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Theorem jensen 26472
Description: Jensen's inequality, a finite extension of the definition of convexity (the last hypothesis). (Contributed by Mario Carneiro, 21-Jun-2015.) (Proof shortened by AV, 27-Jul-2019.)
Hypotheses
Ref Expression
jensen.1 (𝜑𝐷 ⊆ ℝ)
jensen.2 (𝜑𝐹:𝐷⟶ℝ)
jensen.3 ((𝜑 ∧ (𝑎𝐷𝑏𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷)
jensen.4 (𝜑𝐴 ∈ Fin)
jensen.5 (𝜑𝑇:𝐴⟶(0[,)+∞))
jensen.6 (𝜑𝑋:𝐴𝐷)
jensen.7 (𝜑 → 0 < (ℂfld Σg 𝑇))
jensen.8 ((𝜑 ∧ (𝑥𝐷𝑦𝐷𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹𝑥)) + ((1 − 𝑡) · (𝐹𝑦))))
Assertion
Ref Expression
jensen (𝜑 → (((ℂfld Σg (𝑇f · 𝑋)) / (ℂfld Σg 𝑇)) ∈ 𝐷 ∧ (𝐹‘((ℂfld Σg (𝑇f · 𝑋)) / (ℂfld Σg 𝑇))) ≤ ((ℂfld Σg (𝑇f · (𝐹𝑋))) / (ℂfld Σg 𝑇))))
Distinct variable groups:   𝑎,𝑏,𝑡,𝑥,𝑦,𝐴   𝐷,𝑎,𝑏,𝑡,𝑥,𝑦   𝜑,𝑎,𝑏,𝑡,𝑥,𝑦   𝐹,𝑎,𝑏,𝑡,𝑥,𝑦   𝑇,𝑎,𝑏,𝑡,𝑥,𝑦   𝑋,𝑎,𝑏,𝑡,𝑥,𝑦

Proof of Theorem jensen
Dummy variables 𝑐 𝑘 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 jensen.7 . . . . . 6 (𝜑 → 0 < (ℂfld Σg 𝑇))
2 jensen.5 . . . . . . . . 9 (𝜑𝑇:𝐴⟶(0[,)+∞))
32ffnd 6714 . . . . . . . 8 (𝜑𝑇 Fn 𝐴)
4 fnresdm 6665 . . . . . . . 8 (𝑇 Fn 𝐴 → (𝑇𝐴) = 𝑇)
53, 4syl 17 . . . . . . 7 (𝜑 → (𝑇𝐴) = 𝑇)
65oveq2d 7419 . . . . . 6 (𝜑 → (ℂfld Σg (𝑇𝐴)) = (ℂfld Σg 𝑇))
71, 6breqtrrd 5174 . . . . 5 (𝜑 → 0 < (ℂfld Σg (𝑇𝐴)))
8 ssid 4002 . . . . 5 𝐴𝐴
97, 8jctil 521 . . . 4 (𝜑 → (𝐴𝐴 ∧ 0 < (ℂfld Σg (𝑇𝐴))))
10 jensen.4 . . . . 5 (𝜑𝐴 ∈ Fin)
11 sseq1 4005 . . . . . . . . 9 (𝑎 = ∅ → (𝑎𝐴 ↔ ∅ ⊆ 𝐴))
12 reseq2 5973 . . . . . . . . . . . . 13 (𝑎 = ∅ → (𝑇𝑎) = (𝑇 ↾ ∅))
13 res0 5982 . . . . . . . . . . . . 13 (𝑇 ↾ ∅) = ∅
1412, 13eqtrdi 2789 . . . . . . . . . . . 12 (𝑎 = ∅ → (𝑇𝑎) = ∅)
1514oveq2d 7419 . . . . . . . . . . 11 (𝑎 = ∅ → (ℂfld Σg (𝑇𝑎)) = (ℂfld Σg ∅))
16 cnfld0 20953 . . . . . . . . . . . 12 0 = (0g‘ℂfld)
1716gsum0 18598 . . . . . . . . . . 11 (ℂfld Σg ∅) = 0
1815, 17eqtrdi 2789 . . . . . . . . . 10 (𝑎 = ∅ → (ℂfld Σg (𝑇𝑎)) = 0)
1918breq2d 5158 . . . . . . . . 9 (𝑎 = ∅ → (0 < (ℂfld Σg (𝑇𝑎)) ↔ 0 < 0))
2011, 19anbi12d 632 . . . . . . . 8 (𝑎 = ∅ → ((𝑎𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑎))) ↔ (∅ ⊆ 𝐴 ∧ 0 < 0)))
21 reseq2 5973 . . . . . . . . . . 11 (𝑎 = ∅ → ((𝑇f · 𝑋) ↾ 𝑎) = ((𝑇f · 𝑋) ↾ ∅))
2221oveq2d 7419 . . . . . . . . . 10 (𝑎 = ∅ → (ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇f · 𝑋) ↾ ∅)))
2322, 18oveq12d 7421 . . . . . . . . 9 (𝑎 = ∅ → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇f · 𝑋) ↾ ∅)) / 0))
24 reseq2 5973 . . . . . . . . . . . . 13 (𝑎 = ∅ → ((𝑇f · (𝐹𝑋)) ↾ 𝑎) = ((𝑇f · (𝐹𝑋)) ↾ ∅))
2524oveq2d 7419 . . . . . . . . . . . 12 (𝑎 = ∅ → (ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ ∅)))
2625, 18oveq12d 7421 . . . . . . . . . . 11 (𝑎 = ∅ → ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ ∅)) / 0))
2726breq2d 5158 . . . . . . . . . 10 (𝑎 = ∅ → ((𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ↔ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ ∅)) / 0)))
2827rabbidv 3441 . . . . . . . . 9 (𝑎 = ∅ → {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎)))} = {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ ∅)) / 0)})
2923, 28eleq12d 2828 . . . . . . . 8 (𝑎 = ∅ → (((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎)))} ↔ ((ℂfld Σg ((𝑇f · 𝑋) ↾ ∅)) / 0) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ ∅)) / 0)}))
3020, 29imbi12d 345 . . . . . . 7 (𝑎 = ∅ → (((𝑎𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑎))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎)))}) ↔ ((∅ ⊆ 𝐴 ∧ 0 < 0) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ ∅)) / 0) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ ∅)) / 0)})))
3130imbi2d 341 . . . . . 6 (𝑎 = ∅ → ((𝜑 → ((𝑎𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑎))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎)))})) ↔ (𝜑 → ((∅ ⊆ 𝐴 ∧ 0 < 0) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ ∅)) / 0) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ ∅)) / 0)}))))
32 sseq1 4005 . . . . . . . . 9 (𝑎 = 𝑘 → (𝑎𝐴𝑘𝐴))
33 reseq2 5973 . . . . . . . . . . 11 (𝑎 = 𝑘 → (𝑇𝑎) = (𝑇𝑘))
3433oveq2d 7419 . . . . . . . . . 10 (𝑎 = 𝑘 → (ℂfld Σg (𝑇𝑎)) = (ℂfld Σg (𝑇𝑘)))
3534breq2d 5158 . . . . . . . . 9 (𝑎 = 𝑘 → (0 < (ℂfld Σg (𝑇𝑎)) ↔ 0 < (ℂfld Σg (𝑇𝑘))))
3632, 35anbi12d 632 . . . . . . . 8 (𝑎 = 𝑘 → ((𝑎𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑎))) ↔ (𝑘𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑘)))))
37 reseq2 5973 . . . . . . . . . . 11 (𝑎 = 𝑘 → ((𝑇f · 𝑋) ↾ 𝑎) = ((𝑇f · 𝑋) ↾ 𝑘))
3837oveq2d 7419 . . . . . . . . . 10 (𝑎 = 𝑘 → (ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)))
3938, 34oveq12d 7421 . . . . . . . . 9 (𝑎 = 𝑘 → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))))
40 reseq2 5973 . . . . . . . . . . . . 13 (𝑎 = 𝑘 → ((𝑇f · (𝐹𝑋)) ↾ 𝑎) = ((𝑇f · (𝐹𝑋)) ↾ 𝑘))
4140oveq2d 7419 . . . . . . . . . . . 12 (𝑎 = 𝑘 → (ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)))
4241, 34oveq12d 7421 . . . . . . . . . . 11 (𝑎 = 𝑘 → ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))))
4342breq2d 5158 . . . . . . . . . 10 (𝑎 = 𝑘 → ((𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ↔ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))))
4443rabbidv 3441 . . . . . . . . 9 (𝑎 = 𝑘 → {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎)))} = {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})
4539, 44eleq12d 2828 . . . . . . . 8 (𝑎 = 𝑘 → (((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎)))} ↔ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))}))
4636, 45imbi12d 345 . . . . . . 7 (𝑎 = 𝑘 → (((𝑎𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑎))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎)))}) ↔ ((𝑘𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑘))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})))
4746imbi2d 341 . . . . . 6 (𝑎 = 𝑘 → ((𝜑 → ((𝑎𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑎))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎)))})) ↔ (𝜑 → ((𝑘𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑘))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))}))))
48 sseq1 4005 . . . . . . . . 9 (𝑎 = (𝑘 ∪ {𝑐}) → (𝑎𝐴 ↔ (𝑘 ∪ {𝑐}) ⊆ 𝐴))
49 reseq2 5973 . . . . . . . . . . 11 (𝑎 = (𝑘 ∪ {𝑐}) → (𝑇𝑎) = (𝑇 ↾ (𝑘 ∪ {𝑐})))
5049oveq2d 7419 . . . . . . . . . 10 (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld Σg (𝑇𝑎)) = (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))
5150breq2d 5158 . . . . . . . . 9 (𝑎 = (𝑘 ∪ {𝑐}) → (0 < (ℂfld Σg (𝑇𝑎)) ↔ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))))
5248, 51anbi12d 632 . . . . . . . 8 (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑎𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑎))) ↔ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))))
53 reseq2 5973 . . . . . . . . . . 11 (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑇f · 𝑋) ↾ 𝑎) = ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐})))
5453oveq2d 7419 . . . . . . . . . 10 (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))))
5554, 50oveq12d 7421 . . . . . . . . 9 (𝑎 = (𝑘 ∪ {𝑐}) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))))
56 reseq2 5973 . . . . . . . . . . . . 13 (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑇f · (𝐹𝑋)) ↾ 𝑎) = ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐})))
5756oveq2d 7419 . . . . . . . . . . . 12 (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))))
5857, 50oveq12d 7421 . . . . . . . . . . 11 (𝑎 = (𝑘 ∪ {𝑐}) → ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))))
5958breq2d 5158 . . . . . . . . . 10 (𝑎 = (𝑘 ∪ {𝑐}) → ((𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ↔ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))))
6059rabbidv 3441 . . . . . . . . 9 (𝑎 = (𝑘 ∪ {𝑐}) → {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎)))} = {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})
6155, 60eleq12d 2828 . . . . . . . 8 (𝑎 = (𝑘 ∪ {𝑐}) → (((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎)))} ↔ ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))
6252, 61imbi12d 345 . . . . . . 7 (𝑎 = (𝑘 ∪ {𝑐}) → (((𝑎𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑎))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎)))}) ↔ (((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})))
6362imbi2d 341 . . . . . 6 (𝑎 = (𝑘 ∪ {𝑐}) → ((𝜑 → ((𝑎𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑎))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎)))})) ↔ (𝜑 → (((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))))
64 sseq1 4005 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑎𝐴𝐴𝐴))
65 reseq2 5973 . . . . . . . . . . 11 (𝑎 = 𝐴 → (𝑇𝑎) = (𝑇𝐴))
6665oveq2d 7419 . . . . . . . . . 10 (𝑎 = 𝐴 → (ℂfld Σg (𝑇𝑎)) = (ℂfld Σg (𝑇𝐴)))
6766breq2d 5158 . . . . . . . . 9 (𝑎 = 𝐴 → (0 < (ℂfld Σg (𝑇𝑎)) ↔ 0 < (ℂfld Σg (𝑇𝐴))))
6864, 67anbi12d 632 . . . . . . . 8 (𝑎 = 𝐴 → ((𝑎𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑎))) ↔ (𝐴𝐴 ∧ 0 < (ℂfld Σg (𝑇𝐴)))))
69 reseq2 5973 . . . . . . . . . . 11 (𝑎 = 𝐴 → ((𝑇f · 𝑋) ↾ 𝑎) = ((𝑇f · 𝑋) ↾ 𝐴))
7069oveq2d 7419 . . . . . . . . . 10 (𝑎 = 𝐴 → (ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇f · 𝑋) ↾ 𝐴)))
7170, 66oveq12d 7421 . . . . . . . . 9 (𝑎 = 𝐴 → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))))
72 reseq2 5973 . . . . . . . . . . . . 13 (𝑎 = 𝐴 → ((𝑇f · (𝐹𝑋)) ↾ 𝑎) = ((𝑇f · (𝐹𝑋)) ↾ 𝐴))
7372oveq2d 7419 . . . . . . . . . . . 12 (𝑎 = 𝐴 → (ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝐴)))
7473, 66oveq12d 7421 . . . . . . . . . . 11 (𝑎 = 𝐴 → ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))))
7574breq2d 5158 . . . . . . . . . 10 (𝑎 = 𝐴 → ((𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ↔ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴)))))
7675rabbidv 3441 . . . . . . . . 9 (𝑎 = 𝐴 → {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎)))} = {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴)))})
7771, 76eleq12d 2828 . . . . . . . 8 (𝑎 = 𝐴 → (((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎)))} ↔ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴)))}))
7868, 77imbi12d 345 . . . . . . 7 (𝑎 = 𝐴 → (((𝑎𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑎))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎)))}) ↔ ((𝐴𝐴 ∧ 0 < (ℂfld Σg (𝑇𝐴))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴)))})))
7978imbi2d 341 . . . . . 6 (𝑎 = 𝐴 → ((𝜑 → ((𝑎𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑎))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎)))})) ↔ (𝜑 → ((𝐴𝐴 ∧ 0 < (ℂfld Σg (𝑇𝐴))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴)))}))))
80 0re 11211 . . . . . . . . . 10 0 ∈ ℝ
8180ltnri 11318 . . . . . . . . 9 ¬ 0 < 0
8281pm2.21i 119 . . . . . . . 8 (0 < 0 → ((ℂfld Σg ((𝑇f · 𝑋) ↾ ∅)) / 0) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ ∅)) / 0)})
8382adantl 483 . . . . . . 7 ((∅ ⊆ 𝐴 ∧ 0 < 0) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ ∅)) / 0) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ ∅)) / 0)})
8483a1i 11 . . . . . 6 (𝜑 → ((∅ ⊆ 𝐴 ∧ 0 < 0) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ ∅)) / 0) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ ∅)) / 0)}))
85 impexp 452 . . . . . . . . . . . 12 (((𝑘𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑘))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))}) ↔ (𝑘𝐴 → (0 < (ℂfld Σg (𝑇𝑘)) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})))
86 simprl 770 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑘 ∪ {𝑐}) ⊆ 𝐴)
8786unssad 4185 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑘𝐴)
88 simpr 486 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 < (ℂfld Σg (𝑇𝑘))) → 0 < (ℂfld Σg (𝑇𝑘)))
89 jensen.1 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐷 ⊆ ℝ)
9089ad3antrrr 729 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇𝑘)) ∧ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) → 𝐷 ⊆ ℝ)
91 jensen.2 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐹:𝐷⟶ℝ)
9291ad3antrrr 729 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇𝑘)) ∧ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) → 𝐹:𝐷⟶ℝ)
93 simplll 774 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇𝑘)) ∧ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) → 𝜑)
94 jensen.3 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑎𝐷𝑏𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷)
9593, 94sylan 581 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇𝑘)) ∧ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) ∧ (𝑎𝐷𝑏𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷)
9693, 10syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇𝑘)) ∧ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) → 𝐴 ∈ Fin)
9793, 2syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇𝑘)) ∧ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) → 𝑇:𝐴⟶(0[,)+∞))
98 jensen.6 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑋:𝐴𝐷)
9993, 98syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇𝑘)) ∧ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) → 𝑋:𝐴𝐷)
1001ad3antrrr 729 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇𝑘)) ∧ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) → 0 < (ℂfld Σg 𝑇))
101 jensen.8 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑥𝐷𝑦𝐷𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹𝑥)) + ((1 − 𝑡) · (𝐹𝑦))))
10293, 101sylan 581 . . . . . . . . . . . . . . . . . 18 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇𝑘)) ∧ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) ∧ (𝑥𝐷𝑦𝐷𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹𝑥)) + ((1 − 𝑡) · (𝐹𝑦))))
103 simpllr 775 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇𝑘)) ∧ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) → ¬ 𝑐𝑘)
10486adantr 482 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇𝑘)) ∧ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) → (𝑘 ∪ {𝑐}) ⊆ 𝐴)
105 eqid 2733 . . . . . . . . . . . . . . . . . 18 (ℂfld Σg (𝑇𝑘)) = (ℂfld Σg (𝑇𝑘))
106 eqid 2733 . . . . . . . . . . . . . . . . . 18 (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))) = (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))
107 cnring 20951 . . . . . . . . . . . . . . . . . . . . . . 23 fld ∈ Ring
108 ringcmn 20088 . . . . . . . . . . . . . . . . . . . . . . 23 (ℂfld ∈ Ring → ℂfld ∈ CMnd)
109107, 108mp1i 13 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → ℂfld ∈ CMnd)
11010ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝐴 ∈ Fin)
111110, 87ssfid 9262 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑘 ∈ Fin)
112 rege0subm 20985 . . . . . . . . . . . . . . . . . . . . . . 23 (0[,)+∞) ∈ (SubMnd‘ℂfld)
113112a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (0[,)+∞) ∈ (SubMnd‘ℂfld))
1142ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑇:𝐴⟶(0[,)+∞))
115114, 87fssresd 6754 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑇𝑘):𝑘⟶(0[,)+∞))
116 c0ex 11203 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ V
117116a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 0 ∈ V)
118115, 111, 117fdmfifsupp 9368 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑇𝑘) finSupp 0)
11916, 109, 111, 113, 115, 118gsumsubmcl 19778 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (ℂfld Σg (𝑇𝑘)) ∈ (0[,)+∞))
120 elrege0 13426 . . . . . . . . . . . . . . . . . . . . . 22 ((ℂfld Σg (𝑇𝑘)) ∈ (0[,)+∞) ↔ ((ℂfld Σg (𝑇𝑘)) ∈ ℝ ∧ 0 ≤ (ℂfld Σg (𝑇𝑘))))
121120simplbi 499 . . . . . . . . . . . . . . . . . . . . 21 ((ℂfld Σg (𝑇𝑘)) ∈ (0[,)+∞) → (ℂfld Σg (𝑇𝑘)) ∈ ℝ)
122119, 121syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (ℂfld Σg (𝑇𝑘)) ∈ ℝ)
123122adantr 482 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇𝑘)) ∧ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) → (ℂfld Σg (𝑇𝑘)) ∈ ℝ)
124 simprl 770 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇𝑘)) ∧ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) → 0 < (ℂfld Σg (𝑇𝑘)))
125123, 124elrpd 13008 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇𝑘)) ∧ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) → (ℂfld Σg (𝑇𝑘)) ∈ ℝ+)
126 simprr 772 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇𝑘)) ∧ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})
127 fveq2 6887 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) → (𝐹𝑤) = (𝐹‘((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))))
128127breq1d 5156 . . . . . . . . . . . . . . . . . . . . 21 (𝑤 = ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) → ((𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ↔ (𝐹‘((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))))
129128elrab 3681 . . . . . . . . . . . . . . . . . . . 20 (((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))} ↔ (((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ 𝐷 ∧ (𝐹‘((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))))
130126, 129sylib 217 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇𝑘)) ∧ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) → (((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ 𝐷 ∧ (𝐹‘((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))))
131130simpld 496 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇𝑘)) ∧ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ 𝐷)
132130simprd 497 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇𝑘)) ∧ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) → (𝐹‘((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))))
13390, 92, 95, 96, 97, 99, 100, 102, 103, 104, 105, 106, 125, 131, 132jensenlem2 26471 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇𝑘)) ∧ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) → (((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ 𝐷 ∧ (𝐹‘((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))))
134 fveq2 6887 . . . . . . . . . . . . . . . . . . 19 (𝑤 = ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → (𝐹𝑤) = (𝐹‘((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))))
135134breq1d 5156 . . . . . . . . . . . . . . . . . 18 (𝑤 = ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → ((𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ↔ (𝐹‘((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))))
136135elrab 3681 . . . . . . . . . . . . . . . . 17 (((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))} ↔ (((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ 𝐷 ∧ (𝐹‘((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))))
137133, 136sylibr 233 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇𝑘)) ∧ ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})
138137expr 458 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 < (ℂfld Σg (𝑇𝑘))) → (((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))} → ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))
13988, 138embantd 59 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 < (ℂfld Σg (𝑇𝑘))) → ((0 < (ℂfld Σg (𝑇𝑘)) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))}) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))
140 cnfldbas 20932 . . . . . . . . . . . . . . . . . . . . 21 ℂ = (Base‘ℂfld)
141 ringmnd 20056 . . . . . . . . . . . . . . . . . . . . . 22 (ℂfld ∈ Ring → ℂfld ∈ Mnd)
142107, 141mp1i 13 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ℂfld ∈ Mnd)
143110, 86ssfid 9262 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑘 ∪ {𝑐}) ∈ Fin)
144143adantr 482 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑘 ∪ {𝑐}) ∈ Fin)
145 ssun2 4171 . . . . . . . . . . . . . . . . . . . . . . 23 {𝑐} ⊆ (𝑘 ∪ {𝑐})
146 vsnid 4663 . . . . . . . . . . . . . . . . . . . . . . 23 𝑐 ∈ {𝑐}
147145, 146sselii 3977 . . . . . . . . . . . . . . . . . . . . . 22 𝑐 ∈ (𝑘 ∪ {𝑐})
148147a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑐 ∈ (𝑘 ∪ {𝑐}))
149 remulcl 11190 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ)
150149adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
151 rge0ssre 13428 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0[,)+∞) ⊆ ℝ
152 fss 6730 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑇:𝐴⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝑇:𝐴⟶ℝ)
1532, 151, 152sylancl 587 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑇:𝐴⟶ℝ)
15498, 89fssd 6731 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑋:𝐴⟶ℝ)
155 inidm 4216 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴𝐴) = 𝐴
156150, 153, 154, 10, 10, 155off 7682 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑇f · 𝑋):𝐴⟶ℝ)
157 ax-resscn 11162 . . . . . . . . . . . . . . . . . . . . . . . 24 ℝ ⊆ ℂ
158 fss 6730 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑇f · 𝑋):𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → (𝑇f · 𝑋):𝐴⟶ℂ)
159156, 157, 158sylancl 587 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑇f · 𝑋):𝐴⟶ℂ)
160159ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑇f · 𝑋):𝐴⟶ℂ)
16186adantr 482 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑘 ∪ {𝑐}) ⊆ 𝐴)
162160, 161fssresd 6754 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ)
1632ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑇:𝐴⟶(0[,)+∞))
164110adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝐴 ∈ Fin)
165163, 164fexd 7223 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑇 ∈ V)
16698ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑋:𝐴𝐷)
167166, 164fexd 7223 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑋 ∈ V)
168 offres 7964 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑇 ∈ V ∧ 𝑋 ∈ V) → ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐})) = ((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · (𝑋 ↾ (𝑘 ∪ {𝑐}))))
169165, 167, 168syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐})) = ((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · (𝑋 ↾ (𝑘 ∪ {𝑐}))))
170169oveq1d 7418 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐})) supp 0) = (((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · (𝑋 ↾ (𝑘 ∪ {𝑐}))) supp 0))
171151, 157sstri 3989 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0[,)+∞) ⊆ ℂ
172 fss 6730 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑇:𝐴⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℂ) → 𝑇:𝐴⟶ℂ)
173163, 171, 172sylancl 587 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑇:𝐴⟶ℂ)
174173, 161fssresd 6754 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑇 ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ)
175 eldifi 4124 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) → 𝑥 ∈ (𝑘 ∪ {𝑐}))
176175adantl 483 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐})) → 𝑥 ∈ (𝑘 ∪ {𝑐}))
177176fvresd 6907 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐})) → ((𝑇 ↾ (𝑘 ∪ {𝑐}))‘𝑥) = (𝑇𝑥))
178 difun2 4478 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑘 ∪ {𝑐}) ∖ {𝑐}) = (𝑘 ∖ {𝑐})
179 difss 4129 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 ∖ {𝑐}) ⊆ 𝑘
180178, 179eqsstri 4014 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑘 ∪ {𝑐}) ∖ {𝑐}) ⊆ 𝑘
181180sseli 3976 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) → 𝑥𝑘)
182 simpr 486 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 0 = (ℂfld Σg (𝑇𝑘)))
18387adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑘𝐴)
184163, 183feqresmpt 6956 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑇𝑘) = (𝑥𝑘 ↦ (𝑇𝑥)))
185184oveq2d 7419 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (ℂfld Σg (𝑇𝑘)) = (ℂfld Σg (𝑥𝑘 ↦ (𝑇𝑥))))
186111adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑘 ∈ Fin)
187183sselda 3980 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝑘) → 𝑥𝐴)
188163ffvelcdmda 7081 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝐴) → (𝑇𝑥) ∈ (0[,)+∞))
189187, 188syldan 592 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝑘) → (𝑇𝑥) ∈ (0[,)+∞))
190171, 189sselid 3978 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝑘) → (𝑇𝑥) ∈ ℂ)
191186, 190gsumfsum 20996 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (ℂfld Σg (𝑥𝑘 ↦ (𝑇𝑥))) = Σ𝑥𝑘 (𝑇𝑥))
192182, 185, 1913eqtrrd 2778 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → Σ𝑥𝑘 (𝑇𝑥) = 0)
193 elrege0 13426 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑇𝑥) ∈ (0[,)+∞) ↔ ((𝑇𝑥) ∈ ℝ ∧ 0 ≤ (𝑇𝑥)))
194189, 193sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝑘) → ((𝑇𝑥) ∈ ℝ ∧ 0 ≤ (𝑇𝑥)))
195194simpld 496 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝑘) → (𝑇𝑥) ∈ ℝ)
196194simprd 497 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝑘) → 0 ≤ (𝑇𝑥))
197186, 195, 196fsum00 15739 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (Σ𝑥𝑘 (𝑇𝑥) = 0 ↔ ∀𝑥𝑘 (𝑇𝑥) = 0))
198192, 197mpbid 231 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ∀𝑥𝑘 (𝑇𝑥) = 0)
199198r19.21bi 3249 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝑘) → (𝑇𝑥) = 0)
200181, 199sylan2 594 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐})) → (𝑇𝑥) = 0)
201177, 200eqtrd 2773 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐})) → ((𝑇 ↾ (𝑘 ∪ {𝑐}))‘𝑥) = 0)
202174, 201suppss 8173 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇 ↾ (𝑘 ∪ {𝑐})) supp 0) ⊆ {𝑐})
203 mul02 11387 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ ℂ → (0 · 𝑥) = 0)
204203adantl 483 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥 ∈ ℂ) → (0 · 𝑥) = 0)
20589ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝐷 ⊆ ℝ)
206205, 157sstrdi 3992 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝐷 ⊆ ℂ)
207166, 206fssd 6731 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑋:𝐴⟶ℂ)
208207, 161fssresd 6754 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑋 ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ)
209116a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 0 ∈ V)
210202, 204, 174, 208, 144, 209suppssof1 8178 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · (𝑋 ↾ (𝑘 ∪ {𝑐}))) supp 0) ⊆ {𝑐})
211170, 210eqsstrd 4018 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐})) supp 0) ⊆ {𝑐})
212140, 16, 142, 144, 148, 162, 211gsumpt 19821 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) = (((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))‘𝑐))
213148fvresd 6907 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))‘𝑐) = ((𝑇f · 𝑋)‘𝑐))
214163ffnd 6714 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑇 Fn 𝐴)
215166ffnd 6714 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑋 Fn 𝐴)
216161, 148sseldd 3981 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑐𝐴)
217 fnfvof 7681 . . . . . . . . . . . . . . . . . . . . 21 (((𝑇 Fn 𝐴𝑋 Fn 𝐴) ∧ (𝐴 ∈ Fin ∧ 𝑐𝐴)) → ((𝑇f · 𝑋)‘𝑐) = ((𝑇𝑐) · (𝑋𝑐)))
218214, 215, 164, 216, 217syl22anc 838 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇f · 𝑋)‘𝑐) = ((𝑇𝑐) · (𝑋𝑐)))
219212, 213, 2183eqtrd 2777 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) = ((𝑇𝑐) · (𝑋𝑐)))
220140, 16, 142, 144, 148, 174, 202gsumpt 19821 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))) = ((𝑇 ↾ (𝑘 ∪ {𝑐}))‘𝑐))
221148fvresd 6907 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇 ↾ (𝑘 ∪ {𝑐}))‘𝑐) = (𝑇𝑐))
222220, 221eqtrd 2773 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))) = (𝑇𝑐))
223219, 222oveq12d 7421 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) = (((𝑇𝑐) · (𝑋𝑐)) / (𝑇𝑐)))
224207, 216ffvelcdmd 7082 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑋𝑐) ∈ ℂ)
225173, 216ffvelcdmd 7082 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑇𝑐) ∈ ℂ)
226 simplrr 777 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))
227226, 222breqtrd 5172 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 0 < (𝑇𝑐))
228227gt0ne0d 11773 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑇𝑐) ≠ 0)
229224, 225, 228divcan3d 11990 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (((𝑇𝑐) · (𝑋𝑐)) / (𝑇𝑐)) = (𝑋𝑐))
230223, 229eqtrd 2773 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) = (𝑋𝑐))
231166, 216ffvelcdmd 7082 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑋𝑐) ∈ 𝐷)
232230, 231eqeltrd 2834 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ 𝐷)
23391ad3antrrr 729 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝐹:𝐷⟶ℝ)
234233, 231ffvelcdmd 7082 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹‘(𝑋𝑐)) ∈ ℝ)
235234leidd 11775 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹‘(𝑋𝑐)) ≤ (𝐹‘(𝑋𝑐)))
236230fveq2d 6891 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹‘((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) = (𝐹‘(𝑋𝑐)))
237 fco 6737 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐹:𝐷⟶ℝ ∧ 𝑋:𝐴𝐷) → (𝐹𝑋):𝐴⟶ℝ)
23891, 98, 237syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐹𝑋):𝐴⟶ℝ)
239150, 153, 238, 10, 10, 155off 7682 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑇f · (𝐹𝑋)):𝐴⟶ℝ)
240 fss 6730 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑇f · (𝐹𝑋)):𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → (𝑇f · (𝐹𝑋)):𝐴⟶ℂ)
241239, 157, 240sylancl 587 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑇f · (𝐹𝑋)):𝐴⟶ℂ)
242241ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑇f · (𝐹𝑋)):𝐴⟶ℂ)
243242, 161fssresd 6754 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ)
244238ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹𝑋):𝐴⟶ℝ)
245244, 164fexd 7223 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹𝑋) ∈ V)
246 offres 7964 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑇 ∈ V ∧ (𝐹𝑋) ∈ V) → ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐})) = ((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · ((𝐹𝑋) ↾ (𝑘 ∪ {𝑐}))))
247165, 245, 246syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐})) = ((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · ((𝐹𝑋) ↾ (𝑘 ∪ {𝑐}))))
248247oveq1d 7418 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐})) supp 0) = (((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · ((𝐹𝑋) ↾ (𝑘 ∪ {𝑐}))) supp 0))
249 fss 6730 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐹𝑋):𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → (𝐹𝑋):𝐴⟶ℂ)
250244, 157, 249sylancl 587 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹𝑋):𝐴⟶ℂ)
251250, 161fssresd 6754 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝐹𝑋) ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ)
252202, 204, 174, 251, 144, 209suppssof1 8178 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · ((𝐹𝑋) ↾ (𝑘 ∪ {𝑐}))) supp 0) ⊆ {𝑐})
253248, 252eqsstrd 4018 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐})) supp 0) ⊆ {𝑐})
254140, 16, 142, 144, 148, 243, 253gsumpt 19821 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) = (((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))‘𝑐))
255148fvresd 6907 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))‘𝑐) = ((𝑇f · (𝐹𝑋))‘𝑐))
25691ffnd 6714 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐹 Fn 𝐷)
257 fnfco 6752 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹 Fn 𝐷𝑋:𝐴𝐷) → (𝐹𝑋) Fn 𝐴)
258256, 98, 257syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝐹𝑋) Fn 𝐴)
259258ad3antrrr 729 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹𝑋) Fn 𝐴)
260 fnfvof 7681 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑇 Fn 𝐴 ∧ (𝐹𝑋) Fn 𝐴) ∧ (𝐴 ∈ Fin ∧ 𝑐𝐴)) → ((𝑇f · (𝐹𝑋))‘𝑐) = ((𝑇𝑐) · ((𝐹𝑋)‘𝑐)))
261214, 259, 164, 216, 260syl22anc 838 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇f · (𝐹𝑋))‘𝑐) = ((𝑇𝑐) · ((𝐹𝑋)‘𝑐)))
262 fvco3 6985 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑋:𝐴𝐷𝑐𝐴) → ((𝐹𝑋)‘𝑐) = (𝐹‘(𝑋𝑐)))
263166, 216, 262syl2anc 585 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝐹𝑋)‘𝑐) = (𝐹‘(𝑋𝑐)))
264263oveq2d 7419 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇𝑐) · ((𝐹𝑋)‘𝑐)) = ((𝑇𝑐) · (𝐹‘(𝑋𝑐))))
265261, 264eqtrd 2773 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇f · (𝐹𝑋))‘𝑐) = ((𝑇𝑐) · (𝐹‘(𝑋𝑐))))
266254, 255, 2653eqtrd 2777 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) = ((𝑇𝑐) · (𝐹‘(𝑋𝑐))))
267266, 222oveq12d 7421 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) = (((𝑇𝑐) · (𝐹‘(𝑋𝑐))) / (𝑇𝑐)))
268234recnd 11237 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹‘(𝑋𝑐)) ∈ ℂ)
269268, 225, 228divcan3d 11990 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (((𝑇𝑐) · (𝐹‘(𝑋𝑐))) / (𝑇𝑐)) = (𝐹‘(𝑋𝑐)))
270267, 269eqtrd 2773 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) = (𝐹‘(𝑋𝑐)))
271235, 236, 2703brtr4d 5178 . . . . . . . . . . . . . . . 16 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹‘((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))))
272135, 232, 271elrabd 3683 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})
273272a1d 25 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((0 < (ℂfld Σg (𝑇𝑘)) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))}) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))
274120simprbi 498 . . . . . . . . . . . . . . . 16 ((ℂfld Σg (𝑇𝑘)) ∈ (0[,)+∞) → 0 ≤ (ℂfld Σg (𝑇𝑘)))
275119, 274syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 0 ≤ (ℂfld Σg (𝑇𝑘)))
276 leloe 11295 . . . . . . . . . . . . . . . 16 ((0 ∈ ℝ ∧ (ℂfld Σg (𝑇𝑘)) ∈ ℝ) → (0 ≤ (ℂfld Σg (𝑇𝑘)) ↔ (0 < (ℂfld Σg (𝑇𝑘)) ∨ 0 = (ℂfld Σg (𝑇𝑘)))))
27780, 122, 276sylancr 588 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (0 ≤ (ℂfld Σg (𝑇𝑘)) ↔ (0 < (ℂfld Σg (𝑇𝑘)) ∨ 0 = (ℂfld Σg (𝑇𝑘)))))
278275, 277mpbid 231 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (0 < (ℂfld Σg (𝑇𝑘)) ∨ 0 = (ℂfld Σg (𝑇𝑘))))
279139, 273, 278mpjaodan 958 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → ((0 < (ℂfld Σg (𝑇𝑘)) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))}) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))
28087, 279embantd 59 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → ((𝑘𝐴 → (0 < (ℂfld Σg (𝑇𝑘)) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))
28185, 280biimtrid 241 . . . . . . . . . . 11 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (((𝑘𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑘))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))}) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))
282281ex 414 . . . . . . . . . 10 ((𝜑 ∧ ¬ 𝑐𝑘) → (((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → (((𝑘𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑘))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))}) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})))
283282com23 86 . . . . . . . . 9 ((𝜑 ∧ ¬ 𝑐𝑘) → (((𝑘𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑘))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))}) → (((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})))
284283expcom 415 . . . . . . . 8 𝑐𝑘 → (𝜑 → (((𝑘𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑘))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))}) → (((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))))
285284adantl 483 . . . . . . 7 ((𝑘 ∈ Fin ∧ ¬ 𝑐𝑘) → (𝜑 → (((𝑘𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑘))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))}) → (((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))))
286285a2d 29 . . . . . 6 ((𝑘 ∈ Fin ∧ ¬ 𝑐𝑘) → ((𝜑 → ((𝑘𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑘))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})) → (𝜑 → (((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))))
28731, 47, 63, 79, 84, 286findcard2s 9160 . . . . 5 (𝐴 ∈ Fin → (𝜑 → ((𝐴𝐴 ∧ 0 < (ℂfld Σg (𝑇𝐴))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴)))})))
28810, 287mpcom 38 . . . 4 (𝜑 → ((𝐴𝐴 ∧ 0 < (ℂfld Σg (𝑇𝐴))) → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴)))}))
2899, 288mpd 15 . . 3 (𝜑 → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴)))})
290156ffnd 6714 . . . . . 6 (𝜑 → (𝑇f · 𝑋) Fn 𝐴)
291 fnresdm 6665 . . . . . 6 ((𝑇f · 𝑋) Fn 𝐴 → ((𝑇f · 𝑋) ↾ 𝐴) = (𝑇f · 𝑋))
292290, 291syl 17 . . . . 5 (𝜑 → ((𝑇f · 𝑋) ↾ 𝐴) = (𝑇f · 𝑋))
293292oveq2d 7419 . . . 4 (𝜑 → (ℂfld Σg ((𝑇f · 𝑋) ↾ 𝐴)) = (ℂfld Σg (𝑇f · 𝑋)))
294293, 6oveq12d 7421 . . 3 (𝜑 → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))) = ((ℂfld Σg (𝑇f · 𝑋)) / (ℂfld Σg 𝑇)))
2953, 258, 10, 10, 155offn 7677 . . . . . . . 8 (𝜑 → (𝑇f · (𝐹𝑋)) Fn 𝐴)
296 fnresdm 6665 . . . . . . . 8 ((𝑇f · (𝐹𝑋)) Fn 𝐴 → ((𝑇f · (𝐹𝑋)) ↾ 𝐴) = (𝑇f · (𝐹𝑋)))
297295, 296syl 17 . . . . . . 7 (𝜑 → ((𝑇f · (𝐹𝑋)) ↾ 𝐴) = (𝑇f · (𝐹𝑋)))
298297oveq2d 7419 . . . . . 6 (𝜑 → (ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝐴)) = (ℂfld Σg (𝑇f · (𝐹𝑋))))
299298, 6oveq12d 7421 . . . . 5 (𝜑 → ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))) = ((ℂfld Σg (𝑇f · (𝐹𝑋))) / (ℂfld Σg 𝑇)))
300299breq2d 5158 . . . 4 (𝜑 → ((𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))) ↔ (𝐹𝑤) ≤ ((ℂfld Σg (𝑇f · (𝐹𝑋))) / (ℂfld Σg 𝑇))))
301300rabbidv 3441 . . 3 (𝜑 → {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴)))} = {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg (𝑇f · (𝐹𝑋))) / (ℂfld Σg 𝑇))})
302289, 294, 3013eltr3d 2848 . 2 (𝜑 → ((ℂfld Σg (𝑇f · 𝑋)) / (ℂfld Σg 𝑇)) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg (𝑇f · (𝐹𝑋))) / (ℂfld Σg 𝑇))})
303 fveq2 6887 . . . 4 (𝑤 = ((ℂfld Σg (𝑇f · 𝑋)) / (ℂfld Σg 𝑇)) → (𝐹𝑤) = (𝐹‘((ℂfld Σg (𝑇f · 𝑋)) / (ℂfld Σg 𝑇))))
304303breq1d 5156 . . 3 (𝑤 = ((ℂfld Σg (𝑇f · 𝑋)) / (ℂfld Σg 𝑇)) → ((𝐹𝑤) ≤ ((ℂfld Σg (𝑇f · (𝐹𝑋))) / (ℂfld Σg 𝑇)) ↔ (𝐹‘((ℂfld Σg (𝑇f · 𝑋)) / (ℂfld Σg 𝑇))) ≤ ((ℂfld Σg (𝑇f · (𝐹𝑋))) / (ℂfld Σg 𝑇))))
305304elrab 3681 . 2 (((ℂfld Σg (𝑇f · 𝑋)) / (ℂfld Σg 𝑇)) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg (𝑇f · (𝐹𝑋))) / (ℂfld Σg 𝑇))} ↔ (((ℂfld Σg (𝑇f · 𝑋)) / (ℂfld Σg 𝑇)) ∈ 𝐷 ∧ (𝐹‘((ℂfld Σg (𝑇f · 𝑋)) / (ℂfld Σg 𝑇))) ≤ ((ℂfld Σg (𝑇f · (𝐹𝑋))) / (ℂfld Σg 𝑇))))
306302, 305sylib 217 1 (𝜑 → (((ℂfld Σg (𝑇f · 𝑋)) / (ℂfld Σg 𝑇)) ∈ 𝐷 ∧ (𝐹‘((ℂfld Σg (𝑇f · 𝑋)) / (ℂfld Σg 𝑇))) ≤ ((ℂfld Σg (𝑇f · (𝐹𝑋))) / (ℂfld Σg 𝑇))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wral 3062  {crab 3433  Vcvv 3475  cdif 3943  cun 3944  wss 3946  c0 4320  {csn 4626   class class class wbr 5146  cmpt 5229  cres 5676  ccom 5678   Fn wfn 6534  wf 6535  cfv 6539  (class class class)co 7403  f cof 7662   supp csupp 8140  Fincfn 8934  cc 11103  cr 11104  0cc0 11105  1c1 11106   + caddc 11108   · cmul 11110  +∞cpnf 11240   < clt 11243  cle 11244  cmin 11439   / cdiv 11866  [,)cico 13321  [,]cicc 13322  Σcsu 15627   Σg cgsu 17381  Mndcmnd 18620  SubMndcsubmnd 18665  CMndccmn 19640  Ringcrg 20046  fldccnfld 20928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5283  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719  ax-inf2 9631  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182  ax-pre-sup 11183  ax-addf 11184  ax-mulf 11185
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4907  df-int 4949  df-iun 4997  df-iin 4998  df-br 5147  df-opab 5209  df-mpt 5230  df-tr 5264  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-se 5630  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6296  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-isom 6548  df-riota 7359  df-ov 7406  df-oprab 7407  df-mpo 7408  df-of 7664  df-om 7850  df-1st 7969  df-2nd 7970  df-supp 8141  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8365  df-rdg 8404  df-1o 8460  df-er 8698  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-fsupp 9357  df-sup 9432  df-oi 9500  df-card 9929  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11441  df-neg 11442  df-div 11867  df-nn 12208  df-2 12270  df-3 12271  df-4 12272  df-5 12273  df-6 12274  df-7 12275  df-8 12276  df-9 12277  df-n0 12468  df-z 12554  df-dec 12673  df-uz 12818  df-rp 12970  df-ico 13325  df-icc 13326  df-fz 13480  df-fzo 13623  df-seq 13962  df-exp 14023  df-hash 14286  df-cj 15041  df-re 15042  df-im 15043  df-sqrt 15177  df-abs 15178  df-clim 15427  df-sum 15628  df-struct 17075  df-sets 17092  df-slot 17110  df-ndx 17122  df-base 17140  df-ress 17169  df-plusg 17205  df-mulr 17206  df-starv 17207  df-tset 17211  df-ple 17212  df-ds 17214  df-unif 17215  df-0g 17382  df-gsum 17383  df-mre 17525  df-mrc 17526  df-acs 17528  df-mgm 18556  df-sgrp 18605  df-mnd 18621  df-submnd 18667  df-grp 18817  df-minusg 18818  df-mulg 18944  df-subg 18996  df-cntz 19174  df-cmn 19642  df-abl 19643  df-mgp 19979  df-ur 19996  df-ring 20048  df-cring 20049  df-oppr 20138  df-dvdsr 20159  df-unit 20160  df-invr 20190  df-dvr 20203  df-drng 20305  df-subrg 20348  df-cnfld 20929  df-refld 21141
This theorem is referenced by:  amgmlem  26473  amgmwlem  47750
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