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
8079adantr 472 . . 3 ((𝜑 ∧ ∅ ⊆ 𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
81 simp3 1168 . . . . 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 1255 . . . . 5 (((𝑏 ∈ Fin ∧ ¬ 𝑐𝑏) ∧ ((𝜑𝑏𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → ¬ 𝑐𝑏)
83 simpl 474 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → 𝜑)
84 ssun1 3938 . . . . . . . . . . 11 𝑏 ⊆ (𝑏 ∪ {𝑐})
85 sstr2 3768 . . . . . . . . . . 11 (𝑏 ⊆ (𝑏 ∪ {𝑐}) → ((𝑏 ∪ {𝑐}) ⊆ 𝐼𝑏𝐼))
8684, 85ax-mp 5 . . . . . . . . . 10 ((𝑏 ∪ {𝑐}) ⊆ 𝐼𝑏𝐼)
8786adantl 473 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → 𝑏𝐼)
8883, 87jca 507 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝜑𝑏𝐼))
8988adantl 473 . . . . . . 7 ((((𝜑𝑏𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → (𝜑𝑏𝐼))
90 simpl 474 . . . . . . 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 1160 . . . . 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 2009 . . . . . . 7 𝑥((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏)
94 nfcv 2907 . . . . . . . . 9 𝑥𝑆
95 nfcv 2907 . . . . . . . . 9 𝑥 D
96 nfmpt1 4906 . . . . . . . . 9 𝑥(𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)
9794, 95, 96nfov 6872 . . . . . . . 8 𝑥(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴))
98 nfmpt1 4906 . . . . . . . 8 𝑥(𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
9997, 98nfeq 2919 . . . . . . 7 𝑥(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
10093, 99nfan 1998 . . . . . 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 2009 . . . . . . . . 9 𝑖(𝑏 ∪ {𝑐}) ⊆ 𝐼
103101, 102nfan 1998 . . . . . . . 8 𝑖(𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)
104 nfv 2009 . . . . . . . 8 𝑖 ¬ 𝑐𝑏
105103, 104nfan 1998 . . . . . . 7 𝑖((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏)
106 nfcv 2907 . . . . . . . . 9 𝑖𝑆
107 nfcv 2907 . . . . . . . . 9 𝑖 D
108 nfcv 2907 . . . . . . . . . 10 𝑖𝑋
109 nfcv 2907 . . . . . . . . . . 11 𝑖𝑏
110109nfcprod1 14923 . . . . . . . . . 10 𝑖𝑖𝑏 𝐴
111108, 110nfmpt 4905 . . . . . . . . 9 𝑖(𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)
112106, 107, 111nfov 6872 . . . . . . . 8 𝑖(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴))
113 nfcv 2907 . . . . . . . . . . 11 𝑖𝐶
114 nfcv 2907 . . . . . . . . . . 11 𝑖 ·
115 nfcv 2907 . . . . . . . . . . . 12 𝑖(𝑏 ∖ {𝑗})
116115nfcprod1 14923 . . . . . . . . . . 11 𝑖𝑖 ∈ (𝑏 ∖ {𝑗})𝐴
117113, 114, 116nfov 6872 . . . . . . . . . 10 𝑖(𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
118109, 117nfsum 14706 . . . . . . . . 9 𝑖Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
119108, 118nfmpt 4905 . . . . . . . 8 𝑖(𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
120112, 119nfeq 2919 . . . . . . 7 𝑖(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
121105, 120nfan 1998 . . . . . 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 2009 . . . . . . . . 9 𝑗(𝑏 ∪ {𝑐}) ⊆ 𝐼
124122, 123nfan 1998 . . . . . . . 8 𝑗(𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)
125 nfv 2009 . . . . . . . 8 𝑗 ¬ 𝑐𝑏
126124, 125nfan 1998 . . . . . . 7 𝑗((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏)
127 nfcv 2907 . . . . . . . 8 𝑗(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴))
128 nfcv 2907 . . . . . . . . 9 𝑗𝑋
129 nfcv 2907 . . . . . . . . . 10 𝑗𝑏
130129nfsum1 14705 . . . . . . . . 9 𝑗Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
131128, 130nfmpt 4905 . . . . . . . 8 𝑗(𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
132127, 131nfeq 2919 . . . . . . 7 𝑗(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
133126, 132nfan 1998 . . . . . 6 𝑗(((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
134 nfcsb1v 3707 . . . . . 6 𝑖𝑐 / 𝑖𝐴
135 nfcsb1v 3707 . . . . . 6 𝑗𝑐 / 𝑗𝐶
13683ad2antrr 717 . . . . . . . 8 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝜑)
1371363ad2ant1 1163 . . . . . . 7 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑖𝐼𝑥𝑋) → 𝜑)
138 simp2 1167 . . . . . . 7 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑖𝐼𝑥𝑋) → 𝑖𝐼)
139 simp3 1168 . . . . . . 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 1490 . . . . . 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 717 . . . . . . 7 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑏𝐼)
144 ssfi 8387 . . . . . . 7 ((𝐼 ∈ Fin ∧ 𝑏𝐼) → 𝑏 ∈ Fin)
145142, 143, 144syl2anc 579 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑏 ∈ Fin)
146 vex 3353 . . . . . . 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 785 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → ¬ 𝑐𝑏)
149 simpllr 793 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → (𝑏 ∪ {𝑐}) ⊆ 𝐼)
15066ad3antrrr 721 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑆 ∈ {ℝ, ℂ})
151136ad2antrr 717 . . . . . . 7 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝜑)
152143ad2antrr 717 . . . . . . . 8 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝑏𝐼)
153 simpr 477 . . . . . . . 8 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝑗𝑏)
154152, 153sseldd 3762 . . . . . . 7 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝑗𝐼)
155 simplr 785 . . . . . . 7 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝑥𝑋)
156 nfv 2009 . . . . . . . . . 10 𝑖 𝑗𝐼
157 nfv 2009 . . . . . . . . . 10 𝑖 𝑥𝑋
158101, 156, 157nf3an 2000 . . . . . . . . 9 𝑖(𝜑𝑗𝐼𝑥𝑋)
159 nfv 2009 . . . . . . . . 9 𝑖 𝐶 ∈ ℂ
160158, 159nfim 1995 . . . . . . . 8 𝑖((𝜑𝑗𝐼𝑥𝑋) → 𝐶 ∈ ℂ)
161 eleq1w 2827 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑖𝐼𝑗𝐼))
1621613anbi2d 1565 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝜑𝑖𝐼𝑥𝑋) ↔ (𝜑𝑗𝐼𝑥𝑋)))
163 dvmptfprod.bc . . . . . . . . . 10 (𝑖 = 𝑗𝐵 = 𝐶)
164163eleq1d 2829 . . . . . . . . 9 (𝑖 = 𝑗 → (𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ))
165162, 164imbi12d 335 . . . . . . . 8 (𝑖 = 𝑗 → (((𝜑𝑖𝐼𝑥𝑋) → 𝐵 ∈ ℂ) ↔ ((𝜑𝑗𝐼𝑥𝑋) → 𝐶 ∈ ℂ)))
166 dvmptfprod.b . . . . . . . 8 ((𝜑𝑖𝐼𝑥𝑋) → 𝐵 ∈ ℂ)
167160, 165, 166chvar 2368 . . . . . . 7 ((𝜑𝑗𝐼𝑥𝑋) → 𝐶 ∈ ℂ)
168151, 154, 155, 167syl3anc 1490 . . . . . 6 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝐶 ∈ ℂ)
169 simpr 477 . . . . . 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 472 . . . . . . . . 9 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥𝑋) → 𝜑)
171 id 22 . . . . . . . . . . 11 ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑏 ∪ {𝑐}) ⊆ 𝐼)
172146snid 4366 . . . . . . . . . . . . 13 𝑐 ∈ {𝑐}
173 elun2 3943 . . . . . . . . . . . . 13 (𝑐 ∈ {𝑐} → 𝑐 ∈ (𝑏 ∪ {𝑐}))
174172, 173ax-mp 5 . . . . . . . . . . . 12 𝑐 ∈ (𝑏 ∪ {𝑐})
175174a1i 11 . . . . . . . . . . 11 ((𝑏 ∪ {𝑐}) ⊆ 𝐼𝑐 ∈ (𝑏 ∪ {𝑐}))
176171, 175sseldd 3762 . . . . . . . . . 10 ((𝑏 ∪ {𝑐}) ⊆ 𝐼𝑐𝐼)
177176ad2antlr 718 . . . . . . . . 9 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥𝑋) → 𝑐𝐼)
178 simpr 477 . . . . . . . . 9 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥𝑋) → 𝑥𝑋)
179 nfv 2009 . . . . . . . . . . . 12 𝑗 𝑐𝐼
180 nfv 2009 . . . . . . . . . . . 12 𝑗 𝑥𝑋
181122, 179, 180nf3an 2000 . . . . . . . . . . 11 𝑗(𝜑𝑐𝐼𝑥𝑋)
182 nfcv 2907 . . . . . . . . . . . 12 𝑗
183135, 182nfel 2920 . . . . . . . . . . 11 𝑗𝑐 / 𝑗𝐶 ∈ ℂ
184181, 183nfim 1995 . . . . . . . . . 10 𝑗((𝜑𝑐𝐼𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
185 eleq1w 2827 . . . . . . . . . . . 12 (𝑗 = 𝑐 → (𝑗𝐼𝑐𝐼))
1861853anbi2d 1565 . . . . . . . . . . 11 (𝑗 = 𝑐 → ((𝜑𝑗𝐼𝑥𝑋) ↔ (𝜑𝑐𝐼𝑥𝑋)))
187 csbeq1a 3700 . . . . . . . . . . . 12 (𝑗 = 𝑐𝐶 = 𝑐 / 𝑗𝐶)
188187eleq1d 2829 . . . . . . . . . . 11 (𝑗 = 𝑐 → (𝐶 ∈ ℂ ↔ 𝑐 / 𝑗𝐶 ∈ ℂ))
189186, 188imbi12d 335 . . . . . . . . . 10 (𝑗 = 𝑐 → (((𝜑𝑗𝐼𝑥𝑋) → 𝐶 ∈ ℂ) ↔ ((𝜑𝑐𝐼𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)))
190184, 189, 167chvar 2368 . . . . . . . . 9 ((𝜑𝑐𝐼𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
191170, 177, 178, 190syl3anc 1490 . . . . . . . 8 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
192191adantlr 706 . . . . . . 7 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ 𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
193192adantlr 706 . . . . . 6 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
194122, 179nfan 1998 . . . . . . . . . 10 𝑗(𝜑𝑐𝐼)
195 nfcv 2907 . . . . . . . . . . 11 𝑗(𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴))
196128, 135nfmpt 4905 . . . . . . . . . . 11 𝑗(𝑥𝑋𝑐 / 𝑗𝐶)
197195, 196nfeq 2919 . . . . . . . . . 10 𝑗(𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶)
198194, 197nfim 1995 . . . . . . . . 9 𝑗((𝜑𝑐𝐼) → (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶))
199185anbi2d 622 . . . . . . . . . 10 (𝑗 = 𝑐 → ((𝜑𝑗𝐼) ↔ (𝜑𝑐𝐼)))
200 csbeq1a 3700 . . . . . . . . . . . . . 14 (𝑗 = 𝑐𝑗 / 𝑖𝐴 = 𝑐 / 𝑗𝑗 / 𝑖𝐴)
201 csbco 3701 . . . . . . . . . . . . . . 15 𝑐 / 𝑗𝑗 / 𝑖𝐴 = 𝑐 / 𝑖𝐴
202201a1i 11 . . . . . . . . . . . . . 14 (𝑗 = 𝑐𝑐 / 𝑗𝑗 / 𝑖𝐴 = 𝑐 / 𝑖𝐴)
203200, 202eqtrd 2799 . . . . . . . . . . . . 13 (𝑗 = 𝑐𝑗 / 𝑖𝐴 = 𝑐 / 𝑖𝐴)
204203mpteq2dv 4904 . . . . . . . . . . . 12 (𝑗 = 𝑐 → (𝑥𝑋𝑗 / 𝑖𝐴) = (𝑥𝑋𝑐 / 𝑖𝐴))
205204oveq2d 6858 . . . . . . . . . . 11 (𝑗 = 𝑐 → (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)))
206187mpteq2dv 4904 . . . . . . . . . . 11 (𝑗 = 𝑐 → (𝑥𝑋𝐶) = (𝑥𝑋𝑐 / 𝑗𝐶))
207205, 206eqeq12d 2780 . . . . . . . . . 10 (𝑗 = 𝑐 → ((𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶) ↔ (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶)))
208199, 207imbi12d 335 . . . . . . . . 9 (𝑗 = 𝑐 → (((𝜑𝑗𝐼) → (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶)) ↔ ((𝜑𝑐𝐼) → (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶))))
209101, 156nfan 1998 . . . . . . . . . . 11 𝑖(𝜑𝑗𝐼)
210 nfcsb1v 3707 . . . . . . . . . . . . . 14 𝑖𝑗 / 𝑖𝐴
211108, 210nfmpt 4905 . . . . . . . . . . . . 13 𝑖(𝑥𝑋𝑗 / 𝑖𝐴)
212106, 107, 211nfov 6872 . . . . . . . . . . . 12 𝑖(𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴))
213 nfcv 2907 . . . . . . . . . . . 12 𝑖(𝑥𝑋𝐶)
214212, 213nfeq 2919 . . . . . . . . . . 11 𝑖(𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶)
215209, 214nfim 1995 . . . . . . . . . 10 𝑖((𝜑𝑗𝐼) → (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶))
216161anbi2d 622 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝜑𝑖𝐼) ↔ (𝜑𝑗𝐼)))
217 csbeq1a 3700 . . . . . . . . . . . . . 14 (𝑖 = 𝑗𝐴 = 𝑗 / 𝑖𝐴)
218217mpteq2dv 4904 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑥𝑋𝐴) = (𝑥𝑋𝑗 / 𝑖𝐴))
219218oveq2d 6858 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑆 D (𝑥𝑋𝐴)) = (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)))
220163idi 2 . . . . . . . . . . . . 13 (𝑖 = 𝑗𝐵 = 𝐶)
221220mpteq2dv 4904 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑥𝑋𝐵) = (𝑥𝑋𝐶))
222219, 221eqeq12d 2780 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵) ↔ (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶)))
223216, 222imbi12d 335 . . . . . . . . . 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, 224chvar 2368 . . . . . . . . 9 ((𝜑𝑗𝐼) → (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶))
226198, 208, 225chvar 2368 . . . . . . . 8 ((𝜑𝑐𝐼) → (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶))
227176, 226sylan2 586 . . . . . . 7 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶))
228227ad2antrr 717 . . . . . 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 3700 . . . . . 6 (𝑖 = 𝑐𝐴 = 𝑐 / 𝑖𝐴)
230100, 121, 133, 134, 135, 141, 145, 147, 148, 149, 150, 168, 169, 193, 228, 229, 187dvmptfprodlem 40797 . . . . 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 866 . . . 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 1148 . . 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 8408 . 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 384  w3a 1107   = wceq 1652  wnf 1878  wcel 2155  Vcvv 3350  csb 3691  cdif 3729  cun 3730  wss 3732  c0 4079  {csn 4334  {cpr 4336  cmpt 4888  cfv 6068  (class class class)co 6842  Fincfn 8160  cc 10187  cr 10188  0cc0 10189  1c1 10190   · cmul 10194  Σcsu 14701  cprod 14918  t crest 16347  TopOpenctopn 16348  fldccnfld 20019   D cdv 23918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266  ax-pre-sup 10267  ax-addf 10268  ax-mulf 10269
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-fi 8524  df-sup 8555  df-inf 8556  df-oi 8622  df-card 9016  df-cda 9243  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-div 10939  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-q 11990  df-rp 12029  df-xneg 12146  df-xadd 12147  df-xmul 12148  df-icc 12384  df-fz 12534  df-fzo 12674  df-seq 13009  df-exp 13068  df-hash 13322  df-cj 14124  df-re 14125  df-im 14126  df-sqrt 14260  df-abs 14261  df-clim 14504  df-sum 14702  df-prod 14919  df-struct 16132  df-ndx 16133  df-slot 16134  df-base 16136  df-sets 16137  df-ress 16138  df-plusg 16227  df-mulr 16228  df-starv 16229  df-sca 16230  df-vsca 16231  df-ip 16232  df-tset 16233  df-ple 16234  df-ds 16236  df-unif 16237  df-hom 16238  df-cco 16239  df-rest 16349  df-topn 16350  df-0g 16368  df-gsum 16369  df-topgen 16370  df-pt 16371  df-prds 16374  df-xrs 16428  df-qtop 16433  df-imas 16434  df-xps 16436  df-mre 16512  df-mrc 16513  df-acs 16515  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-submnd 17602  df-mulg 17808  df-cntz 18013  df-cmn 18461  df-psmet 20011  df-xmet 20012  df-met 20013  df-bl 20014  df-mopn 20015  df-fbas 20016  df-fg 20017  df-cnfld 20020  df-top 20978  df-topon 20995  df-topsp 21017  df-bases 21030  df-cld 21103  df-ntr 21104  df-cls 21105  df-nei 21182  df-lp 21220  df-perf 21221  df-cn 21311  df-cnp 21312  df-haus 21399  df-tx 21645  df-hmeo 21838  df-fil 21929  df-fm 22021  df-flim 22022  df-flf 22023  df-xms 22404  df-ms 22405  df-tms 22406  df-cncf 22960  df-limc 23921  df-dv 23922
This theorem is referenced by: (None)
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