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Theorem dvmptfprod 40478
 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 3657 . . 3 𝐼𝐼
32jctr 564 . 2 (𝜑 → (𝜑𝐼𝐼))
4 sseq1 3659 . . . . 5 (𝑎 = ∅ → (𝑎𝐼 ↔ ∅ ⊆ 𝐼))
54anbi2d 740 . . . 4 (𝑎 = ∅ → ((𝜑𝑎𝐼) ↔ (𝜑 ∧ ∅ ⊆ 𝐼)))
6 prodeq1 14683 . . . . . . 7 (𝑎 = ∅ → ∏𝑖𝑎 𝐴 = ∏𝑖 ∈ ∅ 𝐴)
76mpteq2dv 4778 . . . . . 6 (𝑎 = ∅ → (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴) = (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴))
87oveq2d 6706 . . . . 5 (𝑎 = ∅ → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)))
9 sumeq1 14463 . . . . . . 7 (𝑎 = ∅ → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
10 difeq1 3754 . . . . . . . . . 10 (𝑎 = ∅ → (𝑎 ∖ {𝑗}) = (∅ ∖ {𝑗}))
1110prodeq1d 14695 . . . . . . . . 9 (𝑎 = ∅ → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)
1211oveq2d 6706 . . . . . . . 8 (𝑎 = ∅ → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
1312sumeq2sdv 14479 . . . . . . 7 (𝑎 = ∅ → Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
149, 13eqtrd 2685 . . . . . 6 (𝑎 = ∅ → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
1514mpteq2dv 4778 . . . . 5 (𝑎 = ∅ → (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
168, 15eqeq12d 2666 . . . 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 333 . . 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 3659 . . . . 5 (𝑎 = 𝑏 → (𝑎𝐼𝑏𝐼))
1918anbi2d 740 . . . 4 (𝑎 = 𝑏 → ((𝜑𝑎𝐼) ↔ (𝜑𝑏𝐼)))
20 prodeq1 14683 . . . . . . 7 (𝑎 = 𝑏 → ∏𝑖𝑎 𝐴 = ∏𝑖𝑏 𝐴)
2120mpteq2dv 4778 . . . . . 6 (𝑎 = 𝑏 → (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴) = (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴))
2221oveq2d 6706 . . . . 5 (𝑎 = 𝑏 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)))
23 sumeq1 14463 . . . . . . 7 (𝑎 = 𝑏 → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
24 difeq1 3754 . . . . . . . . . 10 (𝑎 = 𝑏 → (𝑎 ∖ {𝑗}) = (𝑏 ∖ {𝑗}))
2524prodeq1d 14695 . . . . . . . . 9 (𝑎 = 𝑏 → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
2625oveq2d 6706 . . . . . . . 8 (𝑎 = 𝑏 → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
2726sumeq2sdv 14479 . . . . . . 7 (𝑎 = 𝑏 → Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
2823, 27eqtrd 2685 . . . . . 6 (𝑎 = 𝑏 → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
2928mpteq2dv 4778 . . . . 5 (𝑎 = 𝑏 → (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
3022, 29eqeq12d 2666 . . . 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 333 . . 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 3659 . . . . 5 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎𝐼 ↔ (𝑏 ∪ {𝑐}) ⊆ 𝐼))
3332anbi2d 740 . . . 4 (𝑎 = (𝑏 ∪ {𝑐}) → ((𝜑𝑎𝐼) ↔ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)))
34 prodeq1 14683 . . . . . . 7 (𝑎 = (𝑏 ∪ {𝑐}) → ∏𝑖𝑎 𝐴 = ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)
3534mpteq2dv 4778 . . . . . 6 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴) = (𝑥𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴))
3635oveq2d 6706 . . . . 5 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ (𝑏 ∪ {𝑐})𝐴)))
37 sumeq1 14463 . . . . . . 7 (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
38 difeq1 3754 . . . . . . . . . 10 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑎 ∖ {𝑗}) = ((𝑏 ∪ {𝑐}) ∖ {𝑗}))
3938prodeq1d 14695 . . . . . . . . 9 (𝑎 = (𝑏 ∪ {𝑐}) → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)
4039oveq2d 6706 . . . . . . . 8 (𝑎 = (𝑏 ∪ {𝑐}) → (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))
4140sumeq2sdv 14479 . . . . . . 7 (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))
4237, 41eqtrd 2685 . . . . . 6 (𝑎 = (𝑏 ∪ {𝑐}) → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴))
4342mpteq2dv 4778 . . . . 5 (𝑎 = (𝑏 ∪ {𝑐}) → (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ (𝑏 ∪ {𝑐})(𝐶 · ∏𝑖 ∈ ((𝑏 ∪ {𝑐}) ∖ {𝑗})𝐴)))
4436, 43eqeq12d 2666 . . . 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 333 . . 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 3659 . . . . 5 (𝑎 = 𝐼 → (𝑎𝐼𝐼𝐼))
4746anbi2d 740 . . . 4 (𝑎 = 𝐼 → ((𝜑𝑎𝐼) ↔ (𝜑𝐼𝐼)))
48 prodeq1 14683 . . . . . . 7 (𝑎 = 𝐼 → ∏𝑖𝑎 𝐴 = ∏𝑖𝐼 𝐴)
4948mpteq2dv 4778 . . . . . 6 (𝑎 = 𝐼 → (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴) = (𝑥𝑋 ↦ ∏𝑖𝐼 𝐴))
5049oveq2d 6706 . . . . 5 (𝑎 = 𝐼 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑎 𝐴)) = (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝐼 𝐴)))
51 sumeq1 14463 . . . . . . 7 (𝑎 = 𝐼 → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴))
52 difeq1 3754 . . . . . . . . . . . 12 (𝑎 = 𝐼 → (𝑎 ∖ {𝑗}) = (𝐼 ∖ {𝑗}))
5352prodeq1d 14695 . . . . . . . . . . 11 (𝑎 = 𝐼 → ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴 = ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)
5453oveq2d 6706 . . . . . . . . . 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 2994 . . . . . . . 8 (𝑎 = 𝐼 → ∀𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))
5756sumeq2d 14476 . . . . . . 7 (𝑎 = 𝐼 → Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))
5851, 57eqtrd 2685 . . . . . 6 (𝑎 = 𝐼 → Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴) = Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴))
5958mpteq2dv 4778 . . . . 5 (𝑎 = 𝐼 → (𝑥𝑋 ↦ Σ𝑗𝑎 (𝐶 · ∏𝑖 ∈ (𝑎 ∖ {𝑗})𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝐼 (𝐶 · ∏𝑖 ∈ (𝐼 ∖ {𝑗})𝐴)))
6050, 59eqeq12d 2666 . . . 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 333 . . 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 14717 . . . . . . . 8 𝑖 ∈ ∅ 𝐴 = 1
6362mpteq2i 4774 . . . . . . 7 (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴) = (𝑥𝑋 ↦ 1)
6463oveq2i 6701 . . . . . 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 6700 . . . . . . . 8 (𝐾t 𝑆) = ((TopOpen‘ℂfld) ↾t 𝑆)
7168, 70eqtri 2673 . . . . . . 7 𝐽 = ((TopOpen‘ℂfld) ↾t 𝑆)
7267, 71syl6eleq 2740 . . . . . 6 (𝜑𝑋 ∈ ((TopOpen‘ℂfld) ↾t 𝑆))
73 1cnd 10094 . . . . . 6 (𝜑 → 1 ∈ ℂ)
7466, 72, 73dvmptconst 40447 . . . . 5 (𝜑 → (𝑆 D (𝑥𝑋 ↦ 1)) = (𝑥𝑋 ↦ 0))
75 sum0 14496 . . . . . . . 8 Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴) = 0
7675eqcomi 2660 . . . . . . 7 0 = Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)
7776mpteq2i 4774 . . . . . 6 (𝑥𝑋 ↦ 0) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴))
7877a1i 11 . . . . 5 (𝜑 → (𝑥𝑋 ↦ 0) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
7965, 74, 783eqtrd 2689 . . . 4 (𝜑 → (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
8079adantr 480 . . 3 ((𝜑 ∧ ∅ ⊆ 𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖 ∈ ∅ 𝐴)) = (𝑥𝑋 ↦ Σ𝑗 ∈ ∅ (𝐶 · ∏𝑖 ∈ (∅ ∖ {𝑗})𝐴)))
81 simp3 1083 . . . . 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 1106 . . . . 5 (((𝑏 ∈ Fin ∧ ¬ 𝑐𝑏) ∧ ((𝜑𝑏𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → ¬ 𝑐𝑏)
83 simpl 472 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → 𝜑)
84 ssun1 3809 . . . . . . . . . . 11 𝑏 ⊆ (𝑏 ∪ {𝑐})
85 sstr2 3643 . . . . . . . . . . 11 (𝑏 ⊆ (𝑏 ∪ {𝑐}) → ((𝑏 ∪ {𝑐}) ⊆ 𝐼𝑏𝐼))
8684, 85ax-mp 5 . . . . . . . . . 10 ((𝑏 ∪ {𝑐}) ⊆ 𝐼𝑏𝐼)
8786adantl 481 . . . . . . . . 9 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → 𝑏𝐼)
8883, 87jca 553 . . . . . . . 8 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝜑𝑏𝐼))
8988adantl 481 . . . . . . 7 ((((𝜑𝑏𝐼) → (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ (𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)) → (𝜑𝑏𝐼))
90 simpl 472 . . . . . . 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 1099 . . . . 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 1883 . . . . . . 7 𝑥((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏)
94 nfcv 2793 . . . . . . . . 9 𝑥𝑆
95 nfcv 2793 . . . . . . . . 9 𝑥 D
96 nfmpt1 4780 . . . . . . . . 9 𝑥(𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)
9794, 95, 96nfov 6716 . . . . . . . 8 𝑥(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴))
98 nfmpt1 4780 . . . . . . . 8 𝑥(𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
9997, 98nfeq 2805 . . . . . . 7 𝑥(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
10093, 99nfan 1868 . . . . . 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 1883 . . . . . . . . 9 𝑖(𝑏 ∪ {𝑐}) ⊆ 𝐼
103101, 102nfan 1868 . . . . . . . 8 𝑖(𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)
104 nfv 1883 . . . . . . . 8 𝑖 ¬ 𝑐𝑏
105103, 104nfan 1868 . . . . . . 7 𝑖((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏)
106 nfcv 2793 . . . . . . . . 9 𝑖𝑆
107 nfcv 2793 . . . . . . . . 9 𝑖 D
108 nfcv 2793 . . . . . . . . . 10 𝑖𝑋
109 nfcv 2793 . . . . . . . . . . 11 𝑖𝑏
110109nfcprod1 14684 . . . . . . . . . 10 𝑖𝑖𝑏 𝐴
111108, 110nfmpt 4779 . . . . . . . . 9 𝑖(𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)
112106, 107, 111nfov 6716 . . . . . . . 8 𝑖(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴))
113 nfcv 2793 . . . . . . . . . . 11 𝑖𝐶
114 nfcv 2793 . . . . . . . . . . 11 𝑖 ·
115 nfcv 2793 . . . . . . . . . . . 12 𝑖(𝑏 ∖ {𝑗})
116115nfcprod1 14684 . . . . . . . . . . 11 𝑖𝑖 ∈ (𝑏 ∖ {𝑗})𝐴
117113, 114, 116nfov 6716 . . . . . . . . . 10 𝑖(𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
118109, 117nfsum 14465 . . . . . . . . 9 𝑖Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
119108, 118nfmpt 4779 . . . . . . . 8 𝑖(𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
120112, 119nfeq 2805 . . . . . . 7 𝑖(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
121105, 120nfan 1868 . . . . . 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 1883 . . . . . . . . 9 𝑗(𝑏 ∪ {𝑐}) ⊆ 𝐼
124122, 123nfan 1868 . . . . . . . 8 𝑗(𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼)
125 nfv 1883 . . . . . . . 8 𝑗 ¬ 𝑐𝑏
126124, 125nfan 1868 . . . . . . 7 𝑗((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏)
127 nfcv 2793 . . . . . . . 8 𝑗(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴))
128 nfcv 2793 . . . . . . . . 9 𝑗𝑋
129 nfcv 2793 . . . . . . . . . 10 𝑗𝑏
130129nfsum1 14464 . . . . . . . . 9 𝑗Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)
131128, 130nfmpt 4779 . . . . . . . 8 𝑗(𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
132127, 131nfeq 2805 . . . . . . 7 𝑗(𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))
133126, 132nfan 1868 . . . . . 6 𝑗(((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴)))
134 nfcsb1v 3582 . . . . . 6 𝑖𝑐 / 𝑖𝐴
135 nfcsb1v 3582 . . . . . 6 𝑗𝑐 / 𝑗𝐶
13683ad2antrr 762 . . . . . . . 8 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝜑)
1371363ad2ant1 1102 . . . . . . 7 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑖𝐼𝑥𝑋) → 𝜑)
138 simp2 1082 . . . . . . 7 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑖𝐼𝑥𝑋) → 𝑖𝐼)
139 simp3 1083 . . . . . . 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 1366 . . . . . 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 762 . . . . . . 7 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑏𝐼)
144 ssfi 8221 . . . . . . 7 ((𝐼 ∈ Fin ∧ 𝑏𝐼) → 𝑏 ∈ Fin)
145142, 143, 144syl2anc 694 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑏 ∈ Fin)
146 vex 3234 . . . . . . 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 807 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → ¬ 𝑐𝑏)
149 simpllr 815 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → (𝑏 ∪ {𝑐}) ⊆ 𝐼)
15066ad3antrrr 766 . . . . . 6 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) → 𝑆 ∈ {ℝ, ℂ})
151136ad2antrr 762 . . . . . . 7 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝜑)
152143ad2antrr 762 . . . . . . . 8 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝑏𝐼)
153 simpr 476 . . . . . . . 8 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝑗𝑏)
154152, 153sseldd 3637 . . . . . . 7 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝑗𝐼)
155 simplr 807 . . . . . . 7 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝑥𝑋)
156 nfv 1883 . . . . . . . . . 10 𝑖 𝑗𝐼
157 nfv 1883 . . . . . . . . . 10 𝑖 𝑥𝑋
158101, 156, 157nf3an 1871 . . . . . . . . 9 𝑖(𝜑𝑗𝐼𝑥𝑋)
159 nfv 1883 . . . . . . . . 9 𝑖 𝐶 ∈ ℂ
160158, 159nfim 1865 . . . . . . . 8 𝑖((𝜑𝑗𝐼𝑥𝑋) → 𝐶 ∈ ℂ)
161 eleq1 2718 . . . . . . . . . 10 (𝑖 = 𝑗 → (𝑖𝐼𝑗𝐼))
1621613anbi2d 1444 . . . . . . . . 9 (𝑖 = 𝑗 → ((𝜑𝑖𝐼𝑥𝑋) ↔ (𝜑𝑗𝐼𝑥𝑋)))
163 dvmptfprod.bc . . . . . . . . . 10 (𝑖 = 𝑗𝐵 = 𝐶)
164163eleq1d 2715 . . . . . . . . 9 (𝑖 = 𝑗 → (𝐵 ∈ ℂ ↔ 𝐶 ∈ ℂ))
165162, 164imbi12d 333 . . . . . . . 8 (𝑖 = 𝑗 → (((𝜑𝑖𝐼𝑥𝑋) → 𝐵 ∈ ℂ) ↔ ((𝜑𝑗𝐼𝑥𝑋) → 𝐶 ∈ ℂ)))
166 dvmptfprod.b . . . . . . . 8 ((𝜑𝑖𝐼𝑥𝑋) → 𝐵 ∈ ℂ)
167160, 165, 166chvar 2298 . . . . . . 7 ((𝜑𝑗𝐼𝑥𝑋) → 𝐶 ∈ ℂ)
168151, 154, 155, 167syl3anc 1366 . . . . . 6 ((((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) ∧ 𝑗𝑏) → 𝐶 ∈ ℂ)
169 simpr 476 . . . . . 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 480 . . . . . . . . 9 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥𝑋) → 𝜑)
171 id 22 . . . . . . . . . . 11 ((𝑏 ∪ {𝑐}) ⊆ 𝐼 → (𝑏 ∪ {𝑐}) ⊆ 𝐼)
172146snid 4241 . . . . . . . . . . . . 13 𝑐 ∈ {𝑐}
173 elun2 3814 . . . . . . . . . . . . 13 (𝑐 ∈ {𝑐} → 𝑐 ∈ (𝑏 ∪ {𝑐}))
174172, 173ax-mp 5 . . . . . . . . . . . 12 𝑐 ∈ (𝑏 ∪ {𝑐})
175174a1i 11 . . . . . . . . . . 11 ((𝑏 ∪ {𝑐}) ⊆ 𝐼𝑐 ∈ (𝑏 ∪ {𝑐}))
176171, 175sseldd 3637 . . . . . . . . . 10 ((𝑏 ∪ {𝑐}) ⊆ 𝐼𝑐𝐼)
177176ad2antlr 763 . . . . . . . . 9 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥𝑋) → 𝑐𝐼)
178 simpr 476 . . . . . . . . 9 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥𝑋) → 𝑥𝑋)
179 nfv 1883 . . . . . . . . . . . 12 𝑗 𝑐𝐼
180 nfv 1883 . . . . . . . . . . . 12 𝑗 𝑥𝑋
181122, 179, 180nf3an 1871 . . . . . . . . . . 11 𝑗(𝜑𝑐𝐼𝑥𝑋)
182 nfcv 2793 . . . . . . . . . . . 12 𝑗
183135, 182nfel 2806 . . . . . . . . . . 11 𝑗𝑐 / 𝑗𝐶 ∈ ℂ
184181, 183nfim 1865 . . . . . . . . . 10 𝑗((𝜑𝑐𝐼𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
185 eleq1 2718 . . . . . . . . . . . 12 (𝑗 = 𝑐 → (𝑗𝐼𝑐𝐼))
1861853anbi2d 1444 . . . . . . . . . . 11 (𝑗 = 𝑐 → ((𝜑𝑗𝐼𝑥𝑋) ↔ (𝜑𝑐𝐼𝑥𝑋)))
187 csbeq1a 3575 . . . . . . . . . . . 12 (𝑗 = 𝑐𝐶 = 𝑐 / 𝑗𝐶)
188187eleq1d 2715 . . . . . . . . . . 11 (𝑗 = 𝑐 → (𝐶 ∈ ℂ ↔ 𝑐 / 𝑗𝐶 ∈ ℂ))
189186, 188imbi12d 333 . . . . . . . . . 10 (𝑗 = 𝑐 → (((𝜑𝑗𝐼𝑥𝑋) → 𝐶 ∈ ℂ) ↔ ((𝜑𝑐𝐼𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)))
190184, 189, 167chvar 2298 . . . . . . . . 9 ((𝜑𝑐𝐼𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
191170, 177, 178, 190syl3anc 1366 . . . . . . . 8 (((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ 𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
192191adantlr 751 . . . . . . 7 ((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ 𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
193192adantlr 751 . . . . . 6 (((((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) ∧ ¬ 𝑐𝑏) ∧ (𝑆 D (𝑥𝑋 ↦ ∏𝑖𝑏 𝐴)) = (𝑥𝑋 ↦ Σ𝑗𝑏 (𝐶 · ∏𝑖 ∈ (𝑏 ∖ {𝑗})𝐴))) ∧ 𝑥𝑋) → 𝑐 / 𝑗𝐶 ∈ ℂ)
194122, 179nfan 1868 . . . . . . . . . 10 𝑗(𝜑𝑐𝐼)
195 nfcv 2793 . . . . . . . . . . 11 𝑗(𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴))
196128, 135nfmpt 4779 . . . . . . . . . . 11 𝑗(𝑥𝑋𝑐 / 𝑗𝐶)
197195, 196nfeq 2805 . . . . . . . . . 10 𝑗(𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶)
198194, 197nfim 1865 . . . . . . . . 9 𝑗((𝜑𝑐𝐼) → (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶))
199185anbi2d 740 . . . . . . . . . 10 (𝑗 = 𝑐 → ((𝜑𝑗𝐼) ↔ (𝜑𝑐𝐼)))
200 csbeq1a 3575 . . . . . . . . . . . . . 14 (𝑗 = 𝑐𝑗 / 𝑖𝐴 = 𝑐 / 𝑗𝑗 / 𝑖𝐴)
201 csbco 3576 . . . . . . . . . . . . . . 15 𝑐 / 𝑗𝑗 / 𝑖𝐴 = 𝑐 / 𝑖𝐴
202201a1i 11 . . . . . . . . . . . . . 14 (𝑗 = 𝑐𝑐 / 𝑗𝑗 / 𝑖𝐴 = 𝑐 / 𝑖𝐴)
203200, 202eqtrd 2685 . . . . . . . . . . . . 13 (𝑗 = 𝑐𝑗 / 𝑖𝐴 = 𝑐 / 𝑖𝐴)
204203mpteq2dv 4778 . . . . . . . . . . . 12 (𝑗 = 𝑐 → (𝑥𝑋𝑗 / 𝑖𝐴) = (𝑥𝑋𝑐 / 𝑖𝐴))
205204oveq2d 6706 . . . . . . . . . . 11 (𝑗 = 𝑐 → (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)))
206187mpteq2dv 4778 . . . . . . . . . . 11 (𝑗 = 𝑐 → (𝑥𝑋𝐶) = (𝑥𝑋𝑐 / 𝑗𝐶))
207205, 206eqeq12d 2666 . . . . . . . . . 10 (𝑗 = 𝑐 → ((𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶) ↔ (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶)))
208199, 207imbi12d 333 . . . . . . . . 9 (𝑗 = 𝑐 → (((𝜑𝑗𝐼) → (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶)) ↔ ((𝜑𝑐𝐼) → (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶))))
209101, 156nfan 1868 . . . . . . . . . . 11 𝑖(𝜑𝑗𝐼)
210 nfcsb1v 3582 . . . . . . . . . . . . . 14 𝑖𝑗 / 𝑖𝐴
211108, 210nfmpt 4779 . . . . . . . . . . . . 13 𝑖(𝑥𝑋𝑗 / 𝑖𝐴)
212106, 107, 211nfov 6716 . . . . . . . . . . . 12 𝑖(𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴))
213 nfcv 2793 . . . . . . . . . . . 12 𝑖(𝑥𝑋𝐶)
214212, 213nfeq 2805 . . . . . . . . . . 11 𝑖(𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶)
215209, 214nfim 1865 . . . . . . . . . 10 𝑖((𝜑𝑗𝐼) → (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶))
216161anbi2d 740 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝜑𝑖𝐼) ↔ (𝜑𝑗𝐼)))
217 csbeq1a 3575 . . . . . . . . . . . . . 14 (𝑖 = 𝑗𝐴 = 𝑗 / 𝑖𝐴)
218217mpteq2dv 4778 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (𝑥𝑋𝐴) = (𝑥𝑋𝑗 / 𝑖𝐴))
219218oveq2d 6706 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑆 D (𝑥𝑋𝐴)) = (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)))
220163idi 2 . . . . . . . . . . . . 13 (𝑖 = 𝑗𝐵 = 𝐶)
221220mpteq2dv 4778 . . . . . . . . . . . 12 (𝑖 = 𝑗 → (𝑥𝑋𝐵) = (𝑥𝑋𝐶))
222219, 221eqeq12d 2666 . . . . . . . . . . 11 (𝑖 = 𝑗 → ((𝑆 D (𝑥𝑋𝐴)) = (𝑥𝑋𝐵) ↔ (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶)))
223216, 222imbi12d 333 . . . . . . . . . 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 2298 . . . . . . . . 9 ((𝜑𝑗𝐼) → (𝑆 D (𝑥𝑋𝑗 / 𝑖𝐴)) = (𝑥𝑋𝐶))
226198, 208, 225chvar 2298 . . . . . . . 8 ((𝜑𝑐𝐼) → (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶))
227176, 226sylan2 490 . . . . . . 7 ((𝜑 ∧ (𝑏 ∪ {𝑐}) ⊆ 𝐼) → (𝑆 D (𝑥𝑋𝑐 / 𝑖𝐴)) = (𝑥𝑋𝑐 / 𝑗𝐶))
228227ad2antrr 762 . . . . . 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 3575 . . . . . 6 (𝑖 = 𝑐𝐴 = 𝑐 / 𝑖𝐴)
230100, 121, 133, 134, 135, 141, 145, 147, 148, 149, 150, 168, 169, 193, 228, 229, 187dvmptfprodlem 40477 . . . . 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 1365 . . . 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 1283 . . 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 8242 . 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 383   ∧ w3a 1054   = wceq 1523  Ⅎwnf 1748   ∈ wcel 2030  Vcvv 3231  ⦋csb 3566   ∖ cdif 3604   ∪ cun 3605   ⊆ wss 3607  ∅c0 3948  {csn 4210  {cpr 4212   ↦ cmpt 4762  ‘cfv 5926  (class class class)co 6690  Fincfn 7997  ℂcc 9972  ℝcr 9973  0cc0 9974  1c1 9975   · cmul 9979  Σcsu 14460  ∏cprod 14679   ↾t crest 16128  TopOpenctopn 16129  ℂfldccnfld 19794   D cdv 23672 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-icc 12220  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-prod 14680  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-hom 16013  df-cco 16014  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-pt 16152  df-prds 16155  df-xrs 16209  df-qtop 16214  df-imas 16215  df-xps 16217  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-nei 20950  df-lp 20988  df-perf 20989  df-cn 21079  df-cnp 21080  df-haus 21167  df-tx 21413  df-hmeo 21606  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-xms 22172  df-ms 22173  df-tms 22174  df-cncf 22728  df-limc 23675  df-dv 23676 This theorem is referenced by: (None)
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