Theorem jensen 26138

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	⊢ (𝜑 → 𝐷 ⊆ ℝ)
jensen.2	⊢ (𝜑 → 𝐹:𝐷⟶ℝ)
jensen.3	⊢ ((𝜑 ∧ (𝑎 ∈ 𝐷 ∧ 𝑏 ∈ 𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷)
jensen.4	⊢ (𝜑 → 𝐴 ∈ Fin)
jensen.5	⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞))
jensen.6	⊢ (𝜑 → 𝑋:𝐴⟶𝐷)
jensen.7	⊢ (𝜑 → 0 < (ℂfld Σg 𝑇))
jensen.8	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ∧ 𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦))))

Assertion

Ref	Expression
jensen	⊢ (𝜑 → (((ℂfld Σg (𝑇 ∘f · 𝑋)) / (ℂfld Σg 𝑇)) ∈ 𝐷 ∧ (𝐹‘((ℂfld Σg (𝑇 ∘f · 𝑋)) / (ℂfld Σg 𝑇))) ≤ ((ℂfld Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))) / (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.

Step	Hyp	Ref	Expression
1		jensen.7	. . . . . 6 ⊢ (𝜑 → 0 < (ℂfld Σg 𝑇))
2		jensen.5	. . . . . . . . 9 ⊢ (𝜑 → 𝑇:𝐴⟶(0[,)+∞))
3	2	ffnd 6601	. . . . . . . 8 ⊢ (𝜑 → 𝑇 Fn 𝐴)
4		fnresdm 6551	. . . . . . . 8 ⊢ (𝑇 Fn 𝐴 → (𝑇 ↾ 𝐴) = 𝑇)
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑇 ↾ 𝐴) = 𝑇)
6	5	oveq2d 7291	. . . . . 6 ⊢ (𝜑 → (ℂfld Σg (𝑇 ↾ 𝐴)) = (ℂfld Σg 𝑇))
7	1, 6	breqtrrd 5102	. . . . 5 ⊢ (𝜑 → 0 < (ℂfld Σg (𝑇 ↾ 𝐴)))
8		ssid 3943	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
9	7, 8	jctil 520	. . . 4 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝐴))))
10		jensen.4	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin)
11		sseq1 3946	. . . . . . . . 9 ⊢ (𝑎 = ∅ → (𝑎 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
12		reseq2 5886	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∅ → (𝑇 ↾ 𝑎) = (𝑇 ↾ ∅))
13		res0 5895	. . . . . . . . . . . . 13 ⊢ (𝑇 ↾ ∅) = ∅
14	12, 13	eqtrdi 2794	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → (𝑇 ↾ 𝑎) = ∅)
15	14	oveq2d 7291	. . . . . . . . . . 11 ⊢ (𝑎 = ∅ → (ℂfld Σg (𝑇 ↾ 𝑎)) = (ℂfld Σg ∅))
16		cnfld0 20622	. . . . . . . . . . . 12 ⊢ 0 = (0g‘ℂfld)
17	16	gsum0 18368	. . . . . . . . . . 11 ⊢ (ℂfld Σg ∅) = 0
18	15, 17	eqtrdi 2794	. . . . . . . . . 10 ⊢ (𝑎 = ∅ → (ℂfld Σg (𝑇 ↾ 𝑎)) = 0)
19	18	breq2d 5086	. . . . . . . . 9 ⊢ (𝑎 = ∅ → (0 < (ℂfld Σg (𝑇 ↾ 𝑎)) ↔ 0 < 0))
20	11, 19	anbi12d 631	. . . . . . . 8 ⊢ (𝑎 = ∅ → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑎))) ↔ (∅ ⊆ 𝐴 ∧ 0 < 0)))
21		reseq2 5886	. . . . . . . . . . 11 ⊢ (𝑎 = ∅ → ((𝑇 ∘f · 𝑋) ↾ 𝑎) = ((𝑇 ∘f · 𝑋) ↾ ∅))
22	21	oveq2d 7291	. . . . . . . . . 10 ⊢ (𝑎 = ∅ → (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ ∅)))
23	22, 18	oveq12d 7293	. . . . . . . . 9 ⊢ (𝑎 = ∅ → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) = ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ ∅)) / 0))
24		reseq2 5886	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∅ → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎) = ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅))
25	24	oveq2d 7291	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)))
26	25, 18	oveq12d 7293	. . . . . . . . . . 11 ⊢ (𝑎 = ∅ → ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) = ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) / 0))
27	26	breq2d 5086	. . . . . . . . . 10 ⊢ (𝑎 = ∅ → ((𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) ↔ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) / 0)))
28	27	rabbidv 3414	. . . . . . . . 9 ⊢ (𝑎 = ∅ → {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎)))} = {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) / 0)})
29	23, 28	eleq12d 2833	. . . . . . . 8 ⊢ (𝑎 = ∅ → (((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎)))} ↔ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ ∅)) / 0) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) / 0)}))
30	20, 29	imbi12d 345	. . . . . . 7 ⊢ (𝑎 = ∅ → (((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑎))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎)))}) ↔ ((∅ ⊆ 𝐴 ∧ 0 < 0) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ ∅)) / 0) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) / 0)})))
31	30	imbi2d 341	. . . . . 6 ⊢ (𝑎 = ∅ → ((𝜑 → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑎))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎)))})) ↔ (𝜑 → ((∅ ⊆ 𝐴 ∧ 0 < 0) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ ∅)) / 0) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) / 0)}))))
32		sseq1 3946	. . . . . . . . 9 ⊢ (𝑎 = 𝑘 → (𝑎 ⊆ 𝐴 ↔ 𝑘 ⊆ 𝐴))
33		reseq2 5886	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑘 → (𝑇 ↾ 𝑎) = (𝑇 ↾ 𝑘))
34	33	oveq2d 7291	. . . . . . . . . 10 ⊢ (𝑎 = 𝑘 → (ℂfld Σg (𝑇 ↾ 𝑎)) = (ℂfld Σg (𝑇 ↾ 𝑘)))
35	34	breq2d 5086	. . . . . . . . 9 ⊢ (𝑎 = 𝑘 → (0 < (ℂfld Σg (𝑇 ↾ 𝑎)) ↔ 0 < (ℂfld Σg (𝑇 ↾ 𝑘))))
36	32, 35	anbi12d 631	. . . . . . . 8 ⊢ (𝑎 = 𝑘 → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑎))) ↔ (𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑘)))))
37		reseq2 5886	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑘 → ((𝑇 ∘f · 𝑋) ↾ 𝑎) = ((𝑇 ∘f · 𝑋) ↾ 𝑘))
38	37	oveq2d 7291	. . . . . . . . . 10 ⊢ (𝑎 = 𝑘 → (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)))
39	38, 34	oveq12d 7293	. . . . . . . . 9 ⊢ (𝑎 = 𝑘 → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) = ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))))
40		reseq2 5886	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑘 → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎) = ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘))
41	40	oveq2d 7291	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑘 → (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)))
42	41, 34	oveq12d 7293	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑘 → ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) = ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))))
43	42	breq2d 5086	. . . . . . . . . 10 ⊢ (𝑎 = 𝑘 → ((𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) ↔ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))))
44	43	rabbidv 3414	. . . . . . . . 9 ⊢ (𝑎 = 𝑘 → {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎)))} = {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})
45	39, 44	eleq12d 2833	. . . . . . . 8 ⊢ (𝑎 = 𝑘 → (((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎)))} ↔ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))}))
46	36, 45	imbi12d 345	. . . . . . 7 ⊢ (𝑎 = 𝑘 → (((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑎))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎)))}) ↔ ((𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑘))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})))
47	46	imbi2d 341	. . . . . 6 ⊢ (𝑎 = 𝑘 → ((𝜑 → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑎))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎)))})) ↔ (𝜑 → ((𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑘))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))}))))
48		sseq1 3946	. . . . . . . . 9 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (𝑎 ⊆ 𝐴 ↔ (𝑘 ∪ {𝑐}) ⊆ 𝐴))
49		reseq2 5886	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (𝑇 ↾ 𝑎) = (𝑇 ↾ (𝑘 ∪ {𝑐})))
50	49	oveq2d 7291	. . . . . . . . . 10 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld Σg (𝑇 ↾ 𝑎)) = (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))
51	50	breq2d 5086	. . . . . . . . 9 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (0 < (ℂfld Σg (𝑇 ↾ 𝑎)) ↔ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))))
52	48, 51	anbi12d 631	. . . . . . . 8 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑎))) ↔ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))))
53		reseq2 5886	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑇 ∘f · 𝑋) ↾ 𝑎) = ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐})))
54	53	oveq2d 7291	. . . . . . . . . 10 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))))
55	54, 50	oveq12d 7293	. . . . . . . . 9 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) = ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))))
56		reseq2 5886	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎) = ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐})))
57	56	oveq2d 7291	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))))
58	57, 50	oveq12d 7293	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) = ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))))
59	58	breq2d 5086	. . . . . . . . . 10 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) ↔ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))))
60	59	rabbidv 3414	. . . . . . . . 9 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎)))} = {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})
61	55, 60	eleq12d 2833	. . . . . . . 8 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎)))} ↔ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))
62	52, 61	imbi12d 345	. . . . . . 7 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑎))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎)))}) ↔ (((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})))
63	62	imbi2d 341	. . . . . 6 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((𝜑 → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑎))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎)))})) ↔ (𝜑 → (((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))))
64		sseq1 3946	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
65		reseq2 5886	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑇 ↾ 𝑎) = (𝑇 ↾ 𝐴))
66	65	oveq2d 7291	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (ℂfld Σg (𝑇 ↾ 𝑎)) = (ℂfld Σg (𝑇 ↾ 𝐴)))
67	66	breq2d 5086	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (0 < (ℂfld Σg (𝑇 ↾ 𝑎)) ↔ 0 < (ℂfld Σg (𝑇 ↾ 𝐴))))
68	64, 67	anbi12d 631	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑎))) ↔ (𝐴 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝐴)))))
69		reseq2 5886	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → ((𝑇 ∘f · 𝑋) ↾ 𝑎) = ((𝑇 ∘f · 𝑋) ↾ 𝐴))
70	69	oveq2d 7291	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)))
71	70, 66	oveq12d 7293	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) = ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)) / (ℂfld Σg (𝑇 ↾ 𝐴))))
72		reseq2 5886	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎) = ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴))
73	72	oveq2d 7291	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)))
74	73, 66	oveq12d 7293	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) = ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇 ↾ 𝐴))))
75	74	breq2d 5086	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) ↔ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇 ↾ 𝐴)))))
76	75	rabbidv 3414	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎)))} = {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇 ↾ 𝐴)))})
77	71, 76	eleq12d 2833	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎)))} ↔ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)) / (ℂfld Σg (𝑇 ↾ 𝐴))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇 ↾ 𝐴)))}))
78	68, 77	imbi12d 345	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑎))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎)))}) ↔ ((𝐴 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝐴))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)) / (ℂfld Σg (𝑇 ↾ 𝐴))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇 ↾ 𝐴)))})))
79	78	imbi2d 341	. . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝜑 → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑎))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎)))})) ↔ (𝜑 → ((𝐴 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝐴))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)) / (ℂfld Σg (𝑇 ↾ 𝐴))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇 ↾ 𝐴)))}))))
80		0re 10977	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
81	80	ltnri 11084	. . . . . . . . 9 ⊢ ¬ 0 < 0
82	81	pm2.21i 119	. . . . . . . 8 ⊢ (0 < 0 → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ ∅)) / 0) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) / 0)})
83	82	adantl 482	. . . . . . 7 ⊢ ((∅ ⊆ 𝐴 ∧ 0 < 0) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ ∅)) / 0) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) / 0)})
84	83	a1i 11	. . . . . 6 ⊢ (𝜑 → ((∅ ⊆ 𝐴 ∧ 0 < 0) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ ∅)) / 0) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) / 0)}))
85		impexp 451	. . . . . . . . . . . 12 ⊢ (((𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑘))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))}) ↔ (𝑘 ⊆ 𝐴 → (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})))
86		simprl 768	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑘 ∪ {𝑐}) ⊆ 𝐴)
87	86	unssad 4121	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑘 ⊆ 𝐴)
88		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑘))) → 0 < (ℂfld Σg (𝑇 ↾ 𝑘)))
89		jensen.1	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐷 ⊆ ℝ)
90	89	ad3antrrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) → 𝐷 ⊆ ℝ)
91		jensen.2	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐹:𝐷⟶ℝ)
92	91	ad3antrrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) → 𝐹:𝐷⟶ℝ)
93		simplll 772	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) → 𝜑)
94		jensen.3	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐷 ∧ 𝑏 ∈ 𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷)
95	93, 94	sylan 580	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) ∧ (𝑎 ∈ 𝐷 ∧ 𝑏 ∈ 𝐷)) → (𝑎[,]𝑏) ⊆ 𝐷)
96	93, 10	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) → 𝐴 ∈ Fin)
97	93, 2	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) → 𝑇:𝐴⟶(0[,)+∞))
98		jensen.6	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑋:𝐴⟶𝐷)
99	93, 98	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) → 𝑋:𝐴⟶𝐷)
100	1	ad3antrrr 727	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) → 0 < (ℂfld Σg 𝑇))
101		jensen.8	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ∧ 𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦))))
102	93, 101	sylan 580	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) ∧ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐷 ∧ 𝑡 ∈ (0[,]1))) → (𝐹‘((𝑡 · 𝑥) + ((1 − 𝑡) · 𝑦))) ≤ ((𝑡 · (𝐹‘𝑥)) + ((1 − 𝑡) · (𝐹‘𝑦))))
103		simpllr 773	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) → ¬ 𝑐 ∈ 𝑘)
104	86	adantr 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) → (𝑘 ∪ {𝑐}) ⊆ 𝐴)
105		eqid 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (ℂfld Σg (𝑇 ↾ 𝑘)) = (ℂfld Σg (𝑇 ↾ 𝑘))
106		eqid 2738	. . . . . . . . . . . . . . . . . 18 ⊢ (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))) = (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))
107		cnring 20620	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℂfld ∈ Ring
108		ringcmn 19820	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℂfld ∈ Ring → ℂfld ∈ CMnd)
109	107, 108	mp1i 13	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → ℂfld ∈ CMnd)
110	10	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝐴 ∈ Fin)
111	110, 87	ssfid 9042	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑘 ∈ Fin)
112		rege0subm 20654	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (0[,)+∞) ∈ (SubMnd‘ℂfld)
113	112	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (0[,)+∞) ∈ (SubMnd‘ℂfld))
114	2	ad2antrr 723	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑇:𝐴⟶(0[,)+∞))
115	114, 87	fssresd 6641	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑇 ↾ 𝑘):𝑘⟶(0[,)+∞))
116		c0ex 10969	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 0 ∈ V
117	116	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 0 ∈ V)
118	115, 111, 117	fdmfifsupp 9138	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑇 ↾ 𝑘) finSupp 0)
119	16, 109, 111, 113, 115, 118	gsumsubmcl 19520	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (ℂfld Σg (𝑇 ↾ 𝑘)) ∈ (0[,)+∞))
120		elrege0 13186	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℂfld Σg (𝑇 ↾ 𝑘)) ∈ (0[,)+∞) ↔ ((ℂfld Σg (𝑇 ↾ 𝑘)) ∈ ℝ ∧ 0 ≤ (ℂfld Σg (𝑇 ↾ 𝑘))))
121	120	simplbi 498	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((ℂfld Σg (𝑇 ↾ 𝑘)) ∈ (0[,)+∞) → (ℂfld Σg (𝑇 ↾ 𝑘)) ∈ ℝ)
122	119, 121	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (ℂfld Σg (𝑇 ↾ 𝑘)) ∈ ℝ)
123	122	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) → (ℂfld Σg (𝑇 ↾ 𝑘)) ∈ ℝ)
124		simprl 768	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) → 0 < (ℂfld Σg (𝑇 ↾ 𝑘)))
125	123, 124	elrpd 12769	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) → (ℂfld Σg (𝑇 ↾ 𝑘)) ∈ ℝ+)
126		simprr 770	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})
127		fveq2 6774	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 = ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝐹‘𝑤) = (𝐹‘((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))))
128	127	breq1d 5084	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑤 = ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) → ((𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ↔ (𝐹‘((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))))
129	128	elrab 3624	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))} ↔ (((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ 𝐷 ∧ (𝐹‘((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))))
130	126, 129	sylib 217	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) → (((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ 𝐷 ∧ (𝐹‘((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))))
131	130	simpld 495	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ 𝐷)
132	130	simprd 496	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) → (𝐹‘((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))))
133	90, 92, 95, 96, 97, 99, 100, 102, 103, 104, 105, 106, 125, 131, 132	jensenlem2 26137	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) → (((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ 𝐷 ∧ (𝐹‘((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))))
134		fveq2 6774	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → (𝐹‘𝑤) = (𝐹‘((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))))
135	134	breq1d 5084	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑤 = ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → ((𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ↔ (𝐹‘((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))))
136	135	elrab 3624	. . . . . . . . . . . . . . . . 17 ⊢ (((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))} ↔ (((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ 𝐷 ∧ (𝐹‘((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))))
137	133, 136	sylibr 233	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∧ ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})
138	137	expr 457	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑘))) → (((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))} → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))
139	88, 138	embantd 59	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑘))) → ((0 < (ℂfld Σg (𝑇 ↾ 𝑘)) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))}) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))
140		cnfldbas 20601	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℂ = (Base‘ℂfld)
141		ringmnd 19793	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd)
142	107, 141	mp1i 13	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → ℂfld ∈ Mnd)
143	110, 86	ssfid 9042	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑘 ∪ {𝑐}) ∈ Fin)
144	143	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝑘 ∪ {𝑐}) ∈ Fin)
145		ssun2 4107	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑐} ⊆ (𝑘 ∪ {𝑐})
146		vsnid 4598	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑐 ∈ {𝑐}
147	145, 146	sselii 3918	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑐 ∈ (𝑘 ∪ {𝑐})
148	147	a1i 11	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑐 ∈ (𝑘 ∪ {𝑐}))
149		remulcl 10956	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ)
150	149	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
151		rge0ssre 13188	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (0[,)+∞) ⊆ ℝ
152		fss 6617	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑇:𝐴⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝑇:𝐴⟶ℝ)
153	2, 151, 152	sylancl 586	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑇:𝐴⟶ℝ)
154	98, 89	fssd 6618	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑋:𝐴⟶ℝ)
155		inidm 4152	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∩ 𝐴) = 𝐴
156	150, 153, 154, 10, 10, 155	off 7551	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑇 ∘f · 𝑋):𝐴⟶ℝ)
157		ax-resscn 10928	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℝ ⊆ ℂ
158		fss 6617	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑇 ∘f · 𝑋):𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → (𝑇 ∘f · 𝑋):𝐴⟶ℂ)
159	156, 157, 158	sylancl 586	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑇 ∘f · 𝑋):𝐴⟶ℂ)
160	159	ad3antrrr 727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝑇 ∘f · 𝑋):𝐴⟶ℂ)
161	86	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝑘 ∪ {𝑐}) ⊆ 𝐴)
162	160, 161	fssresd 6641	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ)
163	2	ad3antrrr 727	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑇:𝐴⟶(0[,)+∞))
164	110	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝐴 ∈ Fin)
165	163, 164	fexd 7103	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑇 ∈ V)
166	98	ad3antrrr 727	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑋:𝐴⟶𝐷)
167	166, 164	fexd 7103	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑋 ∈ V)
168		offres 7826	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑇 ∈ V ∧ 𝑋 ∈ V) → ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐})) = ((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · (𝑋 ↾ (𝑘 ∪ {𝑐}))))
169	165, 167, 168	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐})) = ((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · (𝑋 ↾ (𝑘 ∪ {𝑐}))))
170	169	oveq1d 7290	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐})) supp 0) = (((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · (𝑋 ↾ (𝑘 ∪ {𝑐}))) supp 0))
171	151, 157	sstri 3930	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (0[,)+∞) ⊆ ℂ
172		fss 6617	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑇:𝐴⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℂ) → 𝑇:𝐴⟶ℂ)
173	163, 171, 172	sylancl 586	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑇:𝐴⟶ℂ)
174	173, 161	fssresd 6641	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝑇 ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ)
175		eldifi 4061	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) → 𝑥 ∈ (𝑘 ∪ {𝑐}))
176	175	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐})) → 𝑥 ∈ (𝑘 ∪ {𝑐}))
177	176	fvresd 6794	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐})) → ((𝑇 ↾ (𝑘 ∪ {𝑐}))‘𝑥) = (𝑇‘𝑥))
178		difun2 4414	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) = (𝑘 ∖ {𝑐})
179		difss 4066	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∖ {𝑐}) ⊆ 𝑘
180	178, 179	eqsstri 3955	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) ⊆ 𝑘
181	180	sseli 3917	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) → 𝑥 ∈ 𝑘)
182		simpr 485	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 0 = (ℂfld Σg (𝑇 ↾ 𝑘)))
183	87	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑘 ⊆ 𝐴)
184	163, 183	feqresmpt 6838	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝑇 ↾ 𝑘) = (𝑥 ∈ 𝑘 ↦ (𝑇‘𝑥)))
185	184	oveq2d 7291	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (ℂfld Σg (𝑇 ↾ 𝑘)) = (ℂfld Σg (𝑥 ∈ 𝑘 ↦ (𝑇‘𝑥))))
186	111	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑘 ∈ Fin)
187	183	sselda 3921	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ 𝑘) → 𝑥 ∈ 𝐴)
188	163	ffvelrnda 6961	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ 𝐴) → (𝑇‘𝑥) ∈ (0[,)+∞))
189	187, 188	syldan 591	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ 𝑘) → (𝑇‘𝑥) ∈ (0[,)+∞))
190	171, 189	sselid 3919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ 𝑘) → (𝑇‘𝑥) ∈ ℂ)
191	186, 190	gsumfsum 20665	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (ℂfld Σg (𝑥 ∈ 𝑘 ↦ (𝑇‘𝑥))) = Σ𝑥 ∈ 𝑘 (𝑇‘𝑥))
192	182, 185, 191	3eqtrrd 2783	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → Σ𝑥 ∈ 𝑘 (𝑇‘𝑥) = 0)
193		elrege0 13186	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑇‘𝑥) ∈ (0[,)+∞) ↔ ((𝑇‘𝑥) ∈ ℝ ∧ 0 ≤ (𝑇‘𝑥)))
194	189, 193	sylib 217	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ 𝑘) → ((𝑇‘𝑥) ∈ ℝ ∧ 0 ≤ (𝑇‘𝑥)))
195	194	simpld 495	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ 𝑘) → (𝑇‘𝑥) ∈ ℝ)
196	194	simprd 496	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ 𝑘) → 0 ≤ (𝑇‘𝑥))
197	186, 195, 196	fsum00 15510	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (Σ𝑥 ∈ 𝑘 (𝑇‘𝑥) = 0 ↔ ∀𝑥 ∈ 𝑘 (𝑇‘𝑥) = 0))
198	192, 197	mpbid 231	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → ∀𝑥 ∈ 𝑘 (𝑇‘𝑥) = 0)
199	198	r19.21bi 3134	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ 𝑘) → (𝑇‘𝑥) = 0)
200	181, 199	sylan2 593	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐})) → (𝑇‘𝑥) = 0)
201	177, 200	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐})) → ((𝑇 ↾ (𝑘 ∪ {𝑐}))‘𝑥) = 0)
202	174, 201	suppss 8010	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → ((𝑇 ↾ (𝑘 ∪ {𝑐})) supp 0) ⊆ {𝑐})
203		mul02 11153	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ ℂ → (0 · 𝑥) = 0)
204	203	adantl 482	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ ℂ) → (0 · 𝑥) = 0)
205	89	ad3antrrr 727	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝐷 ⊆ ℝ)
206	205, 157	sstrdi 3933	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝐷 ⊆ ℂ)
207	166, 206	fssd 6618	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑋:𝐴⟶ℂ)
208	207, 161	fssresd 6641	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝑋 ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ)
209	116	a1i 11	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 0 ∈ V)
210	202, 204, 174, 208, 144, 209	suppssof1 8015	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · (𝑋 ↾ (𝑘 ∪ {𝑐}))) supp 0) ⊆ {𝑐})
211	170, 210	eqsstrd 3959	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐})) supp 0) ⊆ {𝑐})
212	140, 16, 142, 144, 148, 162, 211	gsumpt 19563	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) = (((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))‘𝑐))
213	148	fvresd 6794	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))‘𝑐) = ((𝑇 ∘f · 𝑋)‘𝑐))
214	163	ffnd 6601	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑇 Fn 𝐴)
215	166	ffnd 6601	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑋 Fn 𝐴)
216	161, 148	sseldd 3922	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑐 ∈ 𝐴)
217		fnfvof 7550	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑇 Fn 𝐴 ∧ 𝑋 Fn 𝐴) ∧ (𝐴 ∈ Fin ∧ 𝑐 ∈ 𝐴)) → ((𝑇 ∘f · 𝑋)‘𝑐) = ((𝑇‘𝑐) · (𝑋‘𝑐)))
218	214, 215, 164, 216, 217	syl22anc 836	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → ((𝑇 ∘f · 𝑋)‘𝑐) = ((𝑇‘𝑐) · (𝑋‘𝑐)))
219	212, 213, 218	3eqtrd 2782	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) = ((𝑇‘𝑐) · (𝑋‘𝑐)))
220	140, 16, 142, 144, 148, 174, 202	gsumpt 19563	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))) = ((𝑇 ↾ (𝑘 ∪ {𝑐}))‘𝑐))
221	148	fvresd 6794	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → ((𝑇 ↾ (𝑘 ∪ {𝑐}))‘𝑐) = (𝑇‘𝑐))
222	220, 221	eqtrd 2778	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))) = (𝑇‘𝑐))
223	219, 222	oveq12d 7293	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) = (((𝑇‘𝑐) · (𝑋‘𝑐)) / (𝑇‘𝑐)))
224	207, 216	ffvelrnd 6962	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝑋‘𝑐) ∈ ℂ)
225	173, 216	ffvelrnd 6962	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝑇‘𝑐) ∈ ℂ)
226		simplrr 775	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))
227	226, 222	breqtrd 5100	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 0 < (𝑇‘𝑐))
228	227	gt0ne0d 11539	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝑇‘𝑐) ≠ 0)
229	224, 225, 228	divcan3d 11756	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (((𝑇‘𝑐) · (𝑋‘𝑐)) / (𝑇‘𝑐)) = (𝑋‘𝑐))
230	223, 229	eqtrd 2778	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) = (𝑋‘𝑐))
231	166, 216	ffvelrnd 6962	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝑋‘𝑐) ∈ 𝐷)
232	230, 231	eqeltrd 2839	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ 𝐷)
233	91	ad3antrrr 727	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝐹:𝐷⟶ℝ)
234	233, 231	ffvelrnd 6962	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝐹‘(𝑋‘𝑐)) ∈ ℝ)
235	234	leidd 11541	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝐹‘(𝑋‘𝑐)) ≤ (𝐹‘(𝑋‘𝑐)))
236	230	fveq2d 6778	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝐹‘((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) = (𝐹‘(𝑋‘𝑐)))
237		fco 6624	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑋:𝐴⟶𝐷) → (𝐹 ∘ 𝑋):𝐴⟶ℝ)
238	91, 98, 237	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝐹 ∘ 𝑋):𝐴⟶ℝ)
239	150, 153, 238, 10, 10, 155	off 7551	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑇 ∘f · (𝐹 ∘ 𝑋)):𝐴⟶ℝ)
240		fss 6617	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑇 ∘f · (𝐹 ∘ 𝑋)):𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → (𝑇 ∘f · (𝐹 ∘ 𝑋)):𝐴⟶ℂ)
241	239, 157, 240	sylancl 586	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝑇 ∘f · (𝐹 ∘ 𝑋)):𝐴⟶ℂ)
242	241	ad3antrrr 727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝑇 ∘f · (𝐹 ∘ 𝑋)):𝐴⟶ℂ)
243	242, 161	fssresd 6641	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ)
244	238	ad3antrrr 727	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝐹 ∘ 𝑋):𝐴⟶ℝ)
245	244, 164	fexd 7103	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝐹 ∘ 𝑋) ∈ V)
246		offres 7826	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑇 ∈ V ∧ (𝐹 ∘ 𝑋) ∈ V) → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐})) = ((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · ((𝐹 ∘ 𝑋) ↾ (𝑘 ∪ {𝑐}))))
247	165, 245, 246	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐})) = ((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · ((𝐹 ∘ 𝑋) ↾ (𝑘 ∪ {𝑐}))))
248	247	oveq1d 7290	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐})) supp 0) = (((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · ((𝐹 ∘ 𝑋) ↾ (𝑘 ∪ {𝑐}))) supp 0))
249		fss 6617	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐹 ∘ 𝑋):𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → (𝐹 ∘ 𝑋):𝐴⟶ℂ)
250	244, 157, 249	sylancl 586	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝐹 ∘ 𝑋):𝐴⟶ℂ)
251	250, 161	fssresd 6641	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → ((𝐹 ∘ 𝑋) ↾ (𝑘 ∪ {𝑐})):(𝑘 ∪ {𝑐})⟶ℂ)
252	202, 204, 174, 251, 144, 209	suppssof1 8015	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · ((𝐹 ∘ 𝑋) ↾ (𝑘 ∪ {𝑐}))) supp 0) ⊆ {𝑐})
253	248, 252	eqsstrd 3959	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐})) supp 0) ⊆ {𝑐})
254	140, 16, 142, 144, 148, 243, 253	gsumpt 19563	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) = (((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))‘𝑐))
255	148	fvresd 6794	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))‘𝑐) = ((𝑇 ∘f · (𝐹 ∘ 𝑋))‘𝑐))
256	91	ffnd 6601	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐹 Fn 𝐷)
257		fnfco 6639	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹 Fn 𝐷 ∧ 𝑋:𝐴⟶𝐷) → (𝐹 ∘ 𝑋) Fn 𝐴)
258	256, 98, 257	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐹 ∘ 𝑋) Fn 𝐴)
259	258	ad3antrrr 727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝐹 ∘ 𝑋) Fn 𝐴)
260		fnfvof 7550	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑇 Fn 𝐴 ∧ (𝐹 ∘ 𝑋) Fn 𝐴) ∧ (𝐴 ∈ Fin ∧ 𝑐 ∈ 𝐴)) → ((𝑇 ∘f · (𝐹 ∘ 𝑋))‘𝑐) = ((𝑇‘𝑐) · ((𝐹 ∘ 𝑋)‘𝑐)))
261	214, 259, 164, 216, 260	syl22anc 836	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → ((𝑇 ∘f · (𝐹 ∘ 𝑋))‘𝑐) = ((𝑇‘𝑐) · ((𝐹 ∘ 𝑋)‘𝑐)))
262		fvco3 6867	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑋:𝐴⟶𝐷 ∧ 𝑐 ∈ 𝐴) → ((𝐹 ∘ 𝑋)‘𝑐) = (𝐹‘(𝑋‘𝑐)))
263	166, 216, 262	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → ((𝐹 ∘ 𝑋)‘𝑐) = (𝐹‘(𝑋‘𝑐)))
264	263	oveq2d 7291	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → ((𝑇‘𝑐) · ((𝐹 ∘ 𝑋)‘𝑐)) = ((𝑇‘𝑐) · (𝐹‘(𝑋‘𝑐))))
265	261, 264	eqtrd 2778	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → ((𝑇 ∘f · (𝐹 ∘ 𝑋))‘𝑐) = ((𝑇‘𝑐) · (𝐹‘(𝑋‘𝑐))))
266	254, 255, 265	3eqtrd 2782	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) = ((𝑇‘𝑐) · (𝐹‘(𝑋‘𝑐))))
267	266, 222	oveq12d 7293	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) = (((𝑇‘𝑐) · (𝐹‘(𝑋‘𝑐))) / (𝑇‘𝑐)))
268	234	recnd 11003	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝐹‘(𝑋‘𝑐)) ∈ ℂ)
269	268, 225, 228	divcan3d 11756	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (((𝑇‘𝑐) · (𝐹‘(𝑋‘𝑐))) / (𝑇‘𝑐)) = (𝐹‘(𝑋‘𝑐)))
270	267, 269	eqtrd 2778	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) = (𝐹‘(𝑋‘𝑐)))
271	235, 236, 270	3brtr4d 5106	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝐹‘((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))))
272	135, 232, 271	elrabd 3626	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})
273	272	a1d 25	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → ((0 < (ℂfld Σg (𝑇 ↾ 𝑘)) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))}) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))
274	120	simprbi 497	. . . . . . . . . . . . . . . 16 ⊢ ((ℂfld Σg (𝑇 ↾ 𝑘)) ∈ (0[,)+∞) → 0 ≤ (ℂfld Σg (𝑇 ↾ 𝑘)))
275	119, 274	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 0 ≤ (ℂfld Σg (𝑇 ↾ 𝑘)))
276		leloe 11061	. . . . . . . . . . . . . . . 16 ⊢ ((0 ∈ ℝ ∧ (ℂfld Σg (𝑇 ↾ 𝑘)) ∈ ℝ) → (0 ≤ (ℂfld Σg (𝑇 ↾ 𝑘)) ↔ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∨ 0 = (ℂfld Σg (𝑇 ↾ 𝑘)))))
277	80, 122, 276	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (0 ≤ (ℂfld Σg (𝑇 ↾ 𝑘)) ↔ (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∨ 0 = (ℂfld Σg (𝑇 ↾ 𝑘)))))
278	275, 277	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) ∨ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))))
279	139, 273, 278	mpjaodan 956	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → ((0 < (ℂfld Σg (𝑇 ↾ 𝑘)) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))}) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))
280	87, 279	embantd 59	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → ((𝑘 ⊆ 𝐴 → (0 < (ℂfld Σg (𝑇 ↾ 𝑘)) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))
281	85, 280	syl5bi 241	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (((𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑘))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))}) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))
282	281	ex 413	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) → (((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → (((𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑘))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))}) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})))
283	282	com23 86	. . . . . . . . 9 ⊢ ((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) → (((𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑘))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))}) → (((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))})))
284	283	expcom 414	. . . . . . . 8 ⊢ (¬ 𝑐 ∈ 𝑘 → (𝜑 → (((𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑘))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))}) → (((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))))
285	284	adantl 482	. . . . . . 7 ⊢ ((𝑘 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑘) → (𝜑 → (((𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑘))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))}) → (((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))))
286	285	a2d 29	. . . . . 6 ⊢ ((𝑘 ∈ Fin ∧ ¬ 𝑐 ∈ 𝑘) → ((𝜑 → ((𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑘))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)) / (ℂfld Σg (𝑇 ↾ 𝑘)))})) → (𝜑 → (((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))) / (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))}))))
287	31, 47, 63, 79, 84, 286	findcard2s 8948	. . . . 5 ⊢ (𝐴 ∈ Fin → (𝜑 → ((𝐴 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝐴))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)) / (ℂfld Σg (𝑇 ↾ 𝐴))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇 ↾ 𝐴)))})))
288	10, 287	mpcom 38	. . . 4 ⊢ (𝜑 → ((𝐴 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝐴))) → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)) / (ℂfld Σg (𝑇 ↾ 𝐴))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇 ↾ 𝐴)))}))
289	9, 288	mpd 15	. . 3 ⊢ (𝜑 → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)) / (ℂfld Σg (𝑇 ↾ 𝐴))) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇 ↾ 𝐴)))})
290	156	ffnd 6601	. . . . . 6 ⊢ (𝜑 → (𝑇 ∘f · 𝑋) Fn 𝐴)
291		fnresdm 6551	. . . . . 6 ⊢ ((𝑇 ∘f · 𝑋) Fn 𝐴 → ((𝑇 ∘f · 𝑋) ↾ 𝐴) = (𝑇 ∘f · 𝑋))
292	290, 291	syl 17	. . . . 5 ⊢ (𝜑 → ((𝑇 ∘f · 𝑋) ↾ 𝐴) = (𝑇 ∘f · 𝑋))
293	292	oveq2d 7291	. . . 4 ⊢ (𝜑 → (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)) = (ℂfld Σg (𝑇 ∘f · 𝑋)))
294	293, 6	oveq12d 7293	. . 3 ⊢ (𝜑 → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)) / (ℂfld Σg (𝑇 ↾ 𝐴))) = ((ℂfld Σg (𝑇 ∘f · 𝑋)) / (ℂfld Σg 𝑇)))
295	3, 258, 10, 10, 155	offn 7546	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ∘f · (𝐹 ∘ 𝑋)) Fn 𝐴)
296		fnresdm 6551	. . . . . . . 8 ⊢ ((𝑇 ∘f · (𝐹 ∘ 𝑋)) Fn 𝐴 → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴) = (𝑇 ∘f · (𝐹 ∘ 𝑋)))
297	295, 296	syl 17	. . . . . . 7 ⊢ (𝜑 → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴) = (𝑇 ∘f · (𝐹 ∘ 𝑋)))
298	297	oveq2d 7291	. . . . . 6 ⊢ (𝜑 → (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) = (ℂfld Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))))
299	298, 6	oveq12d 7293	. . . . 5 ⊢ (𝜑 → ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇 ↾ 𝐴))) = ((ℂfld Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))) / (ℂfld Σg 𝑇)))
300	299	breq2d 5086	. . . 4 ⊢ (𝜑 → ((𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇 ↾ 𝐴))) ↔ (𝐹‘𝑤) ≤ ((ℂfld Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))) / (ℂfld Σg 𝑇))))
301	300	rabbidv 3414	. . 3 ⊢ (𝜑 → {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) / (ℂfld Σg (𝑇 ↾ 𝐴)))} = {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))) / (ℂfld Σg 𝑇))})
302	289, 294, 301	3eltr3d 2853	. 2 ⊢ (𝜑 → ((ℂfld Σg (𝑇 ∘f · 𝑋)) / (ℂfld Σg 𝑇)) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))) / (ℂfld Σg 𝑇))})
303		fveq2 6774	. . . 4 ⊢ (𝑤 = ((ℂfld Σg (𝑇 ∘f · 𝑋)) / (ℂfld Σg 𝑇)) → (𝐹‘𝑤) = (𝐹‘((ℂfld Σg (𝑇 ∘f · 𝑋)) / (ℂfld Σg 𝑇))))
304	303	breq1d 5084	. . 3 ⊢ (𝑤 = ((ℂfld Σg (𝑇 ∘f · 𝑋)) / (ℂfld Σg 𝑇)) → ((𝐹‘𝑤) ≤ ((ℂfld Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))) / (ℂfld Σg 𝑇)) ↔ (𝐹‘((ℂfld Σg (𝑇 ∘f · 𝑋)) / (ℂfld Σg 𝑇))) ≤ ((ℂfld Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))) / (ℂfld Σg 𝑇))))
305	304	elrab 3624	. 2 ⊢ (((ℂfld Σg (𝑇 ∘f · 𝑋)) / (ℂfld Σg 𝑇)) ∈ {𝑤 ∈ 𝐷 ∣ (𝐹‘𝑤) ≤ ((ℂfld Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))) / (ℂfld Σg 𝑇))} ↔ (((ℂfld Σg (𝑇 ∘f · 𝑋)) / (ℂfld Σg 𝑇)) ∈ 𝐷 ∧ (𝐹‘((ℂfld Σg (𝑇 ∘f · 𝑋)) / (ℂfld Σg 𝑇))) ≤ ((ℂfld Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))) / (ℂfld Σg 𝑇))))
306	302, 305	sylib 217	1 ⊢ (𝜑 → (((ℂfld Σg (𝑇 ∘f · 𝑋)) / (ℂfld Σg 𝑇)) ∈ 𝐷 ∧ (𝐹‘((ℂfld Σg (𝑇 ∘f · 𝑋)) / (ℂfld Σg 𝑇))) ≤ ((ℂfld Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))) / (ℂfld Σg 𝑇))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068  Vcvv 3432   ∖ cdif 3884   ∪ cun 3885   ⊆ wss 3887  ∅c0 4256  {csn 4561   class class class wbr 5074   ↦ cmpt 5157   ↾ cres 5591   ∘ ccom 5593   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ∘_f cof 7531   supp csupp 7977  Fincfn 8733  ℂcc 10869  ℝcr 10870  0cc0 10871  1c1 10872   + caddc 10874   · cmul 10876  +∞cpnf 11006   < clt 11009   ≤ cle 11010   − cmin 11205   / cdiv 11632  [,)cico 13081  [,]cicc 13082  Σcsu 15397   Σ_g cgsu 17151  Mndcmnd 18385  SubMndcsubmnd 18429  CMndccmn 19386  Ringcrg 19783  ℂ_fldccnfld 20597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-inf2 9399  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949  ax-addf 10950  ax-mulf 10951

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-of 7533  df-om 7713  df-1st 7831  df-2nd 7832  df-supp 7978  df-tpos 8042  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-fsupp 9129  df-sup 9201  df-oi 9269  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12438  df-uz 12583  df-rp 12731  df-ico 13085  df-icc 13086  df-fz 13240  df-fzo 13383  df-seq 13722  df-exp 13783  df-hash 14045  df-cj 14810  df-re 14811  df-im 14812  df-sqrt 14946  df-abs 14947  df-clim 15197  df-sum 15398  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-starv 16977  df-tset 16981  df-ple 16982  df-ds 16984  df-unif 16985  df-0g 17152  df-gsum 17153  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431  df-grp 18580  df-minusg 18581  df-mulg 18701  df-subg 18752  df-cntz 18923  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-ring 19785  df-cring 19786  df-oppr 19862  df-dvdsr 19883  df-unit 19884  df-invr 19914  df-dvr 19925  df-drng 19993  df-subrg 20022  df-cnfld 20598  df-refld 20810

This theorem is referenced by:  amgmlem  26139  amgmwlem  46506