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Theorem jensen 27046
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 6737 . . . . . . . 8 (𝜑𝑇 Fn 𝐴)
4 fnresdm 6687 . . . . . . . 8 (𝑇 Fn 𝐴 → (𝑇𝐴) = 𝑇)
53, 4syl 17 . . . . . . 7 (𝜑 → (𝑇𝐴) = 𝑇)
65oveq2d 7446 . . . . . 6 (𝜑 → (ℂfld Σg (𝑇𝐴)) = (ℂfld Σg 𝑇))
71, 6breqtrrd 5175 . . . . 5 (𝜑 → 0 < (ℂfld Σg (𝑇𝐴)))
8 ssid 4017 . . . . 5 𝐴𝐴
97, 8jctil 519 . . . 4 (𝜑 → (𝐴𝐴 ∧ 0 < (ℂfld Σg (𝑇𝐴))))
10 jensen.4 . . . . 5 (𝜑𝐴 ∈ Fin)
11 sseq1 4020 . . . . . . . . 9 (𝑎 = ∅ → (𝑎𝐴 ↔ ∅ ⊆ 𝐴))
12 reseq2 5994 . . . . . . . . . . . . 13 (𝑎 = ∅ → (𝑇𝑎) = (𝑇 ↾ ∅))
13 res0 6003 . . . . . . . . . . . . 13 (𝑇 ↾ ∅) = ∅
1412, 13eqtrdi 2790 . . . . . . . . . . . 12 (𝑎 = ∅ → (𝑇𝑎) = ∅)
1514oveq2d 7446 . . . . . . . . . . 11 (𝑎 = ∅ → (ℂfld Σg (𝑇𝑎)) = (ℂfld Σg ∅))
16 cnfld0 21422 . . . . . . . . . . . 12 0 = (0g‘ℂfld)
1716gsum0 18709 . . . . . . . . . . 11 (ℂfld Σg ∅) = 0
1815, 17eqtrdi 2790 . . . . . . . . . 10 (𝑎 = ∅ → (ℂfld Σg (𝑇𝑎)) = 0)
1918breq2d 5159 . . . . . . . . 9 (𝑎 = ∅ → (0 < (ℂfld Σg (𝑇𝑎)) ↔ 0 < 0))
2011, 19anbi12d 632 . . . . . . . 8 (𝑎 = ∅ → ((𝑎𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑎))) ↔ (∅ ⊆ 𝐴 ∧ 0 < 0)))
21 reseq2 5994 . . . . . . . . . . 11 (𝑎 = ∅ → ((𝑇f · 𝑋) ↾ 𝑎) = ((𝑇f · 𝑋) ↾ ∅))
2221oveq2d 7446 . . . . . . . . . 10 (𝑎 = ∅ → (ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇f · 𝑋) ↾ ∅)))
2322, 18oveq12d 7448 . . . . . . . . 9 (𝑎 = ∅ → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇f · 𝑋) ↾ ∅)) / 0))
24 reseq2 5994 . . . . . . . . . . . . 13 (𝑎 = ∅ → ((𝑇f · (𝐹𝑋)) ↾ 𝑎) = ((𝑇f · (𝐹𝑋)) ↾ ∅))
2524oveq2d 7446 . . . . . . . . . . . 12 (𝑎 = ∅ → (ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ ∅)))
2625, 18oveq12d 7448 . . . . . . . . . . 11 (𝑎 = ∅ → ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ ∅)) / 0))
2726breq2d 5159 . . . . . . . . . 10 (𝑎 = ∅ → ((𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ↔ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ ∅)) / 0)))
2827rabbidv 3440 . . . . . . . . 9 (𝑎 = ∅ → {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎)))} = {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ ∅)) / 0)})
2923, 28eleq12d 2832 . . . . . . . 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 344 . . . . . . 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 340 . . . . . 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 4020 . . . . . . . . 9 (𝑎 = 𝑘 → (𝑎𝐴𝑘𝐴))
33 reseq2 5994 . . . . . . . . . . 11 (𝑎 = 𝑘 → (𝑇𝑎) = (𝑇𝑘))
3433oveq2d 7446 . . . . . . . . . 10 (𝑎 = 𝑘 → (ℂfld Σg (𝑇𝑎)) = (ℂfld Σg (𝑇𝑘)))
3534breq2d 5159 . . . . . . . . 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 5994 . . . . . . . . . . 11 (𝑎 = 𝑘 → ((𝑇f · 𝑋) ↾ 𝑎) = ((𝑇f · 𝑋) ↾ 𝑘))
3837oveq2d 7446 . . . . . . . . . 10 (𝑎 = 𝑘 → (ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)))
3938, 34oveq12d 7448 . . . . . . . . 9 (𝑎 = 𝑘 → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))))
40 reseq2 5994 . . . . . . . . . . . . 13 (𝑎 = 𝑘 → ((𝑇f · (𝐹𝑋)) ↾ 𝑎) = ((𝑇f · (𝐹𝑋)) ↾ 𝑘))
4140oveq2d 7446 . . . . . . . . . . . 12 (𝑎 = 𝑘 → (ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)))
4241, 34oveq12d 7448 . . . . . . . . . . 11 (𝑎 = 𝑘 → ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))))
4342breq2d 5159 . . . . . . . . . 10 (𝑎 = 𝑘 → ((𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ↔ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))))
4443rabbidv 3440 . . . . . . . . 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 2832 . . . . . . . 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 344 . . . . . . 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 340 . . . . . 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 4020 . . . . . . . . 9 (𝑎 = (𝑘 ∪ {𝑐}) → (𝑎𝐴 ↔ (𝑘 ∪ {𝑐}) ⊆ 𝐴))
49 reseq2 5994 . . . . . . . . . . 11 (𝑎 = (𝑘 ∪ {𝑐}) → (𝑇𝑎) = (𝑇 ↾ (𝑘 ∪ {𝑐})))
5049oveq2d 7446 . . . . . . . . . 10 (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld Σg (𝑇𝑎)) = (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))
5150breq2d 5159 . . . . . . . . 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 5994 . . . . . . . . . . 11 (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑇f · 𝑋) ↾ 𝑎) = ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐})))
5453oveq2d 7446 . . . . . . . . . 10 (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐}))))
5554, 50oveq12d 7448 . . . . . . . . 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 5994 . . . . . . . . . . . . 13 (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑇f · (𝐹𝑋)) ↾ 𝑎) = ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐})))
5756oveq2d 7446 . . . . . . . . . . . 12 (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))))
5857, 50oveq12d 7448 . . . . . . . . . . 11 (𝑎 = (𝑘 ∪ {𝑐}) → ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))))
5958breq2d 5159 . . . . . . . . . 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 3440 . . . . . . . . 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 2832 . . . . . . . 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 344 . . . . . . 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 340 . . . . . 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 4020 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑎𝐴𝐴𝐴))
65 reseq2 5994 . . . . . . . . . . 11 (𝑎 = 𝐴 → (𝑇𝑎) = (𝑇𝐴))
6665oveq2d 7446 . . . . . . . . . 10 (𝑎 = 𝐴 → (ℂfld Σg (𝑇𝑎)) = (ℂfld Σg (𝑇𝐴)))
6766breq2d 5159 . . . . . . . . 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 5994 . . . . . . . . . . 11 (𝑎 = 𝐴 → ((𝑇f · 𝑋) ↾ 𝑎) = ((𝑇f · 𝑋) ↾ 𝐴))
7069oveq2d 7446 . . . . . . . . . 10 (𝑎 = 𝐴 → (ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇f · 𝑋) ↾ 𝐴)))
7170, 66oveq12d 7448 . . . . . . . . 9 (𝑎 = 𝐴 → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))))
72 reseq2 5994 . . . . . . . . . . . . 13 (𝑎 = 𝐴 → ((𝑇f · (𝐹𝑋)) ↾ 𝑎) = ((𝑇f · (𝐹𝑋)) ↾ 𝐴))
7372oveq2d 7446 . . . . . . . . . . . 12 (𝑎 = 𝐴 → (ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝐴)))
7473, 66oveq12d 7448 . . . . . . . . . . 11 (𝑎 = 𝐴 → ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))))
7574breq2d 5159 . . . . . . . . . 10 (𝑎 = 𝐴 → ((𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ↔ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴)))))
7675rabbidv 3440 . . . . . . . . 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 2832 . . . . . . . 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 344 . . . . . . 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 340 . . . . . 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 11260 . . . . . . . . . 10 0 ∈ ℝ
8180ltnri 11367 . . . . . . . . 9 ¬ 0 < 0
8281pm2.21i 119 . . . . . . . 8 (0 < 0 → ((ℂfld Σg ((𝑇f · 𝑋) ↾ ∅)) / 0) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ ∅)) / 0)})
8382adantl 481 . . . . . . 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 450 . . . . . . . . . . . 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 771 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑘 ∪ {𝑐}) ⊆ 𝐴)
8786unssad 4202 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑘𝐴)
88 simpr 484 . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . . . . . 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 580 . . . . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . . . 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 580 . . . . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . 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 2734 . . . . . . . . . . . . . . . . . 18 (ℂfld Σg (𝑇𝑘)) = (ℂfld Σg (𝑇𝑘))
106 eqid 2734 . . . . . . . . . . . . . . . . . 18 (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))) = (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))
107 cnring 21420 . . . . . . . . . . . . . . . . . . . . . . 23 fld ∈ Ring
108 ringcmn 20295 . . . . . . . . . . . . . . . . . . . . . . 23 (ℂfld ∈ Ring → ℂfld ∈ CMnd)
109107, 108mp1i 13 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → ℂfld ∈ CMnd)
11010ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝐴 ∈ Fin)
111110, 87ssfid 9298 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑘 ∈ Fin)
112 rege0subm 21458 . . . . . . . . . . . . . . . . . . . . . . 23 (0[,)+∞) ∈ (SubMnd‘ℂfld)
113112a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (0[,)+∞) ∈ (SubMnd‘ℂfld))
1142ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑇:𝐴⟶(0[,)+∞))
115114, 87fssresd 6775 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑇𝑘):𝑘⟶(0[,)+∞))
116 c0ex 11252 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ V
117116a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 0 ∈ V)
118115, 111, 117fdmfifsupp 9412 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑇𝑘) finSupp 0)
11916, 109, 111, 113, 115, 118gsumsubmcl 19951 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (ℂfld Σg (𝑇𝑘)) ∈ (0[,)+∞))
120 elrege0 13490 . . . . . . . . . . . . . . . . . . . . . 22 ((ℂfld Σg (𝑇𝑘)) ∈ (0[,)+∞) ↔ ((ℂfld Σg (𝑇𝑘)) ∈ ℝ ∧ 0 ≤ (ℂfld Σg (𝑇𝑘))))
121120simplbi 497 . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . . . . 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 13071 . . . . . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) → (𝐹𝑤) = (𝐹‘((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))))
128127breq1d 5157 . . . . . . . . . . . . . . . . . . . . 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 3694 . . . . . . . . . . . . . . . . . . . 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 218 . . . . . . . . . . . . . . . . . . 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 494 . . . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . . . 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 27045 . . . . . . . . . . . . . . . . 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 6906 . . . . . . . . . . . . . . . . . . 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 5157 . . . . . . . . . . . . . . . . . 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 3694 . . . . . . . . . . . . . . . . 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 234 . . . . . . . . . . . . . . . 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 456 . . . . . . . . . . . . . . 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 21385 . . . . . . . . . . . . . . . . . . . . 21 ℂ = (Base‘ℂfld)
141 ringmnd 20260 . . . . . . . . . . . . . . . . . . . . . 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 9298 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑘 ∪ {𝑐}) ∈ Fin)
144143adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑘 ∪ {𝑐}) ∈ Fin)
145 ssun2 4188 . . . . . . . . . . . . . . . . . . . . . . 23 {𝑐} ⊆ (𝑘 ∪ {𝑐})
146 vsnid 4667 . . . . . . . . . . . . . . . . . . . . . . 23 𝑐 ∈ {𝑐}
147145, 146sselii 3991 . . . . . . . . . . . . . . . . . . . . . 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 11237 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ)
150149adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
151 rge0ssre 13492 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0[,)+∞) ⊆ ℝ
152 fss 6752 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑇:𝐴⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝑇:𝐴⟶ℝ)
1532, 151, 152sylancl 586 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑇:𝐴⟶ℝ)
15498, 89fssd 6753 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑋:𝐴⟶ℝ)
155 inidm 4234 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴𝐴) = 𝐴
156150, 153, 154, 10, 10, 155off 7714 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑇f · 𝑋):𝐴⟶ℝ)
157 ax-resscn 11209 . . . . . . . . . . . . . . . . . . . . . . . 24 ℝ ⊆ ℂ
158 fss 6752 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑇f · 𝑋):𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → (𝑇f · 𝑋):𝐴⟶ℂ)
159156, 157, 158sylancl 586 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑇f · 𝑋):𝐴⟶ℂ)
160159ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑇f · 𝑋):𝐴⟶ℂ)
16186adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑘 ∪ {𝑐}) ⊆ 𝐴)
162160, 161fssresd 6775 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ)
1632ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑇:𝐴⟶(0[,)+∞))
164110adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝐴 ∈ Fin)
165163, 164fexd 7246 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑇 ∈ V)
16698ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑋:𝐴𝐷)
167166, 164fexd 7246 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑋 ∈ V)
168 offres 8006 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑇 ∈ V ∧ 𝑋 ∈ V) → ((𝑇f · 𝑋) ↾ (𝑘 ∪ {𝑐})) = ((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · (𝑋 ↾ (𝑘 ∪ {𝑐}))))
169165, 167, 168syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . 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 7445 . . . . . . . . . . . . . . . . . . . . . 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 4004 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0[,)+∞) ⊆ ℂ
172 fss 6752 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑇:𝐴⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℂ) → 𝑇:𝐴⟶ℂ)
173163, 171, 172sylancl 586 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑇:𝐴⟶ℂ)
174173, 161fssresd 6775 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑇 ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ)
175 eldifi 4140 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) → 𝑥 ∈ (𝑘 ∪ {𝑐}))
176175adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐})) → 𝑥 ∈ (𝑘 ∪ {𝑐}))
177176fvresd 6926 . . . . . . . . . . . . . . . . . . . . . . . . 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 4486 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑘 ∪ {𝑐}) ∖ {𝑐}) = (𝑘 ∖ {𝑐})
179 difss 4145 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 ∖ {𝑐}) ⊆ 𝑘
180178, 179eqsstri 4029 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑘 ∪ {𝑐}) ∖ {𝑐}) ⊆ 𝑘
181180sseli 3990 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) → 𝑥𝑘)
182 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 0 = (ℂfld Σg (𝑇𝑘)))
18387adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑘𝐴)
184163, 183feqresmpt 6977 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑇𝑘) = (𝑥𝑘 ↦ (𝑇𝑥)))
185184oveq2d 7446 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (ℂfld Σg (𝑇𝑘)) = (ℂfld Σg (𝑥𝑘 ↦ (𝑇𝑥))))
186111adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑘 ∈ Fin)
187183sselda 3994 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝑘) → 𝑥𝐴)
188163ffvelcdmda 7103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝐴) → (𝑇𝑥) ∈ (0[,)+∞))
189187, 188syldan 591 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝑘) → (𝑇𝑥) ∈ (0[,)+∞))
190171, 189sselid 3992 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝑘) → (𝑇𝑥) ∈ ℂ)
191186, 190gsumfsum 21469 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2779 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → Σ𝑥𝑘 (𝑇𝑥) = 0)
193 elrege0 13490 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑇𝑥) ∈ (0[,)+∞) ↔ ((𝑇𝑥) ∈ ℝ ∧ 0 ≤ (𝑇𝑥)))
194189, 193sylib 218 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝑘) → ((𝑇𝑥) ∈ ℝ ∧ 0 ≤ (𝑇𝑥)))
195194simpld 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝑘) → (𝑇𝑥) ∈ ℝ)
196194simprd 495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝑘) → 0 ≤ (𝑇𝑥))
197186, 195, 196fsum00 15830 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (Σ𝑥𝑘 (𝑇𝑥) = 0 ↔ ∀𝑥𝑘 (𝑇𝑥) = 0))
198192, 197mpbid 232 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ∀𝑥𝑘 (𝑇𝑥) = 0)
199198r19.21bi 3248 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝑘) → (𝑇𝑥) = 0)
200181, 199sylan2 593 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐})) → (𝑇𝑥) = 0)
201177, 200eqtrd 2774 . . . . . . . . . . . . . . . . . . . . . . . 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 8217 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇 ↾ (𝑘 ∪ {𝑐})) supp 0) ⊆ {𝑐})
203 mul02 11436 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ ℂ → (0 · 𝑥) = 0)
204203adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥 ∈ ℂ) → (0 · 𝑥) = 0)
20589ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝐷 ⊆ ℝ)
206205, 157sstrdi 4007 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝐷 ⊆ ℂ)
207166, 206fssd 6753 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑋:𝐴⟶ℂ)
208207, 161fssresd 6775 . . . . . . . . . . . . . . . . . . . . . . 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 8222 . . . . . . . . . . . . . . . . . . . . . 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 4033 . . . . . . . . . . . . . . . . . . . . 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 19994 . . . . . . . . . . . . . . . . . . . 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 6926 . . . . . . . . . . . . . . . . . . . 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 6737 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑇 Fn 𝐴)
215166ffnd 6737 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑋 Fn 𝐴)
216161, 148sseldd 3995 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑐𝐴)
217 fnfvof 7713 . . . . . . . . . . . . . . . . . . . . 21 (((𝑇 Fn 𝐴𝑋 Fn 𝐴) ∧ (𝐴 ∈ Fin ∧ 𝑐𝐴)) → ((𝑇f · 𝑋)‘𝑐) = ((𝑇𝑐) · (𝑋𝑐)))
218214, 215, 164, 216, 217syl22anc 839 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇f · 𝑋)‘𝑐) = ((𝑇𝑐) · (𝑋𝑐)))
219212, 213, 2183eqtrd 2778 . . . . . . . . . . . . . . . . . . 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 19994 . . . . . . . . . . . . . . . . . . . 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 6926 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇 ↾ (𝑘 ∪ {𝑐}))‘𝑐) = (𝑇𝑐))
222220, 221eqtrd 2774 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))) = (𝑇𝑐))
223219, 222oveq12d 7448 . . . . . . . . . . . . . . . . . 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 7104 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑋𝑐) ∈ ℂ)
225173, 216ffvelcdmd 7104 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑇𝑐) ∈ ℂ)
226 simplrr 778 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))
227226, 222breqtrd 5173 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 0 < (𝑇𝑐))
228227gt0ne0d 11824 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑇𝑐) ≠ 0)
229224, 225, 228divcan3d 12045 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (((𝑇𝑐) · (𝑋𝑐)) / (𝑇𝑐)) = (𝑋𝑐))
230223, 229eqtrd 2774 . . . . . . . . . . . . . . . . 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 7104 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑋𝑐) ∈ 𝐷)
232230, 231eqeltrd 2838 . . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝐹:𝐷⟶ℝ)
234233, 231ffvelcdmd 7104 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹‘(𝑋𝑐)) ∈ ℝ)
235234leidd 11826 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹‘(𝑋𝑐)) ≤ (𝐹‘(𝑋𝑐)))
236230fveq2d 6910 . . . . . . . . . . . . . . . . 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 6760 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐹:𝐷⟶ℝ ∧ 𝑋:𝐴𝐷) → (𝐹𝑋):𝐴⟶ℝ)
23891, 98, 237syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐹𝑋):𝐴⟶ℝ)
239150, 153, 238, 10, 10, 155off 7714 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑇f · (𝐹𝑋)):𝐴⟶ℝ)
240 fss 6752 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑇f · (𝐹𝑋)):𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → (𝑇f · (𝐹𝑋)):𝐴⟶ℂ)
241239, 157, 240sylancl 586 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑇f · (𝐹𝑋)):𝐴⟶ℂ)
242241ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑇f · (𝐹𝑋)):𝐴⟶ℂ)
243242, 161fssresd 6775 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ)
244238ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹𝑋):𝐴⟶ℝ)
245244, 164fexd 7246 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹𝑋) ∈ V)
246 offres 8006 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑇 ∈ V ∧ (𝐹𝑋) ∈ V) → ((𝑇f · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐})) = ((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · ((𝐹𝑋) ↾ (𝑘 ∪ {𝑐}))))
247165, 245, 246syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . 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 7445 . . . . . . . . . . . . . . . . . . . . . 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 6752 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐹𝑋):𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → (𝐹𝑋):𝐴⟶ℂ)
250244, 157, 249sylancl 586 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹𝑋):𝐴⟶ℂ)
251250, 161fssresd 6775 . . . . . . . . . . . . . . . . . . . . . . 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 8222 . . . . . . . . . . . . . . . . . . . . . 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 4033 . . . . . . . . . . . . . . . . . . . . 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 19994 . . . . . . . . . . . . . . . . . . . 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 6926 . . . . . . . . . . . . . . . . . . . 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 6737 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐹 Fn 𝐷)
257 fnfco 6773 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹 Fn 𝐷𝑋:𝐴𝐷) → (𝐹𝑋) Fn 𝐴)
258256, 98, 257syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝐹𝑋) Fn 𝐴)
259258ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹𝑋) Fn 𝐴)
260 fnfvof 7713 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑇 Fn 𝐴 ∧ (𝐹𝑋) Fn 𝐴) ∧ (𝐴 ∈ Fin ∧ 𝑐𝐴)) → ((𝑇f · (𝐹𝑋))‘𝑐) = ((𝑇𝑐) · ((𝐹𝑋)‘𝑐)))
261214, 259, 164, 216, 260syl22anc 839 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇f · (𝐹𝑋))‘𝑐) = ((𝑇𝑐) · ((𝐹𝑋)‘𝑐)))
262 fvco3 7007 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑋:𝐴𝐷𝑐𝐴) → ((𝐹𝑋)‘𝑐) = (𝐹‘(𝑋𝑐)))
263166, 216, 262syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝐹𝑋)‘𝑐) = (𝐹‘(𝑋𝑐)))
264263oveq2d 7446 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇𝑐) · ((𝐹𝑋)‘𝑐)) = ((𝑇𝑐) · (𝐹‘(𝑋𝑐))))
265261, 264eqtrd 2774 . . . . . . . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . . . . . . . . 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 7448 . . . . . . . . . . . . . . . . . 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 11286 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹‘(𝑋𝑐)) ∈ ℂ)
269268, 225, 228divcan3d 12045 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (((𝑇𝑐) · (𝐹‘(𝑋𝑐))) / (𝑇𝑐)) = (𝐹‘(𝑋𝑐)))
270267, 269eqtrd 2774 . . . . . . . . . . . . . . . . 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 5179 . . . . . . . . . . . . . . . 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 3696 . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . . 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 11344 . . . . . . . . . . . . . . . 16 ((0 ∈ ℝ ∧ (ℂfld Σg (𝑇𝑘)) ∈ ℝ) → (0 ≤ (ℂfld Σg (𝑇𝑘)) ↔ (0 < (ℂfld Σg (𝑇𝑘)) ∨ 0 = (ℂfld Σg (𝑇𝑘)))))
27780, 122, 276sylancr 587 . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (0 < (ℂfld Σg (𝑇𝑘)) ∨ 0 = (ℂfld Σg (𝑇𝑘))))
279139, 273, 278mpjaodan 960 . . . . . . . . . . . . 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 242 . . . . . . . . . . 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 412 . . . . . . . . . 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 413 . . . . . . . 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 481 . . . . . . 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 9203 . . . . 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 6737 . . . . . 6 (𝜑 → (𝑇f · 𝑋) Fn 𝐴)
291 fnresdm 6687 . . . . . 6 ((𝑇f · 𝑋) Fn 𝐴 → ((𝑇f · 𝑋) ↾ 𝐴) = (𝑇f · 𝑋))
292290, 291syl 17 . . . . 5 (𝜑 → ((𝑇f · 𝑋) ↾ 𝐴) = (𝑇f · 𝑋))
293292oveq2d 7446 . . . 4 (𝜑 → (ℂfld Σg ((𝑇f · 𝑋) ↾ 𝐴)) = (ℂfld Σg (𝑇f · 𝑋)))
294293, 6oveq12d 7448 . . 3 (𝜑 → ((ℂfld Σg ((𝑇f · 𝑋) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))) = ((ℂfld Σg (𝑇f · 𝑋)) / (ℂfld Σg 𝑇)))
2953, 258, 10, 10, 155offn 7709 . . . . . . . 8 (𝜑 → (𝑇f · (𝐹𝑋)) Fn 𝐴)
296 fnresdm 6687 . . . . . . . 8 ((𝑇f · (𝐹𝑋)) Fn 𝐴 → ((𝑇f · (𝐹𝑋)) ↾ 𝐴) = (𝑇f · (𝐹𝑋)))
297295, 296syl 17 . . . . . . 7 (𝜑 → ((𝑇f · (𝐹𝑋)) ↾ 𝐴) = (𝑇f · (𝐹𝑋)))
298297oveq2d 7446 . . . . . 6 (𝜑 → (ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝐴)) = (ℂfld Σg (𝑇f · (𝐹𝑋))))
299298, 6oveq12d 7448 . . . . 5 (𝜑 → ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))) = ((ℂfld Σg (𝑇f · (𝐹𝑋))) / (ℂfld Σg 𝑇)))
300299breq2d 5159 . . . 4 (𝜑 → ((𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))) ↔ (𝐹𝑤) ≤ ((ℂfld Σg (𝑇f · (𝐹𝑋))) / (ℂfld Σg 𝑇))))
301300rabbidv 3440 . . 3 (𝜑 → {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇f · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴)))} = {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg (𝑇f · (𝐹𝑋))) / (ℂfld Σg 𝑇))})
302289, 294, 3013eltr3d 2852 . 2 (𝜑 → ((ℂfld Σg (𝑇f · 𝑋)) / (ℂfld Σg 𝑇)) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg (𝑇f · (𝐹𝑋))) / (ℂfld Σg 𝑇))})
303 fveq2 6906 . . . 4 (𝑤 = ((ℂfld Σg (𝑇f · 𝑋)) / (ℂfld Σg 𝑇)) → (𝐹𝑤) = (𝐹‘((ℂfld Σg (𝑇f · 𝑋)) / (ℂfld Σg 𝑇))))
304303breq1d 5157 . . 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 3694 . 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 218 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 206  wa 395  wo 847  w3a 1086   = wceq 1536  wcel 2105  wral 3058  {crab 3432  Vcvv 3477  cdif 3959  cun 3960  wss 3962  c0 4338  {csn 4630   class class class wbr 5147  cmpt 5230  cres 5690  ccom 5692   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  f cof 7694   supp csupp 8183  Fincfn 8983  cc 11150  cr 11151  0cc0 11152  1c1 11153   + caddc 11155   · cmul 11157  +∞cpnf 11289   < clt 11292  cle 11293  cmin 11489   / cdiv 11917  [,)cico 13385  [,]cicc 13386  Σcsu 15718   Σg cgsu 17486  Mndcmnd 18759  SubMndcsubmnd 18807  CMndccmn 19812  Ringcrg 20250  fldccnfld 21381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-inf2 9678  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230  ax-addf 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-tpos 8249  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-sup 9479  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-rp 13032  df-ico 13389  df-icc 13390  df-fz 13544  df-fzo 13691  df-seq 14039  df-exp 14099  df-hash 14366  df-cj 15134  df-re 15135  df-im 15136  df-sqrt 15270  df-abs 15271  df-clim 15520  df-sum 15719  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-starv 17312  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-0g 17487  df-gsum 17488  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-submnd 18809  df-grp 18966  df-minusg 18967  df-mulg 19098  df-subg 19153  df-cntz 19347  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-cring 20253  df-oppr 20350  df-dvdsr 20373  df-unit 20374  df-invr 20404  df-dvr 20417  df-subrng 20562  df-subrg 20586  df-drng 20747  df-cnfld 21382  df-refld 21640
This theorem is referenced by:  amgmlem  27047  amgmwlem  49032
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