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Theorem jensen 24760
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[,)+∞))
3 ffn 6083 . . . . . . . . 9 (𝑇:𝐴⟶(0[,)+∞) → 𝑇 Fn 𝐴)
42, 3syl 17 . . . . . . . 8 (𝜑𝑇 Fn 𝐴)
5 fnresdm 6038 . . . . . . . 8 (𝑇 Fn 𝐴 → (𝑇𝐴) = 𝑇)
64, 5syl 17 . . . . . . 7 (𝜑 → (𝑇𝐴) = 𝑇)
76oveq2d 6706 . . . . . 6 (𝜑 → (ℂfld Σg (𝑇𝐴)) = (ℂfld Σg 𝑇))
81, 7breqtrrd 4713 . . . . 5 (𝜑 → 0 < (ℂfld Σg (𝑇𝐴)))
9 ssid 3657 . . . . 5 𝐴𝐴
108, 9jctil 559 . . . 4 (𝜑 → (𝐴𝐴 ∧ 0 < (ℂfld Σg (𝑇𝐴))))
11 jensen.4 . . . . 5 (𝜑𝐴 ∈ Fin)
12 sseq1 3659 . . . . . . . . 9 (𝑎 = ∅ → (𝑎𝐴 ↔ ∅ ⊆ 𝐴))
13 reseq2 5423 . . . . . . . . . . . . 13 (𝑎 = ∅ → (𝑇𝑎) = (𝑇 ↾ ∅))
14 res0 5432 . . . . . . . . . . . . 13 (𝑇 ↾ ∅) = ∅
1513, 14syl6eq 2701 . . . . . . . . . . . 12 (𝑎 = ∅ → (𝑇𝑎) = ∅)
1615oveq2d 6706 . . . . . . . . . . 11 (𝑎 = ∅ → (ℂfld Σg (𝑇𝑎)) = (ℂfld Σg ∅))
17 cnfld0 19818 . . . . . . . . . . . 12 0 = (0g‘ℂfld)
1817gsum0 17325 . . . . . . . . . . 11 (ℂfld Σg ∅) = 0
1916, 18syl6eq 2701 . . . . . . . . . 10 (𝑎 = ∅ → (ℂfld Σg (𝑇𝑎)) = 0)
2019breq2d 4697 . . . . . . . . 9 (𝑎 = ∅ → (0 < (ℂfld Σg (𝑇𝑎)) ↔ 0 < 0))
2112, 20anbi12d 747 . . . . . . . 8 (𝑎 = ∅ → ((𝑎𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑎))) ↔ (∅ ⊆ 𝐴 ∧ 0 < 0)))
22 reseq2 5423 . . . . . . . . . . 11 (𝑎 = ∅ → ((𝑇𝑓 · 𝑋) ↾ 𝑎) = ((𝑇𝑓 · 𝑋) ↾ ∅))
2322oveq2d 6706 . . . . . . . . . 10 (𝑎 = ∅ → (ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ ∅)))
2423, 19oveq12d 6708 . . . . . . . . 9 (𝑎 = ∅ → ((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ ∅)) / 0))
25 reseq2 5423 . . . . . . . . . . . . 13 (𝑎 = ∅ → ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑎) = ((𝑇𝑓 · (𝐹𝑋)) ↾ ∅))
2625oveq2d 6706 . . . . . . . . . . . 12 (𝑎 = ∅ → (ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ ∅)))
2726, 19oveq12d 6708 . . . . . . . . . . 11 (𝑎 = ∅ → ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ ∅)) / 0))
2827breq2d 4697 . . . . . . . . . 10 (𝑎 = ∅ → ((𝐹𝑤) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ↔ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ ∅)) / 0)))
2928rabbidv 3220 . . . . . . . . 9 (𝑎 = ∅ → {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎)))} = {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ ∅)) / 0)})
3024, 29eleq12d 2724 . . . . . . . 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)}))
3121, 30imbi12d 333 . . . . . . 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)})))
3231imbi2d 329 . . . . . 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)}))))
33 sseq1 3659 . . . . . . . . 9 (𝑎 = 𝑘 → (𝑎𝐴𝑘𝐴))
34 reseq2 5423 . . . . . . . . . . 11 (𝑎 = 𝑘 → (𝑇𝑎) = (𝑇𝑘))
3534oveq2d 6706 . . . . . . . . . 10 (𝑎 = 𝑘 → (ℂfld Σg (𝑇𝑎)) = (ℂfld Σg (𝑇𝑘)))
3635breq2d 4697 . . . . . . . . 9 (𝑎 = 𝑘 → (0 < (ℂfld Σg (𝑇𝑎)) ↔ 0 < (ℂfld Σg (𝑇𝑘))))
3733, 36anbi12d 747 . . . . . . . 8 (𝑎 = 𝑘 → ((𝑎𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑎))) ↔ (𝑘𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑘)))))
38 reseq2 5423 . . . . . . . . . . 11 (𝑎 = 𝑘 → ((𝑇𝑓 · 𝑋) ↾ 𝑎) = ((𝑇𝑓 · 𝑋) ↾ 𝑘))
3938oveq2d 6706 . . . . . . . . . 10 (𝑎 = 𝑘 → (ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ 𝑘)))
4039, 35oveq12d 6708 . . . . . . . . 9 (𝑎 = 𝑘 → ((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))))
41 reseq2 5423 . . . . . . . . . . . . 13 (𝑎 = 𝑘 → ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑎) = ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑘))
4241oveq2d 6706 . . . . . . . . . . . 12 (𝑎 = 𝑘 → (ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑘)))
4342, 35oveq12d 6708 . . . . . . . . . . 11 (𝑎 = 𝑘 → ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))))
4443breq2d 4697 . . . . . . . . . 10 (𝑎 = 𝑘 → ((𝐹𝑤) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ↔ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))))
4544rabbidv 3220 . . . . . . . . 9 (𝑎 = 𝑘 → {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎)))} = {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))})
4640, 45eleq12d 2724 . . . . . . . 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)))}))
4737, 46imbi12d 333 . . . . . . 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)))})))
4847imbi2d 329 . . . . . 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)))}))))
49 sseq1 3659 . . . . . . . . 9 (𝑎 = (𝑘 ∪ {𝑐}) → (𝑎𝐴 ↔ (𝑘 ∪ {𝑐}) ⊆ 𝐴))
50 reseq2 5423 . . . . . . . . . . 11 (𝑎 = (𝑘 ∪ {𝑐}) → (𝑇𝑎) = (𝑇 ↾ (𝑘 ∪ {𝑐})))
5150oveq2d 6706 . . . . . . . . . 10 (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld Σg (𝑇𝑎)) = (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))
5251breq2d 4697 . . . . . . . . 9 (𝑎 = (𝑘 ∪ {𝑐}) → (0 < (ℂfld Σg (𝑇𝑎)) ↔ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))))
5349, 52anbi12d 747 . . . . . . . 8 (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑎𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑎))) ↔ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))))
54 reseq2 5423 . . . . . . . . . . 11 (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑇𝑓 · 𝑋) ↾ 𝑎) = ((𝑇𝑓 · 𝑋) ↾ (𝑘 ∪ {𝑐})))
5554oveq2d 6706 . . . . . . . . . 10 (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ (𝑘 ∪ {𝑐}))))
5655, 51oveq12d 6708 . . . . . . . . 9 (𝑎 = (𝑘 ∪ {𝑐}) → ((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))))
57 reseq2 5423 . . . . . . . . . . . . 13 (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑎) = ((𝑇𝑓 · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐})))
5857oveq2d 6706 . . . . . . . . . . . 12 (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))))
5958, 51oveq12d 6708 . . . . . . . . . . 11 (𝑎 = (𝑘 ∪ {𝑐}) → ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))))
6059breq2d 4697 . . . . . . . . . 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}))))))
6160rabbidv 3220 . . . . . . . . 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}))))})
6256, 61eleq12d 2724 . . . . . . . 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}))))}))
6353, 62imbi12d 333 . . . . . . 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}))))})))
6463imbi2d 329 . . . . . 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}))))}))))
65 sseq1 3659 . . . . . . . . 9 (𝑎 = 𝐴 → (𝑎𝐴𝐴𝐴))
66 reseq2 5423 . . . . . . . . . . 11 (𝑎 = 𝐴 → (𝑇𝑎) = (𝑇𝐴))
6766oveq2d 6706 . . . . . . . . . 10 (𝑎 = 𝐴 → (ℂfld Σg (𝑇𝑎)) = (ℂfld Σg (𝑇𝐴)))
6867breq2d 4697 . . . . . . . . 9 (𝑎 = 𝐴 → (0 < (ℂfld Σg (𝑇𝑎)) ↔ 0 < (ℂfld Σg (𝑇𝐴))))
6965, 68anbi12d 747 . . . . . . . 8 (𝑎 = 𝐴 → ((𝑎𝐴 ∧ 0 < (ℂfld Σg (𝑇𝑎))) ↔ (𝐴𝐴 ∧ 0 < (ℂfld Σg (𝑇𝐴)))))
70 reseq2 5423 . . . . . . . . . . 11 (𝑎 = 𝐴 → ((𝑇𝑓 · 𝑋) ↾ 𝑎) = ((𝑇𝑓 · 𝑋) ↾ 𝐴))
7170oveq2d 6706 . . . . . . . . . 10 (𝑎 = 𝐴 → (ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ 𝐴)))
7271, 67oveq12d 6708 . . . . . . . . 9 (𝑎 = 𝐴 → ((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))))
73 reseq2 5423 . . . . . . . . . . . . 13 (𝑎 = 𝐴 → ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑎) = ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝐴))
7473oveq2d 6706 . . . . . . . . . . . 12 (𝑎 = 𝐴 → (ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝐴)))
7574, 67oveq12d 6708 . . . . . . . . . . 11 (𝑎 = 𝐴 → ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) = ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))))
7675breq2d 4697 . . . . . . . . . 10 (𝑎 = 𝐴 → ((𝐹𝑤) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎))) ↔ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴)))))
7776rabbidv 3220 . . . . . . . . 9 (𝑎 = 𝐴 → {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇𝑎)))} = {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴)))})
7872, 77eleq12d 2724 . . . . . . . 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)))}))
7969, 78imbi12d 333 . . . . . . 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)))})))
8079imbi2d 329 . . . . . 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)))}))))
81 0re 10078 . . . . . . . . . 10 0 ∈ ℝ
8281ltnri 10184 . . . . . . . . 9 ¬ 0 < 0
8382pm2.21i 116 . . . . . . . 8 (0 < 0 → ((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ ∅)) / 0) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ ∅)) / 0)})
8483adantl 481 . . . . . . 7 ((∅ ⊆ 𝐴 ∧ 0 < 0) → ((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ ∅)) / 0) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ ∅)) / 0)})
8584a1i 11 . . . . . 6 (𝜑 → ((∅ ⊆ 𝐴 ∧ 0 < 0) → ((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ ∅)) / 0) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ ∅)) / 0)}))
86 impexp 461 . . . . . . . . . . . 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)))})))
87 simprl 809 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑘 ∪ {𝑐}) ⊆ 𝐴)
8887unssad 3823 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑘𝐴)
89 simpr 476 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 < (ℂfld Σg (𝑇𝑘))) → 0 < (ℂfld Σg (𝑇𝑘)))
90 jensen.1 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐷 ⊆ ℝ)
9190ad3antrrr 766 . . . . . . . . . . . . . . . . . 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)
92 jensen.2 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐹:𝐷⟶ℝ)
9392ad3antrrr 766 . . . . . . . . . . . . . . . . . 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)
94 simplll 813 . . . . . . . . . . . . . . . . . . 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)
95 jensen.3 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑎𝐷𝑏𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷)
9694, 95sylan 487 . . . . . . . . . . . . . . . . . 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)
9794, 11syl 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)
9894, 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[,)+∞))
99 jensen.6 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑋:𝐴𝐷)
10094, 99syl 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)
1011ad3antrrr 766 . . . . . . . . . . . . . . . . . 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))
102 jensen.8 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑥𝐷𝑦𝐷𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹𝑥)) + ((1 − 𝑡) · (𝐹𝑦))))
10394, 102sylan 487 . . . . . . . . . . . . . . . . . 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))))
104 simpllr 815 . . . . . . . . . . . . . . . . . 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)
10587adantr 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)
106 eqid 2651 . . . . . . . . . . . . . . . . . 18 (ℂfld Σg (𝑇𝑘)) = (ℂfld Σg (𝑇𝑘))
107 eqid 2651 . . . . . . . . . . . . . . . . . 18 (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))) = (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))
108 cnring 19816 . . . . . . . . . . . . . . . . . . . . . . 23 fld ∈ Ring
109 ringcmn 18627 . . . . . . . . . . . . . . . . . . . . . . 23 (ℂfld ∈ Ring → ℂfld ∈ CMnd)
110108, 109mp1i 13 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → ℂfld ∈ CMnd)
11111ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝐴 ∈ Fin)
112 ssfi 8221 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ Fin ∧ 𝑘𝐴) → 𝑘 ∈ Fin)
113111, 88, 112syl2anc 694 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑘 ∈ Fin)
114 rege0subm 19850 . . . . . . . . . . . . . . . . . . . . . . 23 (0[,)+∞) ∈ (SubMnd‘ℂfld)
115114a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (0[,)+∞) ∈ (SubMnd‘ℂfld))
1162ad2antrr 762 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑇:𝐴⟶(0[,)+∞))
117116, 88fssresd 6109 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑇𝑘):𝑘⟶(0[,)+∞))
118 c0ex 10072 . . . . . . . . . . . . . . . . . . . . . . . 24 0 ∈ V
119118a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 0 ∈ V)
120117, 113, 119fdmfifsupp 8326 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑇𝑘) finSupp 0)
12117, 110, 113, 115, 117, 120gsumsubmcl 18365 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (ℂfld Σg (𝑇𝑘)) ∈ (0[,)+∞))
122 elrege0 12316 . . . . . . . . . . . . . . . . . . . . . 22 ((ℂfld Σg (𝑇𝑘)) ∈ (0[,)+∞) ↔ ((ℂfld Σg (𝑇𝑘)) ∈ ℝ ∧ 0 ≤ (ℂfld Σg (𝑇𝑘))))
123122simplbi 475 . . . . . . . . . . . . . . . . . . . . 21 ((ℂfld Σg (𝑇𝑘)) ∈ (0[,)+∞) → (ℂfld Σg (𝑇𝑘)) ∈ ℝ)
124121, 123syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (ℂfld Σg (𝑇𝑘)) ∈ ℝ)
125124adantr 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)
126 simprl 809 . . . . . . . . . . . . . . . . . . 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)))
127125, 126elrpd 11907 . . . . . . . . . . . . . . . . . 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+)
128 simprr 811 . . . . . . . . . . . . . . . . . . . 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)))})
129 fveq2 6229 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤 = ((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘))) → (𝐹𝑤) = (𝐹‘((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇𝑘)))))
130129breq1d 4695 . . . . . . . . . . . . . . . . . . . . 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)))))
131130elrab 3396 . . . . . . . . . . . . . . . . . . . 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)))))
132128, 131sylib 208 . . . . . . . . . . . . . . . . . . 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)))))
133132simpld 474 . . . . . . . . . . . . . . . . . 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)
134132simprd 478 . . . . . . . . . . . . . . . . . 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))))
13591, 93, 96, 97, 98, 100, 101, 103, 104, 105, 106, 107, 127, 133, 134jensenlem2 24759 . . . . . . . . . . . . . . . . 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}))))))
136 fveq2 6229 . . . . . . . . . . . . . . . . . . 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}))))))
137136breq1d 4695 . . . . . . . . . . . . . . . . . 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}))))))
138137elrab 3396 . . . . . . . . . . . . . . . . 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}))))))
139135, 138sylibr 224 . . . . . . . . . . . . . . . 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}))))})
140139expr 642 . . . . . . . . . . . . . . 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}))))}))
14189, 140embantd 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}))))}))
142 cnfldbas 19798 . . . . . . . . . . . . . . . . . . . . 21 ℂ = (Base‘ℂfld)
143 ringmnd 18602 . . . . . . . . . . . . . . . . . . . . . 22 (ℂfld ∈ Ring → ℂfld ∈ Mnd)
144108, 143mp1i 13 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ℂfld ∈ Mnd)
145 ssfi 8221 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐴 ∈ Fin ∧ (𝑘 ∪ {𝑐}) ⊆ 𝐴) → (𝑘 ∪ {𝑐}) ∈ Fin)
146111, 87, 145syl2anc 694 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑘 ∪ {𝑐}) ∈ Fin)
147146adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑘 ∪ {𝑐}) ∈ Fin)
148 ssun2 3810 . . . . . . . . . . . . . . . . . . . . . . 23 {𝑐} ⊆ (𝑘 ∪ {𝑐})
149 vsnid 4242 . . . . . . . . . . . . . . . . . . . . . . 23 𝑐 ∈ {𝑐}
150148, 149sselii 3633 . . . . . . . . . . . . . . . . . . . . . 22 𝑐 ∈ (𝑘 ∪ {𝑐})
151150a1i 11 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑐 ∈ (𝑘 ∪ {𝑐}))
152 remulcl 10059 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ)
153152adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
154 rge0ssre 12318 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0[,)+∞) ⊆ ℝ
155 fss 6094 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑇:𝐴⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝑇:𝐴⟶ℝ)
1562, 154, 155sylancl 695 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑇:𝐴⟶ℝ)
15799, 90fssd 6095 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑋:𝐴⟶ℝ)
158 inidm 3855 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐴𝐴) = 𝐴
159153, 156, 157, 11, 11, 158off 6954 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑇𝑓 · 𝑋):𝐴⟶ℝ)
160 ax-resscn 10031 . . . . . . . . . . . . . . . . . . . . . . . 24 ℝ ⊆ ℂ
161 fss 6094 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑇𝑓 · 𝑋):𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → (𝑇𝑓 · 𝑋):𝐴⟶ℂ)
162159, 160, 161sylancl 695 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑇𝑓 · 𝑋):𝐴⟶ℂ)
163162ad3antrrr 766 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑇𝑓 · 𝑋):𝐴⟶ℂ)
16487adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑘 ∪ {𝑐}) ⊆ 𝐴)
165163, 164fssresd 6109 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇𝑓 · 𝑋) ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ)
1662ad3antrrr 766 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑇:𝐴⟶(0[,)+∞))
167111adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝐴 ∈ Fin)
168 fex 6530 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑇:𝐴⟶(0[,)+∞) ∧ 𝐴 ∈ Fin) → 𝑇 ∈ V)
169166, 167, 168syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑇 ∈ V)
17099ad3antrrr 766 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑋:𝐴𝐷)
171 fex 6530 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑋:𝐴𝐷𝐴 ∈ Fin) → 𝑋 ∈ V)
172170, 167, 171syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑋 ∈ V)
173 offres 7205 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑇 ∈ V ∧ 𝑋 ∈ V) → ((𝑇𝑓 · 𝑋) ↾ (𝑘 ∪ {𝑐})) = ((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘𝑓 · (𝑋 ↾ (𝑘 ∪ {𝑐}))))
174169, 172, 173syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . 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}))))
175174oveq1d 6705 . . . . . . . . . . . . . . . . . . . . . 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))
176154, 160sstri 3645 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (0[,)+∞) ⊆ ℂ
177 fss 6094 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑇:𝐴⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℂ) → 𝑇:𝐴⟶ℂ)
178166, 176, 177sylancl 695 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑇:𝐴⟶ℂ)
179178, 164fssresd 6109 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑇 ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ)
180 eldifi 3765 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) → 𝑥 ∈ (𝑘 ∪ {𝑐}))
181180adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐})) → 𝑥 ∈ (𝑘 ∪ {𝑐}))
182 fvres 6245 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ (𝑘 ∪ {𝑐}) → ((𝑇 ↾ (𝑘 ∪ {𝑐}))‘𝑥) = (𝑇𝑥))
183181, 182syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 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))
184 difun2 4081 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑘 ∪ {𝑐}) ∖ {𝑐}) = (𝑘 ∖ {𝑐})
185 difss 3770 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑘 ∖ {𝑐}) ⊆ 𝑘
186184, 185eqsstri 3668 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑘 ∪ {𝑐}) ∖ {𝑐}) ⊆ 𝑘
187186sseli 3632 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) → 𝑥𝑘)
188 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 0 = (ℂfld Σg (𝑇𝑘)))
18988adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑘𝐴)
190166, 189feqresmpt 6289 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑇𝑘) = (𝑥𝑘 ↦ (𝑇𝑥)))
191190oveq2d 6706 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (ℂfld Σg (𝑇𝑘)) = (ℂfld Σg (𝑥𝑘 ↦ (𝑇𝑥))))
192113adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑘 ∈ Fin)
193189sselda 3636 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝑘) → 𝑥𝐴)
194166ffvelrnda 6399 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝐴) → (𝑇𝑥) ∈ (0[,)+∞))
195193, 194syldan 486 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝑘) → (𝑇𝑥) ∈ (0[,)+∞))
196176, 195sseldi 3634 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝑘) → (𝑇𝑥) ∈ ℂ)
197192, 196gsumfsum 19861 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (ℂfld Σg (𝑥𝑘 ↦ (𝑇𝑥))) = Σ𝑥𝑘 (𝑇𝑥))
198188, 191, 1973eqtrrd 2690 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → Σ𝑥𝑘 (𝑇𝑥) = 0)
199 elrege0 12316 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑇𝑥) ∈ (0[,)+∞) ↔ ((𝑇𝑥) ∈ ℝ ∧ 0 ≤ (𝑇𝑥)))
200195, 199sylib 208 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝑘) → ((𝑇𝑥) ∈ ℝ ∧ 0 ≤ (𝑇𝑥)))
201200simpld 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝑘) → (𝑇𝑥) ∈ ℝ)
202200simprd 478 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝑘) → 0 ≤ (𝑇𝑥))
203192, 201, 202fsum00 14574 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (Σ𝑥𝑘 (𝑇𝑥) = 0 ↔ ∀𝑥𝑘 (𝑇𝑥) = 0))
204198, 203mpbid 222 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ∀𝑥𝑘 (𝑇𝑥) = 0)
205204r19.21bi 2961 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥𝑘) → (𝑇𝑥) = 0)
206187, 205sylan2 490 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐})) → (𝑇𝑥) = 0)
207183, 206eqtrd 2685 . . . . . . . . . . . . . . . . . . . . . . . 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)
208179, 207suppss 7370 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇 ↾ (𝑘 ∪ {𝑐})) supp 0) ⊆ {𝑐})
209 mul02 10252 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 ∈ ℂ → (0 · 𝑥) = 0)
210209adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) ∧ 𝑥 ∈ ℂ) → (0 · 𝑥) = 0)
21190ad3antrrr 766 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝐷 ⊆ ℝ)
212211, 160syl6ss 3648 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝐷 ⊆ ℂ)
213170, 212fssd 6095 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑋:𝐴⟶ℂ)
214213, 164fssresd 6109 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑋 ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ)
215118a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 0 ∈ V)
216208, 210, 179, 214, 147, 215suppssof1 7373 . . . . . . . . . . . . . . . . . . . . . 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})
217175, 216eqsstrd 3672 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (((𝑇𝑓 · 𝑋) ↾ (𝑘 ∪ {𝑐})) supp 0) ⊆ {𝑐})
218142, 17, 144, 147, 151, 165, 217gsumpt 18407 . . . . . . . . . . . . . . . . . . . 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))
219 fvres 6245 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 ∈ (𝑘 ∪ {𝑐}) → (((𝑇𝑓 · 𝑋) ↾ (𝑘 ∪ {𝑐}))‘𝑐) = ((𝑇𝑓 · 𝑋)‘𝑐))
220151, 219syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (((𝑇𝑓 · 𝑋) ↾ (𝑘 ∪ {𝑐}))‘𝑐) = ((𝑇𝑓 · 𝑋)‘𝑐))
221166, 3syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑇 Fn 𝐴)
222 ffn 6083 . . . . . . . . . . . . . . . . . . . . . 22 (𝑋:𝐴𝐷𝑋 Fn 𝐴)
223170, 222syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑋 Fn 𝐴)
224164, 151sseldd 3637 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝑐𝐴)
225 fnfvof 6953 . . . . . . . . . . . . . . . . . . . . 21 (((𝑇 Fn 𝐴𝑋 Fn 𝐴) ∧ (𝐴 ∈ Fin ∧ 𝑐𝐴)) → ((𝑇𝑓 · 𝑋)‘𝑐) = ((𝑇𝑐) · (𝑋𝑐)))
226221, 223, 167, 224, 225syl22anc 1367 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇𝑓 · 𝑋)‘𝑐) = ((𝑇𝑐) · (𝑋𝑐)))
227218, 220, 2263eqtrd 2689 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ (𝑘 ∪ {𝑐}))) = ((𝑇𝑐) · (𝑋𝑐)))
228142, 17, 144, 147, 151, 179, 208gsumpt 18407 . . . . . . . . . . . . . . . . . . . 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))
229 fvres 6245 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 ∈ (𝑘 ∪ {𝑐}) → ((𝑇 ↾ (𝑘 ∪ {𝑐}))‘𝑐) = (𝑇𝑐))
230151, 229syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇 ↾ (𝑘 ∪ {𝑐}))‘𝑐) = (𝑇𝑐))
231228, 230eqtrd 2685 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))) = (𝑇𝑐))
232227, 231oveq12d 6708 . . . . . . . . . . . . . . . . . 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)))
233213, 224ffvelrnd 6400 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑋𝑐) ∈ ℂ)
234178, 224ffvelrnd 6400 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑇𝑐) ∈ ℂ)
235 simplrr 818 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))
236235, 231breqtrd 4711 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 0 < (𝑇𝑐))
237236gt0ne0d 10630 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑇𝑐) ≠ 0)
238233, 234, 237divcan3d 10844 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (((𝑇𝑐) · (𝑋𝑐)) / (𝑇𝑐)) = (𝑋𝑐))
239232, 238eqtrd 2685 . . . . . . . . . . . . . . . . 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))
240170, 224ffvelrnd 6400 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑋𝑐) ∈ 𝐷)
241239, 240eqeltrd 2730 . . . . . . . . . . . . . . . 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)
24292ad3antrrr 766 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → 𝐹:𝐷⟶ℝ)
243242, 240ffvelrnd 6400 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹‘(𝑋𝑐)) ∈ ℝ)
244243leidd 10632 . . . . . . . . . . . . . . . . 17 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹‘(𝑋𝑐)) ≤ (𝐹‘(𝑋𝑐)))
245239fveq2d 6233 . . . . . . . . . . . . . . . . 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)))
246 fco 6096 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐹:𝐷⟶ℝ ∧ 𝑋:𝐴𝐷) → (𝐹𝑋):𝐴⟶ℝ)
24792, 99, 246syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑 → (𝐹𝑋):𝐴⟶ℝ)
248153, 156, 247, 11, 11, 158off 6954 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝑇𝑓 · (𝐹𝑋)):𝐴⟶ℝ)
249 fss 6094 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑇𝑓 · (𝐹𝑋)):𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → (𝑇𝑓 · (𝐹𝑋)):𝐴⟶ℂ)
250248, 160, 249sylancl 695 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑇𝑓 · (𝐹𝑋)):𝐴⟶ℂ)
251250ad3antrrr 766 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝑇𝑓 · (𝐹𝑋)):𝐴⟶ℂ)
252251, 164fssresd 6109 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇𝑓 · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ)
253247ad3antrrr 766 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹𝑋):𝐴⟶ℝ)
254 fex 6530 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐹𝑋):𝐴⟶ℝ ∧ 𝐴 ∈ Fin) → (𝐹𝑋) ∈ V)
255253, 167, 254syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹𝑋) ∈ V)
256 offres 7205 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑇 ∈ V ∧ (𝐹𝑋) ∈ V) → ((𝑇𝑓 · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐})) = ((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘𝑓 · ((𝐹𝑋) ↾ (𝑘 ∪ {𝑐}))))
257169, 255, 256syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . 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}))))
258257oveq1d 6705 . . . . . . . . . . . . . . . . . . . . . 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))
259 fss 6094 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐹𝑋):𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → (𝐹𝑋):𝐴⟶ℂ)
260253, 160, 259sylancl 695 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹𝑋):𝐴⟶ℂ)
261260, 164fssresd 6109 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝐹𝑋) ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ)
262208, 210, 179, 261, 147, 215suppssof1 7373 . . . . . . . . . . . . . . . . . . . . . 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})
263258, 262eqsstrd 3672 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (((𝑇𝑓 · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐})) supp 0) ⊆ {𝑐})
264142, 17, 144, 147, 151, 252, 263gsumpt 18407 . . . . . . . . . . . . . . . . . . . 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))
265 fvres 6245 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 ∈ (𝑘 ∪ {𝑐}) → (((𝑇𝑓 · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))‘𝑐) = ((𝑇𝑓 · (𝐹𝑋))‘𝑐))
266151, 265syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (((𝑇𝑓 · (𝐹𝑋)) ↾ (𝑘 ∪ {𝑐}))‘𝑐) = ((𝑇𝑓 · (𝐹𝑋))‘𝑐))
267 ffn 6083 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹:𝐷⟶ℝ → 𝐹 Fn 𝐷)
26892, 267syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐹 Fn 𝐷)
269 fnfco 6107 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹 Fn 𝐷𝑋:𝐴𝐷) → (𝐹𝑋) Fn 𝐴)
270268, 99, 269syl2anc 694 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝐹𝑋) Fn 𝐴)
271270ad3antrrr 766 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹𝑋) Fn 𝐴)
272 fnfvof 6953 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑇 Fn 𝐴 ∧ (𝐹𝑋) Fn 𝐴) ∧ (𝐴 ∈ Fin ∧ 𝑐𝐴)) → ((𝑇𝑓 · (𝐹𝑋))‘𝑐) = ((𝑇𝑐) · ((𝐹𝑋)‘𝑐)))
273221, 271, 167, 224, 272syl22anc 1367 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇𝑓 · (𝐹𝑋))‘𝑐) = ((𝑇𝑐) · ((𝐹𝑋)‘𝑐)))
274 fvco3 6314 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑋:𝐴𝐷𝑐𝐴) → ((𝐹𝑋)‘𝑐) = (𝐹‘(𝑋𝑐)))
275170, 224, 274syl2anc 694 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝐹𝑋)‘𝑐) = (𝐹‘(𝑋𝑐)))
276275oveq2d 6706 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇𝑐) · ((𝐹𝑋)‘𝑐)) = ((𝑇𝑐) · (𝐹‘(𝑋𝑐))))
277273, 276eqtrd 2685 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → ((𝑇𝑓 · (𝐹𝑋))‘𝑐) = ((𝑇𝑐) · (𝐹‘(𝑋𝑐))))
278264, 266, 2773eqtrd 2689 . . . . . . . . . . . . . . . . . . 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))))
279278, 231oveq12d 6708 . . . . . . . . . . . . . . . . . 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)))
280243recnd 10106 . . . . . . . . . . . . . . . . . . 19 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (𝐹‘(𝑋𝑐)) ∈ ℂ)
281280, 234, 237divcan3d 10844 . . . . . . . . . . . . . . . . . 18 ((((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇𝑘))) → (((𝑇𝑐) · (𝐹‘(𝑋𝑐))) / (𝑇𝑐)) = (𝐹‘(𝑋𝑐)))
282279, 281eqtrd 2685 . . . . . . . . . . . . . . . . 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)))
283244, 245, 2823brtr4d 4717 . . . . . . . . . . . . . . . 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})))))
284241, 283, 138sylanbrc 699 . . . . . . . . . . . . . . 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}))))})
285284a1d 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}))))}))
286122simprbi 479 . . . . . . . . . . . . . . . 16 ((ℂfld Σg (𝑇𝑘)) ∈ (0[,)+∞) → 0 ≤ (ℂfld Σg (𝑇𝑘)))
287121, 286syl 17 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 0 ≤ (ℂfld Σg (𝑇𝑘)))
288 leloe 10162 . . . . . . . . . . . . . . . 16 ((0 ∈ ℝ ∧ (ℂfld Σg (𝑇𝑘)) ∈ ℝ) → (0 ≤ (ℂfld Σg (𝑇𝑘)) ↔ (0 < (ℂfld Σg (𝑇𝑘)) ∨ 0 = (ℂfld Σg (𝑇𝑘)))))
28981, 124, 288sylancr 696 . . . . . . . . . . . . . . 15 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (0 ≤ (ℂfld Σg (𝑇𝑘)) ↔ (0 < (ℂfld Σg (𝑇𝑘)) ∨ 0 = (ℂfld Σg (𝑇𝑘)))))
290287, 289mpbid 222 . . . . . . . . . . . . . 14 (((𝜑 ∧ ¬ 𝑐𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (0 < (ℂfld Σg (𝑇𝑘)) ∨ 0 = (ℂfld Σg (𝑇𝑘))))
291141, 285, 290mpjaodan 844 . . . . . . . . . . . . 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}))))}))
29288, 291embantd 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}))))}))
29386, 292syl5bi 232 . . . . . . . . . . 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}))))}))
294293ex 449 . . . . . . . . . 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}))))})))
295294com23 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}))))})))
296295expcom 450 . . . . . . . 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}))))}))))
297296adantl 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}))))}))))
298297a2d 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}))))}))))
29932, 48, 64, 80, 85, 298findcard2s 8242 . . . . 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)))})))
30011, 299mpcom 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)))}))
30110, 300mpd 15 . . 3 (𝜑 → ((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴)))})
302 ffn 6083 . . . . . . 7 ((𝑇𝑓 · 𝑋):𝐴⟶ℝ → (𝑇𝑓 · 𝑋) Fn 𝐴)
303159, 302syl 17 . . . . . 6 (𝜑 → (𝑇𝑓 · 𝑋) Fn 𝐴)
304 fnresdm 6038 . . . . . 6 ((𝑇𝑓 · 𝑋) Fn 𝐴 → ((𝑇𝑓 · 𝑋) ↾ 𝐴) = (𝑇𝑓 · 𝑋))
305303, 304syl 17 . . . . 5 (𝜑 → ((𝑇𝑓 · 𝑋) ↾ 𝐴) = (𝑇𝑓 · 𝑋))
306305oveq2d 6706 . . . 4 (𝜑 → (ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ 𝐴)) = (ℂfld Σg (𝑇𝑓 · 𝑋)))
307306, 7oveq12d 6708 . . 3 (𝜑 → ((ℂfld Σg ((𝑇𝑓 · 𝑋) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))) = ((ℂfld Σg (𝑇𝑓 · 𝑋)) / (ℂfld Σg 𝑇)))
3084, 270, 11, 11, 158offn 6950 . . . . . . . 8 (𝜑 → (𝑇𝑓 · (𝐹𝑋)) Fn 𝐴)
309 fnresdm 6038 . . . . . . . 8 ((𝑇𝑓 · (𝐹𝑋)) Fn 𝐴 → ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝐴) = (𝑇𝑓 · (𝐹𝑋)))
310308, 309syl 17 . . . . . . 7 (𝜑 → ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝐴) = (𝑇𝑓 · (𝐹𝑋)))
311310oveq2d 6706 . . . . . 6 (𝜑 → (ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝐴)) = (ℂfld Σg (𝑇𝑓 · (𝐹𝑋))))
312311, 7oveq12d 6708 . . . . 5 (𝜑 → ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))) = ((ℂfld Σg (𝑇𝑓 · (𝐹𝑋))) / (ℂfld Σg 𝑇)))
313312breq2d 4697 . . . 4 (𝜑 → ((𝐹𝑤) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴))) ↔ (𝐹𝑤) ≤ ((ℂfld Σg (𝑇𝑓 · (𝐹𝑋))) / (ℂfld Σg 𝑇))))
314313rabbidv 3220 . . 3 (𝜑 → {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg ((𝑇𝑓 · (𝐹𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇𝐴)))} = {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg (𝑇𝑓 · (𝐹𝑋))) / (ℂfld Σg 𝑇))})
315301, 307, 3143eltr3d 2744 . 2 (𝜑 → ((ℂfld Σg (𝑇𝑓 · 𝑋)) / (ℂfld Σg 𝑇)) ∈ {𝑤𝐷 ∣ (𝐹𝑤) ≤ ((ℂfld Σg (𝑇𝑓 · (𝐹𝑋))) / (ℂfld Σg 𝑇))})
316 fveq2 6229 . . . 4 (𝑤 = ((ℂfld Σg (𝑇𝑓 · 𝑋)) / (ℂfld Σg 𝑇)) → (𝐹𝑤) = (𝐹‘((ℂfld Σg (𝑇𝑓 · 𝑋)) / (ℂfld Σg 𝑇))))
317316breq1d 4695 . . 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))))
318317elrab 3396 . 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))))
319315, 318sylib 208 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 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231  cdif 3604  cun 3605  wss 3607  c0 3948  {csn 4210   class class class wbr 4685  cmpt 4762  cres 5145  ccom 5147   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  𝑓 cof 6937   supp csupp 7340  Fincfn 7997  cc 9972  cr 9973  0cc0 9974  1c1 9975   + caddc 9977   · cmul 9979  +∞cpnf 10109   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  [,)cico 12215  [,]cicc 12216  Σcsu 14460   Σg cgsu 16148  Mndcmnd 17341  SubMndcsubmnd 17381  CMndccmn 18239  Ringcrg 18593  fldccnfld 19794
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-tpos 7397  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-rp 11871  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-0g 16149  df-gsum 16150  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-grp 17472  df-minusg 17473  df-mulg 17588  df-subg 17638  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-cring 18596  df-oppr 18669  df-dvdsr 18687  df-unit 18688  df-invr 18718  df-dvr 18729  df-drng 18797  df-subrg 18826  df-cnfld 19795  df-refld 19999
This theorem is referenced by:  amgmlem  24761  amgmwlem  42876
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