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Theorem dvmptfprod 44176
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 3966 . . 3 𝐼𝐼
32jctr 525 . 2 (𝜑 → (𝜑𝐼𝐼))
4 sseq1 3969 . . . . 5 (𝑎 = ∅ → (𝑎𝐼 ↔ ∅ ⊆ 𝐼))
54anbi2d 629 . . . 4 (𝑎 = ∅ → ((𝜑𝑎𝐼) ↔ (𝜑 ∧ ∅ ⊆ 𝐼)))
6 prodeq1 15792 . . . . . . 7 (𝑎 = ∅ → ∏𝑖𝑎 𝐴 = ∏𝑖 ∈ ∅ 𝐴)
76mpteq2dv 5207 . . . . . 6 (𝑎 = ∅ → (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴) = (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴))
87oveq2d 7373 . . . . 5 (𝑎 = ∅ → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)))
9 sumeq1 15573 . . . . . . 7 (𝑎 = ∅ → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
10 difeq1 4075 . . . . . . . . . 10 (𝑎 = ∅ → (𝑎 ∖ {𝑗}) = (∅ ∖ {𝑗}))
1110prodeq1d 15804 . . . . . . . . 9 (𝑎 = ∅ → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)
1211oveq2d 7373 . . . . . . . 8 (𝑎 = ∅ → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
1312sumeq2sdv 15589 . . . . . . 7 (𝑎 = ∅ → Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
149, 13eqtrd 2776 . . . . . 6 (𝑎 = ∅ → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
1514mpteq2dv 5207 . . . . 5 (𝑎 = ∅ → (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
168, 15eqeq12d 2752 . . . 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 344 . . 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 3969 . . . . 5 (𝑎 = 𝑏 → (𝑎𝐼𝑏𝐼))
1918anbi2d 629 . . . 4 (𝑎 = 𝑏 → ((𝜑𝑎𝐼) ↔ (𝜑𝑏𝐼)))
20 prodeq1 15792 . . . . . . 7 (𝑎 = 𝑏 → ∏𝑖𝑎 𝐴 = ∏𝑖𝑏 𝐴)
2120mpteq2dv 5207 . . . . . 6 (𝑎 = 𝑏 → (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴) = (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴))
2221oveq2d 7373 . . . . 5 (𝑎 = 𝑏 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)))
23 sumeq1 15573 . . . . . . 7 (𝑎 = 𝑏 → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
24 difeq1 4075 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝑎 ∖ {𝑗}) = (𝑏 ∖ {𝑗}))
2524prodeq1d 15804 . . . . . . . . 9 (𝑎 = 𝑏 → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
2625oveq2d 7373 . . . . . . . 8 (𝑎 = 𝑏 → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
2726sumeq2sdv 15589 . . . . . . 7 (𝑎 = 𝑏 → Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
2823, 27eqtrd 2776 . . . . . 6 (𝑎 = 𝑏 → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
2928mpteq2dv 5207 . . . . 5 (𝑎 = 𝑏 → (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
3022, 29eqeq12d 2752 . . . 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 344 . . 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 3969 . . . . 5 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎𝐼 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝐼))
3332anbi2d 629 . . . 4 (𝑎 = (𝑏 ∪ {𝑐}) → ((𝜑𝑎𝐼) ↔ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)))
34 prodeq1 15792 . . . . . . 7 (𝑎 = (𝑏 ∪ {𝑐}) → ∏𝑖𝑎 𝐴 = ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)
3534mpteq2dv 5207 . . . . . 6 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴) = (𝑥𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴))
3635oveq2d 7373 . . . . 5 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)))
37 sumeq1 15573 . . . . . . 7 (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
38 difeq1 4075 . . . . . . . . . 10 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ∖ {𝑗}) = ((𝑏 ∪ {𝑐}) ∖ {𝑗}))
3938prodeq1d 15804 . . . . . . . . 9 (𝑎 = (𝑏 ∪ {𝑐}) → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)
4039oveq2d 7373 . . . . . . . 8 (𝑎 = (𝑏 ∪ {𝑐}) → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))
4140sumeq2sdv 15589 . . . . . . 7 (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))
4237, 41eqtrd 2776 . . . . . 6 (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))
4342mpteq2dv 5207 . . . . 5 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)))
4436, 43eqeq12d 2752 . . . 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 344 . . 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 3969 . . . . 5 (𝑎 = 𝐼 → (𝑎𝐼𝐼𝐼))
4746anbi2d 629 . . . 4 (𝑎 = 𝐼 → ((𝜑𝑎𝐼) ↔ (𝜑𝐼𝐼)))
48 prodeq1 15792 . . . . . . 7 (𝑎 = 𝐼 → ∏𝑖𝑎 𝐴 = ∏𝑖𝐼 𝐴)
4948mpteq2dv 5207 . . . . . 6 (𝑎 = 𝐼 → (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴) = (𝑥𝑋 ↦ ∏𝑖𝐼 𝐴))
5049oveq2d 7373 . . . . 5 (𝑎 = 𝐼 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝐼 𝐴)))
51 sumeq1 15573 . . . . . . 7 (𝑎 = 𝐼 → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
52 difeq1 4075 . . . . . . . . . . . 12 (𝑎 = 𝐼 → (𝑎 ∖ {𝑗}) = (𝐼 ∖ {𝑗}))
5352prodeq1d 15804 . . . . . . . . . . 11 (𝑎 = 𝐼 → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)
5453oveq2d 7373 . . . . . . . . . 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 3142 . . . . . . . 8 (𝑎 = 𝐼 → ∀𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))
5756sumeq2d 15587 . . . . . . 7 (𝑎 = 𝐼 → Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))
5851, 57eqtrd 2776 . . . . . 6 (𝑎 = 𝐼 → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))
5958mpteq2dv 5207 . . . . 5 (𝑎 = 𝐼 → (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)))
6050, 59eqeq12d 2752 . . . 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 344 . . 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 15826 . . . . . . . 8 𝑖 ∈ ∅ 𝐴 = 1
6362mpteq2i 5210 . . . . . . 7 (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴) = (𝑥𝑋 ↦ 1)
6463oveq2i 7368 . . . . . 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 7367 . . . . . . . 8 (𝐾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
7168, 70eqtri 2764 . . . . . . 7 𝐽 = ((TopOpen‘ℂfld) ↾t 𝑆)
7267, 71eleqtrdi 2848 . . . . . 6 (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
73 1cnd 11150 . . . . . 6 (𝜑 → 1 ∈ ℂ)
7466, 72, 73dvmptconst 44146 . . . . 5 (𝜑 → (𝑆 D (𝑥𝑋 ↦ 1)) = (𝑥𝑋 ↦ 0))
75 sum0 15606 . . . . . . . 8 Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴) = 0
7675eqcomi 2745 . . . . . . 7 0 = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)
7776mpteq2i 5210 . . . . . 6 (𝑥𝑋 ↦ 0) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
7877a1i 11 . . . . 5 (𝜑 → (𝑥𝑋 ↦ 0) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
7965, 74, 783eqtrd 2780 . . . 4 (𝜑 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
8079adantr 481 . . 3 ((𝜑 ∧ ∅ ⊆ 𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
81 simp3 1138 . . . . 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 1198 . . . . 5 (((𝑏 ∈ Fin ∧ ¬ 𝑐𝑏) ∧ ((𝜑𝑏𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → ¬ 𝑐𝑏)
83 simpl 483 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → 𝜑)
84 ssun1 4132 . . . . . . . . . . 11 𝑏 ⊆ (𝑏 ∪ {𝑐})
85 sstr2 3951 . . . . . . . . . . 11 (𝑏 ⊆ (𝑏 ∪ {𝑐}) → ((𝑏 ∪ {𝑐}) ⊆ 𝐼𝑏𝐼))
8684, 85ax-mp 5 . . . . . . . . . 10 ((𝑏 ∪ {𝑐}) ⊆ 𝐼𝑏𝐼)
8786adantl 482 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → 𝑏𝐼)
8883, 87jca 512 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝜑𝑏𝐼))
8988adantl 482 . . . . . . 7 ((((𝜑𝑏𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → (𝜑𝑏𝐼))
90 simpl 483 . . . . . . 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 1130 . . . . 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 1917 . . . . . . 7 𝑥((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏)
94 nfcv 2907 . . . . . . . . 9 𝑥𝑆
95 nfcv 2907 . . . . . . . . 9 𝑥 D
96 nfmpt1 5213 . . . . . . . . 9 𝑥(𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)
9794, 95, 96nfov 7387 . . . . . . . 8 𝑥(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴))
98 nfmpt1 5213 . . . . . . . 8 𝑥(𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
9997, 98nfeq 2920 . . . . . . 7 𝑥(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
10093, 99nfan 1902 . . . . . 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 1917 . . . . . . . . 9 𝑖(𝑏 ∪ {𝑐}) ⊆ 𝐼
103101, 102nfan 1902 . . . . . . . 8 𝑖(𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)
104 nfv 1917 . . . . . . . 8 𝑖 ¬ 𝑐𝑏
105103, 104nfan 1902 . . . . . . 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 15793 . . . . . . . . . 10 𝑖𝑖𝑏 𝐴
111108, 110nfmpt 5212 . . . . . . . . 9 𝑖(𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)
112106, 107, 111nfov 7387 . . . . . . . 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 15793 . . . . . . . . . . 11 𝑖𝑖 ∈ (𝑏 ∖ {𝑗})𝐴
117113, 114, 116nfov 7387 . . . . . . . . . 10 𝑖(𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
118109, 117nfsum 15575 . . . . . . . . 9 𝑖Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
119108, 118nfmpt 5212 . . . . . . . 8 𝑖(𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
120112, 119nfeq 2920 . . . . . . 7 𝑖(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
121105, 120nfan 1902 . . . . . 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 1917 . . . . . . . . 9 𝑗(𝑏 ∪ {𝑐}) ⊆ 𝐼
124122, 123nfan 1902 . . . . . . . 8 𝑗(𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)
125 nfv 1917 . . . . . . . 8 𝑗 ¬ 𝑐𝑏
126124, 125nfan 1902 . . . . . . 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 15574 . . . . . . . . 9 𝑗Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
131128, 130nfmpt 5212 . . . . . . . 8 𝑗(𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
132127, 131nfeq 2920 . . . . . . 7 𝑗(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
133126, 132nfan 1902 . . . . . 6 𝑗(((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
134 nfcsb1v 3880 . . . . . 6 𝑖𝑐 / 𝑖𝐴
135 nfcsb1v 3880 . . . . . 6 𝑗𝑐 / 𝑗𝐶
13683ad2antrr 724 . . . . . . . 8 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝜑)
1371363ad2ant1 1133 . . . . . . 7 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑖𝐼𝑥𝑋) → 𝜑)
138 simp2 1137 . . . . . . 7 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑖𝐼𝑥𝑋) → 𝑖𝐼)
139 simp3 1138 . . . . . . 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 1371 . . . . . 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 724 . . . . . . 7 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑏𝐼)
144 ssfi 9117 . . . . . . 7 ((𝐼 ∈ Fin ∧ 𝑏𝐼) → 𝑏 ∈ Fin)
145142, 143, 144syl2anc 584 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑏 ∈ Fin)
146 vex 3449 . . . . . . 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 767 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → ¬ 𝑐𝑏)
149 simpllr 774 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → (𝑏 ∪ {𝑐}) ⊆ 𝐼)
15066ad3antrrr 728 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑆 ∈ {ℝ, ℂ})
151136ad2antrr 724 . . . . . . 7 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝜑)
152143ad2antrr 724 . . . . . . . 8 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝑏𝐼)
153 simpr 485 . . . . . . . 8 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝑗𝑏)
154152, 153sseldd 3945 . . . . . . 7 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝑗𝐼)
155 simplr 767 . . . . . . 7 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝑥𝑋)
156 nfv 1917 . . . . . . . . . 10 𝑖 𝑗𝐼
157 nfv 1917 . . . . . . . . . 10 𝑖 𝑥𝑋
158101, 156, 157nf3an 1904 . . . . . . . . 9 𝑖(𝜑𝑗𝐼𝑥𝑋)
159 nfv 1917 . . . . . . . . 9 𝑖 𝐶 ∈ ℂ
160158, 159nfim 1899 . . . . . . . 8 𝑖((𝜑𝑗𝐼𝑥𝑋) → 𝐶 ∈ ℂ)
161 eleq1w 2820 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑖𝐼𝑗𝐼))
1621613anbi2d 1441 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝜑𝑖𝐼𝑥𝑋) ↔ (𝜑𝑗𝐼𝑥𝑋)))
163 dvmptfprod.bc . . . . . . . . . 10 (𝑖 = 𝑗𝐵 = 𝐶)
164163eleq1d 2822 . . . . . . . . 9 (𝑖 = 𝑗 → (𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ))
165162, 164imbi12d 344 . . . . . . . 8 (𝑖 = 𝑗 → (((𝜑𝑖𝐼𝑥𝑋) → 𝐵 ∈ ℂ) ↔ ((𝜑𝑗𝐼𝑥𝑋) → 𝐶 ∈ ℂ)))
166 dvmptfprod.b . . . . . . . 8 ((𝜑𝑖𝐼𝑥𝑋) → 𝐵 ∈ ℂ)
167160, 165, 166chvarfv 2233 . . . . . . 7 ((𝜑𝑗𝐼𝑥𝑋) → 𝐶 ∈ ℂ)
168151, 154, 155, 167syl3anc 1371 . . . . . 6 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝐶 ∈ ℂ)
169 simpr 485 . . . . . 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 481 . . . . . . . . 9 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥𝑋) → 𝜑)
171 id 22 . . . . . . . . . . 11 ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑏 ∪ {𝑐}) ⊆ 𝐼)
172146snid 4622 . . . . . . . . . . . . 13 𝑐 ∈ {𝑐}
173 elun2 4137 . . . . . . . . . . . . 13 (𝑐 ∈ {𝑐} → 𝑐 ∈ (𝑏 ∪ {𝑐}))
174172, 173ax-mp 5 . . . . . . . . . . . 12 𝑐 ∈ (𝑏 ∪ {𝑐})
175174a1i 11 . . . . . . . . . . 11 ((𝑏 ∪ {𝑐}) ⊆ 𝐼𝑐 ∈ (𝑏 ∪ {𝑐}))
176171, 175sseldd 3945 . . . . . . . . . 10 ((𝑏 ∪ {𝑐}) ⊆ 𝐼𝑐𝐼)
177176ad2antlr 725 . . . . . . . . 9 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥𝑋) → 𝑐𝐼)
178 simpr 485 . . . . . . . . 9 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥𝑋) → 𝑥𝑋)
179 nfv 1917 . . . . . . . . . . . 12 𝑗 𝑐𝐼
180 nfv 1917 . . . . . . . . . . . 12 𝑗 𝑥𝑋
181122, 179, 180nf3an 1904 . . . . . . . . . . 11 𝑗(𝜑𝑐𝐼𝑥𝑋)
182 nfcv 2907 . . . . . . . . . . . 12 𝑗
183135, 182nfel 2921 . . . . . . . . . . 11 𝑗𝑐 / 𝑗𝐶 ∈ ℂ
184181, 183nfim 1899 . . . . . . . . . 10 𝑗((𝜑𝑐𝐼𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
185 eleq1w 2820 . . . . . . . . . . . 12 (𝑗 = 𝑐 → (𝑗𝐼𝑐𝐼))
1861853anbi2d 1441 . . . . . . . . . . 11 (𝑗 = 𝑐 → ((𝜑𝑗𝐼𝑥𝑋) ↔ (𝜑𝑐𝐼𝑥𝑋)))
187 csbeq1a 3869 . . . . . . . . . . . 12 (𝑗 = 𝑐𝐶 = 𝑐 / 𝑗𝐶)
188187eleq1d 2822 . . . . . . . . . . 11 (𝑗 = 𝑐 → (𝐶 ∈ ℂ ↔ 𝑐 / 𝑗𝐶 ∈ ℂ))
189186, 188imbi12d 344 . . . . . . . . . 10 (𝑗 = 𝑐 → (((𝜑𝑗𝐼𝑥𝑋) → 𝐶 ∈ ℂ) ↔ ((𝜑𝑐𝐼𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)))
190184, 189, 167chvarfv 2233 . . . . . . . . 9 ((𝜑𝑐𝐼𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
191170, 177, 178, 190syl3anc 1371 . . . . . . . 8 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
192191adantlr 713 . . . . . . 7 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ 𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
193192adantlr 713 . . . . . 6 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
194122, 179nfan 1902 . . . . . . . . . 10 𝑗(𝜑𝑐𝐼)
195 nfcv 2907 . . . . . . . . . . 11 𝑗(𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴))
196128, 135nfmpt 5212 . . . . . . . . . . 11 𝑗(𝑥𝑋𝑐 / 𝑗𝐶)
197195, 196nfeq 2920 . . . . . . . . . 10 𝑗(𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶)
198194, 197nfim 1899 . . . . . . . . 9 𝑗((𝜑𝑐𝐼) → (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶))
199185anbi2d 629 . . . . . . . . . 10 (𝑗 = 𝑐 → ((𝜑𝑗𝐼) ↔ (𝜑𝑐𝐼)))
200 csbeq1a 3869 . . . . . . . . . . . . . 14 (𝑗 = 𝑐𝑗 / 𝑖𝐴 = 𝑐 / 𝑗𝑗 / 𝑖𝐴)
201 csbcow 3870 . . . . . . . . . . . . . . 15 𝑐 / 𝑗𝑗 / 𝑖𝐴 = 𝑐 / 𝑖𝐴
202201a1i 11 . . . . . . . . . . . . . 14 (𝑗 = 𝑐𝑐 / 𝑗𝑗 / 𝑖𝐴 = 𝑐 / 𝑖𝐴)
203200, 202eqtrd 2776 . . . . . . . . . . . . 13 (𝑗 = 𝑐𝑗 / 𝑖𝐴 = 𝑐 / 𝑖𝐴)
204203mpteq2dv 5207 . . . . . . . . . . . 12 (𝑗 = 𝑐 → (𝑥𝑋𝑗 / 𝑖𝐴) = (𝑥𝑋𝑐 / 𝑖𝐴))
205204oveq2d 7373 . . . . . . . . . . 11 (𝑗 = 𝑐 → (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)))
206187mpteq2dv 5207 . . . . . . . . . . 11 (𝑗 = 𝑐 → (𝑥𝑋𝐶) = (𝑥𝑋𝑐 / 𝑗𝐶))
207205, 206eqeq12d 2752 . . . . . . . . . 10 (𝑗 = 𝑐 → ((𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶) ↔ (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶)))
208199, 207imbi12d 344 . . . . . . . . 9 (𝑗 = 𝑐 → (((𝜑𝑗𝐼) → (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶)) ↔ ((𝜑𝑐𝐼) → (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶))))
209101, 156nfan 1902 . . . . . . . . . . 11 𝑖(𝜑𝑗𝐼)
210 nfcsb1v 3880 . . . . . . . . . . . . . 14 𝑖𝑗 / 𝑖𝐴
211108, 210nfmpt 5212 . . . . . . . . . . . . 13 𝑖(𝑥𝑋𝑗 / 𝑖𝐴)
212106, 107, 211nfov 7387 . . . . . . . . . . . 12 𝑖(𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴))
213 nfcv 2907 . . . . . . . . . . . 12 𝑖(𝑥𝑋𝐶)
214212, 213nfeq 2920 . . . . . . . . . . 11 𝑖(𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶)
215209, 214nfim 1899 . . . . . . . . . 10 𝑖((𝜑𝑗𝐼) → (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶))
216161anbi2d 629 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝜑𝑖𝐼) ↔ (𝜑𝑗𝐼)))
217 csbeq1a 3869 . . . . . . . . . . . . . 14 (𝑖 = 𝑗𝐴 = 𝑗 / 𝑖𝐴)
218217mpteq2dv 5207 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑥𝑋𝐴) = (𝑥𝑋𝑗 / 𝑖𝐴))
219218oveq2d 7373 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑆 D (𝑥𝑋𝐴)) = (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)))
220163idi 1 . . . . . . . . . . . . 13 (𝑖 = 𝑗𝐵 = 𝐶)
221220mpteq2dv 5207 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑥𝑋𝐵) = (𝑥𝑋𝐶))
222219, 221eqeq12d 2752 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵) ↔ (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶)))
223216, 222imbi12d 344 . . . . . . . . . 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 2233 . . . . . . . . 9 ((𝜑𝑗𝐼) → (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶))
226198, 208, 225chvarfv 2233 . . . . . . . 8 ((𝜑𝑐𝐼) → (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶))
227176, 226sylan2 593 . . . . . . 7 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶))
228227ad2antrr 724 . . . . . 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 3869 . . . . . 6 (𝑖 = 𝑐𝐴 = 𝑐 / 𝑖𝐴)
230100, 121, 133, 134, 135, 141, 145, 147, 148, 149, 150, 168, 169, 193, 228, 229, 187dvmptfprodlem 44175 . . . . 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 836 . . . 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 1119 . . 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 9109 . 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 396  w3a 1087   = wceq 1541  wnf 1785  wcel 2106  Vcvv 3445  csb 3855  cdif 3907  cun 3908  wss 3910  c0 4282  {csn 4586  {cpr 4588  cmpt 5188  cfv 6496  (class class class)co 7357  Fincfn 8883  cc 11049  cr 11050  0cc0 11051  1c1 11052   · cmul 11056  Σcsu 15570  cprod 15788  t crest 17302  TopOpenctopn 17303  fldccnfld 20796   D cdv 25227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-fi 9347  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-icc 13271  df-fz 13425  df-fzo 13568  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-clim 15370  df-sum 15571  df-prod 15789  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-rest 17304  df-topn 17305  df-0g 17323  df-gsum 17324  df-topgen 17325  df-pt 17326  df-prds 17329  df-xrs 17384  df-qtop 17389  df-imas 17390  df-xps 17392  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-submnd 18602  df-mulg 18873  df-cntz 19097  df-cmn 19564  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-cnfld 20797  df-top 22243  df-topon 22260  df-topsp 22282  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-lp 22487  df-perf 22488  df-cn 22578  df-cnp 22579  df-haus 22666  df-tx 22913  df-hmeo 23106  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-xms 23673  df-ms 23674  df-tms 23675  df-cncf 24241  df-limc 25230  df-dv 25231
This theorem is referenced by: (None)
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