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Theorem dvmptfprod 39466
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 3603 . . 3 𝐼𝐼
32jctr 564 . 2 (𝜑 → (𝜑𝐼𝐼))
4 sseq1 3605 . . . . 5 (𝑎 = ∅ → (𝑎𝐼 ↔ ∅ ⊆ 𝐼))
54anbi2d 739 . . . 4 (𝑎 = ∅ → ((𝜑𝑎𝐼) ↔ (𝜑 ∧ ∅ ⊆ 𝐼)))
6 prodeq1 14564 . . . . . . 7 (𝑎 = ∅ → ∏𝑖𝑎 𝐴 = ∏𝑖 ∈ ∅ 𝐴)
76mpteq2dv 4705 . . . . . 6 (𝑎 = ∅ → (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴) = (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴))
87oveq2d 6620 . . . . 5 (𝑎 = ∅ → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)))
9 sumeq1 14353 . . . . . . 7 (𝑎 = ∅ → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
10 difeq1 3699 . . . . . . . . . 10 (𝑎 = ∅ → (𝑎 ∖ {𝑗}) = (∅ ∖ {𝑗}))
1110prodeq1d 14576 . . . . . . . . 9 (𝑎 = ∅ → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)
1211oveq2d 6620 . . . . . . . 8 (𝑎 = ∅ → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
1312sumeq2sdv 14368 . . . . . . 7 (𝑎 = ∅ → Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
149, 13eqtrd 2655 . . . . . 6 (𝑎 = ∅ → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
1514mpteq2dv 4705 . . . . 5 (𝑎 = ∅ → (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
168, 15eqeq12d 2636 . . . 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 334 . . 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 3605 . . . . 5 (𝑎 = 𝑏 → (𝑎𝐼𝑏𝐼))
1918anbi2d 739 . . . 4 (𝑎 = 𝑏 → ((𝜑𝑎𝐼) ↔ (𝜑𝑏𝐼)))
20 prodeq1 14564 . . . . . . 7 (𝑎 = 𝑏 → ∏𝑖𝑎 𝐴 = ∏𝑖𝑏 𝐴)
2120mpteq2dv 4705 . . . . . 6 (𝑎 = 𝑏 → (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴) = (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴))
2221oveq2d 6620 . . . . 5 (𝑎 = 𝑏 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)))
23 sumeq1 14353 . . . . . . 7 (𝑎 = 𝑏 → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
24 difeq1 3699 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝑎 ∖ {𝑗}) = (𝑏 ∖ {𝑗}))
2524prodeq1d 14576 . . . . . . . . 9 (𝑎 = 𝑏 → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
2625oveq2d 6620 . . . . . . . 8 (𝑎 = 𝑏 → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
2726sumeq2sdv 14368 . . . . . . 7 (𝑎 = 𝑏 → Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
2823, 27eqtrd 2655 . . . . . 6 (𝑎 = 𝑏 → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
2928mpteq2dv 4705 . . . . 5 (𝑎 = 𝑏 → (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
3022, 29eqeq12d 2636 . . . 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 334 . . 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 3605 . . . . 5 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎𝐼 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝐼))
3332anbi2d 739 . . . 4 (𝑎 = (𝑏 ∪ {𝑐}) → ((𝜑𝑎𝐼) ↔ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)))
34 prodeq1 14564 . . . . . . 7 (𝑎 = (𝑏 ∪ {𝑐}) → ∏𝑖𝑎 𝐴 = ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)
3534mpteq2dv 4705 . . . . . 6 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴) = (𝑥𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴))
3635oveq2d 6620 . . . . 5 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)))
37 sumeq1 14353 . . . . . . 7 (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
38 difeq1 3699 . . . . . . . . . 10 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ∖ {𝑗}) = ((𝑏 ∪ {𝑐}) ∖ {𝑗}))
3938prodeq1d 14576 . . . . . . . . 9 (𝑎 = (𝑏 ∪ {𝑐}) → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)
4039oveq2d 6620 . . . . . . . 8 (𝑎 = (𝑏 ∪ {𝑐}) → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))
4140sumeq2sdv 14368 . . . . . . 7 (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))
4237, 41eqtrd 2655 . . . . . 6 (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))
4342mpteq2dv 4705 . . . . 5 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)))
4436, 43eqeq12d 2636 . . . 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 334 . . 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 3605 . . . . 5 (𝑎 = 𝐼 → (𝑎𝐼𝐼𝐼))
4746anbi2d 739 . . . 4 (𝑎 = 𝐼 → ((𝜑𝑎𝐼) ↔ (𝜑𝐼𝐼)))
48 prodeq1 14564 . . . . . . 7 (𝑎 = 𝐼 → ∏𝑖𝑎 𝐴 = ∏𝑖𝐼 𝐴)
4948mpteq2dv 4705 . . . . . 6 (𝑎 = 𝐼 → (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴) = (𝑥𝑋 ↦ ∏𝑖𝐼 𝐴))
5049oveq2d 6620 . . . . 5 (𝑎 = 𝐼 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝐼 𝐴)))
51 sumeq1 14353 . . . . . . 7 (𝑎 = 𝐼 → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
52 difeq1 3699 . . . . . . . . . . . 12 (𝑎 = 𝐼 → (𝑎 ∖ {𝑗}) = (𝐼 ∖ {𝑗}))
5352prodeq1d 14576 . . . . . . . . . . 11 (𝑎 = 𝐼 → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)
5453oveq2d 6620 . . . . . . . . . 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 2959 . . . . . . . 8 (𝑎 = 𝐼 → ∀𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))
5756sumeq2d 14366 . . . . . . 7 (𝑎 = 𝐼 → Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))
5851, 57eqtrd 2655 . . . . . 6 (𝑎 = 𝐼 → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))
5958mpteq2dv 4705 . . . . 5 (𝑎 = 𝐼 → (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)))
6050, 59eqeq12d 2636 . . . 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 334 . . 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 14598 . . . . . . . 8 𝑖 ∈ ∅ 𝐴 = 1
6362mpteq2i 4701 . . . . . . 7 (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴) = (𝑥𝑋 ↦ 1)
6463oveq2i 6615 . . . . . 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 6614 . . . . . . . 8 (𝐾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
7168, 70eqtri 2643 . . . . . . 7 𝐽 = ((TopOpen‘ℂfld) ↾t 𝑆)
7267, 71syl6eleq 2708 . . . . . 6 (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
73 1cnd 10000 . . . . . 6 (𝜑 → 1 ∈ ℂ)
7466, 72, 73dvmptconst 39434 . . . . 5 (𝜑 → (𝑆 D (𝑥𝑋 ↦ 1)) = (𝑥𝑋 ↦ 0))
75 sum0 14385 . . . . . . . 8 Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴) = 0
7675eqcomi 2630 . . . . . . 7 0 = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)
7776mpteq2i 4701 . . . . . 6 (𝑥𝑋 ↦ 0) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
7877a1i 11 . . . . 5 (𝜑 → (𝑥𝑋 ↦ 0) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
7965, 74, 783eqtrd 2659 . . . 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 1061 . . . . 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 1084 . . . . 5 (((𝑏 ∈ Fin ∧ ¬ 𝑐𝑏) ∧ ((𝜑𝑏𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → ¬ 𝑐𝑏)
83 simpl 473 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → 𝜑)
84 ssun1 3754 . . . . . . . . . . 11 𝑏 ⊆ (𝑏 ∪ {𝑐})
85 sstr2 3590 . . . . . . . . . . 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 554 . . . . . . . 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 473 . . . . . . 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 1077 . . . . 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 1840 . . . . . . 7 𝑥((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏)
94 nfcv 2761 . . . . . . . . 9 𝑥𝑆
95 nfcv 2761 . . . . . . . . 9 𝑥 D
96 nfmpt1 4707 . . . . . . . . 9 𝑥(𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)
9794, 95, 96nfov 6630 . . . . . . . 8 𝑥(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴))
98 nfmpt1 4707 . . . . . . . 8 𝑥(𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
9997, 98nfeq 2772 . . . . . . 7 𝑥(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
10093, 99nfan 1825 . . . . . 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 1840 . . . . . . . . 9 𝑖(𝑏 ∪ {𝑐}) ⊆ 𝐼
103101, 102nfan 1825 . . . . . . . 8 𝑖(𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)
104 nfv 1840 . . . . . . . 8 𝑖 ¬ 𝑐𝑏
105103, 104nfan 1825 . . . . . . 7 𝑖((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏)
106 nfcv 2761 . . . . . . . . 9 𝑖𝑆
107 nfcv 2761 . . . . . . . . 9 𝑖 D
108 nfcv 2761 . . . . . . . . . 10 𝑖𝑋
109 nfcv 2761 . . . . . . . . . . 11 𝑖𝑏
110109nfcprod1 14565 . . . . . . . . . 10 𝑖𝑖𝑏 𝐴
111108, 110nfmpt 4706 . . . . . . . . 9 𝑖(𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)
112106, 107, 111nfov 6630 . . . . . . . 8 𝑖(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴))
113 nfcv 2761 . . . . . . . . . . 11 𝑖𝐶
114 nfcv 2761 . . . . . . . . . . 11 𝑖 ·
115 nfcv 2761 . . . . . . . . . . . 12 𝑖(𝑏 ∖ {𝑗})
116115nfcprod1 14565 . . . . . . . . . . 11 𝑖𝑖 ∈ (𝑏 ∖ {𝑗})𝐴
117113, 114, 116nfov 6630 . . . . . . . . . 10 𝑖(𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
118109, 117nfsum 14355 . . . . . . . . 9 𝑖Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
119108, 118nfmpt 4706 . . . . . . . 8 𝑖(𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
120112, 119nfeq 2772 . . . . . . 7 𝑖(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
121105, 120nfan 1825 . . . . . 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 1840 . . . . . . . . 9 𝑗(𝑏 ∪ {𝑐}) ⊆ 𝐼
124122, 123nfan 1825 . . . . . . . 8 𝑗(𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)
125 nfv 1840 . . . . . . . 8 𝑗 ¬ 𝑐𝑏
126124, 125nfan 1825 . . . . . . 7 𝑗((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏)
127 nfcv 2761 . . . . . . . 8 𝑗(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴))
128 nfcv 2761 . . . . . . . . 9 𝑗𝑋
129 nfcv 2761 . . . . . . . . . 10 𝑗𝑏
130129nfsum1 14354 . . . . . . . . 9 𝑗Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
131128, 130nfmpt 4706 . . . . . . . 8 𝑗(𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
132127, 131nfeq 2772 . . . . . . 7 𝑗(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
133126, 132nfan 1825 . . . . . 6 𝑗(((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
134 nfcsb1v 3530 . . . . . 6 𝑖𝑐 / 𝑖𝐴
135 nfcsb1v 3530 . . . . . 6 𝑗𝑐 / 𝑗𝐶
13683ad2antrr 761 . . . . . . . 8 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝜑)
1371363ad2ant1 1080 . . . . . . 7 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑖𝐼𝑥𝑋) → 𝜑)
138 simp2 1060 . . . . . . 7 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑖𝐼𝑥𝑋) → 𝑖𝐼)
139 simp3 1061 . . . . . . 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 1323 . . . . . 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 761 . . . . . . 7 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑏𝐼)
144 ssfi 8124 . . . . . . 7 ((𝐼 ∈ Fin ∧ 𝑏𝐼) → 𝑏 ∈ Fin)
145142, 143, 144syl2anc 692 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑏 ∈ Fin)
146 vex 3189 . . . . . . 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 791 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → ¬ 𝑐𝑏)
149 simpllr 798 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → (𝑏 ∪ {𝑐}) ⊆ 𝐼)
15066ad3antrrr 765 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑆 ∈ {ℝ, ℂ})
151136ad2antrr 761 . . . . . . 7 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝜑)
152143ad2antrr 761 . . . . . . . 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 3584 . . . . . . 7 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝑗𝐼)
155 simplr 791 . . . . . . 7 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝑥𝑋)
156 nfv 1840 . . . . . . . . . 10 𝑖 𝑗𝐼
157 nfv 1840 . . . . . . . . . 10 𝑖 𝑥𝑋
158101, 156, 157nf3an 1828 . . . . . . . . 9 𝑖(𝜑𝑗𝐼𝑥𝑋)
159 nfv 1840 . . . . . . . . 9 𝑖 𝐶 ∈ ℂ
160158, 159nfim 1822 . . . . . . . 8 𝑖((𝜑𝑗𝐼𝑥𝑋) → 𝐶 ∈ ℂ)
161 eleq1 2686 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑖𝐼𝑗𝐼))
1621613anbi2d 1401 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝜑𝑖𝐼𝑥𝑋) ↔ (𝜑𝑗𝐼𝑥𝑋)))
163 dvmptfprod.bc . . . . . . . . . 10 (𝑖 = 𝑗𝐵 = 𝐶)
164163eleq1d 2683 . . . . . . . . 9 (𝑖 = 𝑗 → (𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ))
165162, 164imbi12d 334 . . . . . . . 8 (𝑖 = 𝑗 → (((𝜑𝑖𝐼𝑥𝑋) → 𝐵 ∈ ℂ) ↔ ((𝜑𝑗𝐼𝑥𝑋) → 𝐶 ∈ ℂ)))
166 dvmptfprod.b . . . . . . . 8 ((𝜑𝑖𝐼𝑥𝑋) → 𝐵 ∈ ℂ)
167160, 165, 166chvar 2261 . . . . . . 7 ((𝜑𝑗𝐼𝑥𝑋) → 𝐶 ∈ ℂ)
168151, 154, 155, 167syl3anc 1323 . . . . . 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 481 . . . . . . . . 9 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥𝑋) → 𝜑)
171 id 22 . . . . . . . . . . 11 ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑏 ∪ {𝑐}) ⊆ 𝐼)
172146snid 4179 . . . . . . . . . . . . 13 𝑐 ∈ {𝑐}
173 elun2 3759 . . . . . . . . . . . . 13 (𝑐 ∈ {𝑐} → 𝑐 ∈ (𝑏 ∪ {𝑐}))
174172, 173ax-mp 5 . . . . . . . . . . . 12 𝑐 ∈ (𝑏 ∪ {𝑐})
175174a1i 11 . . . . . . . . . . 11 ((𝑏 ∪ {𝑐}) ⊆ 𝐼𝑐 ∈ (𝑏 ∪ {𝑐}))
176171, 175sseldd 3584 . . . . . . . . . 10 ((𝑏 ∪ {𝑐}) ⊆ 𝐼𝑐𝐼)
177176ad2antlr 762 . . . . . . . . 9 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥𝑋) → 𝑐𝐼)
178 simpr 477 . . . . . . . . 9 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥𝑋) → 𝑥𝑋)
179 nfv 1840 . . . . . . . . . . . 12 𝑗 𝑐𝐼
180 nfv 1840 . . . . . . . . . . . 12 𝑗 𝑥𝑋
181122, 179, 180nf3an 1828 . . . . . . . . . . 11 𝑗(𝜑𝑐𝐼𝑥𝑋)
182 nfcv 2761 . . . . . . . . . . . 12 𝑗
183135, 182nfel 2773 . . . . . . . . . . 11 𝑗𝑐 / 𝑗𝐶 ∈ ℂ
184181, 183nfim 1822 . . . . . . . . . 10 𝑗((𝜑𝑐𝐼𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
185 eleq1 2686 . . . . . . . . . . . 12 (𝑗 = 𝑐 → (𝑗𝐼𝑐𝐼))
1861853anbi2d 1401 . . . . . . . . . . 11 (𝑗 = 𝑐 → ((𝜑𝑗𝐼𝑥𝑋) ↔ (𝜑𝑐𝐼𝑥𝑋)))
187 csbeq1a 3523 . . . . . . . . . . . 12 (𝑗 = 𝑐𝐶 = 𝑐 / 𝑗𝐶)
188187eleq1d 2683 . . . . . . . . . . 11 (𝑗 = 𝑐 → (𝐶 ∈ ℂ ↔ 𝑐 / 𝑗𝐶 ∈ ℂ))
189186, 188imbi12d 334 . . . . . . . . . 10 (𝑗 = 𝑐 → (((𝜑𝑗𝐼𝑥𝑋) → 𝐶 ∈ ℂ) ↔ ((𝜑𝑐𝐼𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)))
190184, 189, 167chvar 2261 . . . . . . . . 9 ((𝜑𝑐𝐼𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
191170, 177, 178, 190syl3anc 1323 . . . . . . . 8 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
192191adantlr 750 . . . . . . 7 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ 𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
193192adantlr 750 . . . . . 6 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
194122, 179nfan 1825 . . . . . . . . . 10 𝑗(𝜑𝑐𝐼)
195 nfcv 2761 . . . . . . . . . . 11 𝑗(𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴))
196128, 135nfmpt 4706 . . . . . . . . . . 11 𝑗(𝑥𝑋𝑐 / 𝑗𝐶)
197195, 196nfeq 2772 . . . . . . . . . 10 𝑗(𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶)
198194, 197nfim 1822 . . . . . . . . 9 𝑗((𝜑𝑐𝐼) → (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶))
199185anbi2d 739 . . . . . . . . . 10 (𝑗 = 𝑐 → ((𝜑𝑗𝐼) ↔ (𝜑𝑐𝐼)))
200 csbeq1a 3523 . . . . . . . . . . . . . 14 (𝑗 = 𝑐𝑗 / 𝑖𝐴 = 𝑐 / 𝑗𝑗 / 𝑖𝐴)
201 csbco 3524 . . . . . . . . . . . . . . 15 𝑐 / 𝑗𝑗 / 𝑖𝐴 = 𝑐 / 𝑖𝐴
202201a1i 11 . . . . . . . . . . . . . 14 (𝑗 = 𝑐𝑐 / 𝑗𝑗 / 𝑖𝐴 = 𝑐 / 𝑖𝐴)
203200, 202eqtrd 2655 . . . . . . . . . . . . 13 (𝑗 = 𝑐𝑗 / 𝑖𝐴 = 𝑐 / 𝑖𝐴)
204203mpteq2dv 4705 . . . . . . . . . . . 12 (𝑗 = 𝑐 → (𝑥𝑋𝑗 / 𝑖𝐴) = (𝑥𝑋𝑐 / 𝑖𝐴))
205204oveq2d 6620 . . . . . . . . . . 11 (𝑗 = 𝑐 → (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)))
206187mpteq2dv 4705 . . . . . . . . . . 11 (𝑗 = 𝑐 → (𝑥𝑋𝐶) = (𝑥𝑋𝑐 / 𝑗𝐶))
207205, 206eqeq12d 2636 . . . . . . . . . 10 (𝑗 = 𝑐 → ((𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶) ↔ (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶)))
208199, 207imbi12d 334 . . . . . . . . 9 (𝑗 = 𝑐 → (((𝜑𝑗𝐼) → (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶)) ↔ ((𝜑𝑐𝐼) → (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶))))
209101, 156nfan 1825 . . . . . . . . . . 11 𝑖(𝜑𝑗𝐼)
210 nfcsb1v 3530 . . . . . . . . . . . . . 14 𝑖𝑗 / 𝑖𝐴
211108, 210nfmpt 4706 . . . . . . . . . . . . 13 𝑖(𝑥𝑋𝑗 / 𝑖𝐴)
212106, 107, 211nfov 6630 . . . . . . . . . . . 12 𝑖(𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴))
213 nfcv 2761 . . . . . . . . . . . 12 𝑖(𝑥𝑋𝐶)
214212, 213nfeq 2772 . . . . . . . . . . 11 𝑖(𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶)
215209, 214nfim 1822 . . . . . . . . . 10 𝑖((𝜑𝑗𝐼) → (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶))
216161anbi2d 739 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝜑𝑖𝐼) ↔ (𝜑𝑗𝐼)))
217 csbeq1a 3523 . . . . . . . . . . . . . 14 (𝑖 = 𝑗𝐴 = 𝑗 / 𝑖𝐴)
218217mpteq2dv 4705 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑥𝑋𝐴) = (𝑥𝑋𝑗 / 𝑖𝐴))
219218oveq2d 6620 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑆 D (𝑥𝑋𝐴)) = (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)))
220163idi 2 . . . . . . . . . . . . 13 (𝑖 = 𝑗𝐵 = 𝐶)
221220mpteq2dv 4705 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑥𝑋𝐵) = (𝑥𝑋𝐶))
222219, 221eqeq12d 2636 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵) ↔ (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶)))
223216, 222imbi12d 334 . . . . . . . . . 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 2261 . . . . . . . . 9 ((𝜑𝑗𝐼) → (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶))
226198, 208, 225chvar 2261 . . . . . . . 8 ((𝜑𝑐𝐼) → (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶))
227176, 226sylan2 491 . . . . . . 7 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶))
228227ad2antrr 761 . . . . . 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 3523 . . . . . 6 (𝑖 = 𝑐𝐴 = 𝑐 / 𝑖𝐴)
230100, 121, 133, 134, 135, 141, 145, 147, 148, 149, 150, 168, 169, 193, 228, 229, 187dvmptfprodlem 39465 . . . . 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 1322 . . . 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 1261 . . 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 8145 . 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 1036   = wceq 1480  wnf 1705  wcel 1987  Vcvv 3186  csb 3514  cdif 3552  cun 3553  wss 3555  c0 3891  {csn 4148  {cpr 4150  cmpt 4673  cfv 5847  (class class class)co 6604  Fincfn 7899  cc 9878  cr 9879  0cc0 9880  1c1 9881   · cmul 9885  Σcsu 14350  cprod 14560  t crest 16002  TopOpenctopn 16003  fldccnfld 19665   D cdv 23533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957  ax-pre-sup 9958  ax-addf 9959  ax-mulf 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-fi 8261  df-sup 8292  df-inf 8293  df-oi 8359  df-card 8709  df-cda 8934  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-q 11733  df-rp 11777  df-xneg 11890  df-xadd 11891  df-xmul 11892  df-icc 12124  df-fz 12269  df-fzo 12407  df-seq 12742  df-exp 12801  df-hash 13058  df-cj 13773  df-re 13774  df-im 13775  df-sqrt 13909  df-abs 13910  df-clim 14153  df-sum 14351  df-prod 14561  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-starv 15877  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-unif 15886  df-hom 15887  df-cco 15888  df-rest 16004  df-topn 16005  df-0g 16023  df-gsum 16024  df-topgen 16025  df-pt 16026  df-prds 16029  df-xrs 16083  df-qtop 16088  df-imas 16089  df-xps 16091  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-submnd 17257  df-mulg 17462  df-cntz 17671  df-cmn 18116  df-psmet 19657  df-xmet 19658  df-met 19659  df-bl 19660  df-mopn 19661  df-fbas 19662  df-fg 19663  df-cnfld 19666  df-top 20621  df-bases 20622  df-topon 20623  df-topsp 20624  df-cld 20733  df-ntr 20734  df-cls 20735  df-nei 20812  df-lp 20850  df-perf 20851  df-cn 20941  df-cnp 20942  df-haus 21029  df-tx 21275  df-hmeo 21468  df-fil 21560  df-fm 21652  df-flim 21653  df-flf 21654  df-xms 22035  df-ms 22036  df-tms 22037  df-cncf 22589  df-limc 23536  df-dv 23537
This theorem is referenced by: (None)
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