Theorem jensen 26899

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 6689	. . . . . . . 8 ⊢ (𝜑 → 𝑇 Fn 𝐴)
4		fnresdm 6637	. . . . . . . 8 ⊢ (𝑇 Fn 𝐴 → (𝑇 ↾ 𝐴) = 𝑇)
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑇 ↾ 𝐴) = 𝑇)
6	5	oveq2d 7403	. . . . . 6 ⊢ (𝜑 → (ℂfld Σg (𝑇 ↾ 𝐴)) = (ℂfld Σg 𝑇))
7	1, 6	breqtrrd 5135	. . . . 5 ⊢ (𝜑 → 0 < (ℂfld Σg (𝑇 ↾ 𝐴)))
8		ssid 3969	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
9	7, 8	jctil 519	. . . 4 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝐴))))
10		jensen.4	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin)
11		sseq1 3972	. . . . . . . . 9 ⊢ (𝑎 = ∅ → (𝑎 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
12		reseq2 5945	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∅ → (𝑇 ↾ 𝑎) = (𝑇 ↾ ∅))
13		res0 5954	. . . . . . . . . . . . 13 ⊢ (𝑇 ↾ ∅) = ∅
14	12, 13	eqtrdi 2780	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → (𝑇 ↾ 𝑎) = ∅)
15	14	oveq2d 7403	. . . . . . . . . . 11 ⊢ (𝑎 = ∅ → (ℂfld Σg (𝑇 ↾ 𝑎)) = (ℂfld Σg ∅))
16		cnfld0 21304	. . . . . . . . . . . 12 ⊢ 0 = (0g‘ℂfld)
17	16	gsum0 18611	. . . . . . . . . . 11 ⊢ (ℂfld Σg ∅) = 0
18	15, 17	eqtrdi 2780	. . . . . . . . . 10 ⊢ (𝑎 = ∅ → (ℂfld Σg (𝑇 ↾ 𝑎)) = 0)
19	18	breq2d 5119	. . . . . . . . 9 ⊢ (𝑎 = ∅ → (0 < (ℂfld Σg (𝑇 ↾ 𝑎)) ↔ 0 < 0))
20	11, 19	anbi12d 632	. . . . . . . 8 ⊢ (𝑎 = ∅ → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑎))) ↔ (∅ ⊆ 𝐴 ∧ 0 < 0)))
21		reseq2 5945	. . . . . . . . . . 11 ⊢ (𝑎 = ∅ → ((𝑇 ∘f · 𝑋) ↾ 𝑎) = ((𝑇 ∘f · 𝑋) ↾ ∅))
22	21	oveq2d 7403	. . . . . . . . . 10 ⊢ (𝑎 = ∅ → (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ ∅)))
23	22, 18	oveq12d 7405	. . . . . . . . 9 ⊢ (𝑎 = ∅ → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) = ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ ∅)) / 0))
24		reseq2 5945	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∅ → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎) = ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅))
25	24	oveq2d 7403	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)))
26	25, 18	oveq12d 7405	. . . . . . . . . . 11 ⊢ (𝑎 = ∅ → ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) = ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) / 0))
27	26	breq2d 5119	. . . . . . . . . 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 3413	. . . . . . . . 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 2822	. . . . . . . 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 344	. . . . . . 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 340	. . . . . 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 3972	. . . . . . . . 9 ⊢ (𝑎 = 𝑘 → (𝑎 ⊆ 𝐴 ↔ 𝑘 ⊆ 𝐴))
33		reseq2 5945	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑘 → (𝑇 ↾ 𝑎) = (𝑇 ↾ 𝑘))
34	33	oveq2d 7403	. . . . . . . . . 10 ⊢ (𝑎 = 𝑘 → (ℂfld Σg (𝑇 ↾ 𝑎)) = (ℂfld Σg (𝑇 ↾ 𝑘)))
35	34	breq2d 5119	. . . . . . . . 9 ⊢ (𝑎 = 𝑘 → (0 < (ℂfld Σg (𝑇 ↾ 𝑎)) ↔ 0 < (ℂfld Σg (𝑇 ↾ 𝑘))))
36	32, 35	anbi12d 632	. . . . . . . 8 ⊢ (𝑎 = 𝑘 → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑎))) ↔ (𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑘)))))
37		reseq2 5945	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑘 → ((𝑇 ∘f · 𝑋) ↾ 𝑎) = ((𝑇 ∘f · 𝑋) ↾ 𝑘))
38	37	oveq2d 7403	. . . . . . . . . 10 ⊢ (𝑎 = 𝑘 → (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)))
39	38, 34	oveq12d 7405	. . . . . . . . 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 5945	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑘 → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎) = ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘))
41	40	oveq2d 7403	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑘 → (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)))
42	41, 34	oveq12d 7405	. . . . . . . . . . 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 5119	. . . . . . . . . 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 3413	. . . . . . . . 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 2822	. . . . . . . 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 344	. . . . . . 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 340	. . . . . 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 3972	. . . . . . . . 9 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (𝑎 ⊆ 𝐴 ↔ (𝑘 ∪ {𝑐}) ⊆ 𝐴))
49		reseq2 5945	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (𝑇 ↾ 𝑎) = (𝑇 ↾ (𝑘 ∪ {𝑐})))
50	49	oveq2d 7403	. . . . . . . . . 10 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld Σg (𝑇 ↾ 𝑎)) = (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))
51	50	breq2d 5119	. . . . . . . . 9 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (0 < (ℂfld Σg (𝑇 ↾ 𝑎)) ↔ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))))
52	48, 51	anbi12d 632	. . . . . . . 8 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑎))) ↔ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))))
53		reseq2 5945	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑇 ∘f · 𝑋) ↾ 𝑎) = ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐})))
54	53	oveq2d 7403	. . . . . . . . . 10 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))))
55	54, 50	oveq12d 7405	. . . . . . . . 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 5945	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎) = ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐})))
57	56	oveq2d 7403	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))))
58	57, 50	oveq12d 7405	. . . . . . . . . . 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 5119	. . . . . . . . . 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 3413	. . . . . . . . 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 2822	. . . . . . . 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 344	. . . . . . 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 340	. . . . . 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 3972	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
65		reseq2 5945	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑇 ↾ 𝑎) = (𝑇 ↾ 𝐴))
66	65	oveq2d 7403	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (ℂfld Σg (𝑇 ↾ 𝑎)) = (ℂfld Σg (𝑇 ↾ 𝐴)))
67	66	breq2d 5119	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (0 < (ℂfld Σg (𝑇 ↾ 𝑎)) ↔ 0 < (ℂfld Σg (𝑇 ↾ 𝐴))))
68	64, 67	anbi12d 632	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑎))) ↔ (𝐴 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝐴)))))
69		reseq2 5945	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → ((𝑇 ∘f · 𝑋) ↾ 𝑎) = ((𝑇 ∘f · 𝑋) ↾ 𝐴))
70	69	oveq2d 7403	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)))
71	70, 66	oveq12d 7405	. . . . . . . . 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 5945	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎) = ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴))
73	72	oveq2d 7403	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)))
74	73, 66	oveq12d 7405	. . . . . . . . . . 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 5119	. . . . . . . . . 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 3413	. . . . . . . . 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 2822	. . . . . . . 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 344	. . . . . . 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 340	. . . . . 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 11176	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
81	80	ltnri 11283	. . . . . . . . 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 481	. . . . . . 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 450	. . . . . . . . . . . 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 770	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑘 ∪ {𝑐}) ⊆ 𝐴)
87	86	unssad 4156	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑘 ⊆ 𝐴)
88		simpr 484	. . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . . . . . 18 ⊢ (ℂfld Σg (𝑇 ↾ 𝑘)) = (ℂfld Σg (𝑇 ↾ 𝑘))
106		eqid 2729	. . . . . . . . . . . . . . . . . 18 ⊢ (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))) = (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))
107		cnring 21302	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℂfld ∈ Ring
108		ringcmn 20191	. . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝐴 ∈ Fin)
111	110, 87	ssfid 9212	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑘 ∈ Fin)
112		rege0subm 21340	. . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑇:𝐴⟶(0[,)+∞))
115	114, 87	fssresd 6727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑇 ↾ 𝑘):𝑘⟶(0[,)+∞))
116		c0ex 11168	. . . . . . . . . . . . . . . . . . . . . . . 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 9326	. . . . . . . . . . . . . . . . . . . . . 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 19849	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (ℂfld Σg (𝑇 ↾ 𝑘)) ∈ (0[,)+∞))
120		elrege0 13415	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℂfld Σg (𝑇 ↾ 𝑘)) ∈ (0[,)+∞) ↔ ((ℂfld Σg (𝑇 ↾ 𝑘)) ∈ ℝ ∧ 0 ≤ (ℂfld Σg (𝑇 ↾ 𝑘))))
121	120	simplbi 497	. . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . . . . . 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 12992	. . . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . . . . . 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 6858	. . . . . . . . . . . . . . . . . . . . . 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 5117	. . . . . . . . . . . . . . . . . . . . 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 3659	. . . . . . . . . . . . . . . . . . . 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 218	. . . . . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . . . . 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 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)))})) → (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 26898	. . . . . . . . . . . . . . . . 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 6858	. . . . . . . . . . . . . . . . . . 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 5117	. . . . . . . . . . . . . . . . . 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 3659	. . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . 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 456	. . . . . . . . . . . . . . 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 21268	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℂ = (Base‘ℂfld)
141		ringmnd 20152	. . . . . . . . . . . . . . . . . . . . . 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 9212	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑘 ∪ {𝑐}) ∈ Fin)
144	143	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝑘 ∪ {𝑐}) ∈ Fin)
145		ssun2 4142	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑐} ⊆ (𝑘 ∪ {𝑐})
146		vsnid 4627	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑐 ∈ {𝑐}
147	145, 146	sselii 3943	. . . . . . . . . . . . . . . . . . . . . 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 11153	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ)
150	149	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
151		rge0ssre 13417	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (0[,)+∞) ⊆ ℝ
152		fss 6704	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑇:𝐴⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝑇:𝐴⟶ℝ)
153	2, 151, 152	sylancl 586	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑇:𝐴⟶ℝ)
154	98, 89	fssd 6705	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑋:𝐴⟶ℝ)
155		inidm 4190	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∩ 𝐴) = 𝐴
156	150, 153, 154, 10, 10, 155	off 7671	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑇 ∘f · 𝑋):𝐴⟶ℝ)
157		ax-resscn 11125	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℝ ⊆ ℂ
158		fss 6704	. . . . . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . 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 6727	. . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑇:𝐴⟶(0[,)+∞))
164	110	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝐴 ∈ Fin)
165	163, 164	fexd 7201	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑇 ∈ V)
166	98	ad3antrrr 730	. . . . . . . . . . . . . . . . . . . . . . . . 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 7201	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑋 ∈ V)
168		offres 7962	. . . . . . . . . . . . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . . . . . . . . . . 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 3956	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (0[,)+∞) ⊆ ℂ
172		fss 6704	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6727	. . . . . . . . . . . . . . . . . . . . . . . 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 4094	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) → 𝑥 ∈ (𝑘 ∪ {𝑐}))
176	175	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6878	. . . . . . . . . . . . . . . . . . . . . . . . 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 4444	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) = (𝑘 ∖ {𝑐})
179		difss 4099	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∖ {𝑐}) ⊆ 𝑘
180	178, 179	eqsstri 3993	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) ⊆ 𝑘
181	180	sseli 3942	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) → 𝑥 ∈ 𝑘)
182		simpr 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑘 ⊆ 𝐴)
184	163, 183	feqresmpt 6930	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7403	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑘 ∈ Fin)
187	183	sselda 3946	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) ∧ 𝑥 ∈ 𝑘) → 𝑥 ∈ 𝐴)
188	163	ffvelcdmda 7056	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3944	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 21351	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2769	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13415	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑇‘𝑥) ∈ (0[,)+∞) ↔ ((𝑇‘𝑥) ∈ ℝ ∧ 0 ≤ (𝑇‘𝑥)))
194	189, 193	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 15764	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 3229	. . . . . . . . . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . . . . . . 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 8173	. . . . . . . . . . . . . . . . . . . . . . 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 11352	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ ℂ → (0 · 𝑥) = 0)
204	203	adantl 481	. . . . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝐷 ⊆ ℝ)
206	205, 157	sstrdi 3959	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝐷 ⊆ ℂ)
207	166, 206	fssd 6705	. . . . . . . . . . . . . . . . . . . . . . . 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 6727	. . . . . . . . . . . . . . . . . . . . . . 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 8178	. . . . . . . . . . . . . . . . . . . . . 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 3981	. . . . . . . . . . . . . . . . . . . . 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 19892	. . . . . . . . . . . . . . . . . . . 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 6878	. . . . . . . . . . . . . . . . . . . 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 6689	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑇 Fn 𝐴)
215	166	ffnd 6689	. . . . . . . . . . . . . . . . . . . . 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 3947	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑐 ∈ 𝐴)
217		fnfvof 7670	. . . . . . . . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . . . . 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 19892	. . . . . . . . . . . . . . . . . . . 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 6878	. . . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . 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 7405	. . . . . . . . . . . . . . . . . 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	ffvelcdmd 7057	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝑋‘𝑐) ∈ ℂ)
225	173, 216	ffvelcdmd 7057	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝑇‘𝑐) ∈ ℂ)
226		simplrr 777	. . . . . . . . . . . . . . . . . . . . 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 5133	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 0 < (𝑇‘𝑐))
228	227	gt0ne0d 11742	. . . . . . . . . . . . . . . . . . 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 11963	. . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . 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	ffvelcdmd 7057	. . . . . . . . . . . . . . . . 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 2828	. . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝐹:𝐷⟶ℝ)
234	233, 231	ffvelcdmd 7057	. . . . . . . . . . . . . . . . . 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 11744	. . . . . . . . . . . . . . . . 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 6862	. . . . . . . . . . . . . . . . 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 6712	. . . . . . . . . . . . . . . . . . . . . . . . . 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 7671	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑇 ∘f · (𝐹 ∘ 𝑋)):𝐴⟶ℝ)
240		fss 6704	. . . . . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . . 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 6727	. . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . . . . . 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 7201	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝐹 ∘ 𝑋) ∈ V)
246		offres 7962	. . . . . . . . . . . . . . . . . . . . . . . 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 7402	. . . . . . . . . . . . . . . . . . . . . 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 6704	. . . . . . . . . . . . . . . . . . . . . . . . 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 6727	. . . . . . . . . . . . . . . . . . . . . . 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 8178	. . . . . . . . . . . . . . . . . . . . . 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 3981	. . . . . . . . . . . . . . . . . . . . 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 19892	. . . . . . . . . . . . . . . . . . . 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 6878	. . . . . . . . . . . . . . . . . . . 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 6689	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐹 Fn 𝐷)
257		fnfco 6725	. . . . . . . . . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝐹 ∘ 𝑋) Fn 𝐴)
260		fnfvof 7670	. . . . . . . . . . . . . . . . . . . . . 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 838	. . . . . . . . . . . . . . . . . . . . 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 6960	. . . . . . . . . . . . . . . . . . . . . . 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 7403	. . . . . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . . . . 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 7405	. . . . . . . . . . . . . . . . . 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 11202	. . . . . . . . . . . . . . . . . . 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 11963	. . . . . . . . . . . . . . . . . 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 2764	. . . . . . . . . . . . . . . . 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 5139	. . . . . . . . . . . . . . . 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 3661	. . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . 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 11260	. . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . 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 960	. . . . . . . . . . . . 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	biimtrid 242	. . . . . . . . . . 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 412	. . . . . . . . . 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 413	. . . . . . . 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 481	. . . . . . 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 9129	. . . . 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 6689	. . . . . 6 ⊢ (𝜑 → (𝑇 ∘f · 𝑋) Fn 𝐴)
291		fnresdm 6637	. . . . . 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 7403	. . . 4 ⊢ (𝜑 → (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)) = (ℂfld Σg (𝑇 ∘f · 𝑋)))
294	293, 6	oveq12d 7405	. . 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 7666	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ∘f · (𝐹 ∘ 𝑋)) Fn 𝐴)
296		fnresdm 6637	. . . . . . . 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 7403	. . . . . 6 ⊢ (𝜑 → (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) = (ℂfld Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))))
299	298, 6	oveq12d 7405	. . . . 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 5119	. . . 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 3413	. . 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 2842	. 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 6858	. . . 4 ⊢ (𝑤 = ((ℂfld Σg (𝑇 ∘f · 𝑋)) / (ℂfld Σg 𝑇)) → (𝐹‘𝑤) = (𝐹‘((ℂfld Σg (𝑇 ∘f · 𝑋)) / (ℂfld Σg 𝑇))))
304	303	breq1d 5117	. . 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 3659	. 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 218	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 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3405  Vcvv 3447   ∖ cdif 3911   ∪ cun 3912   ⊆ wss 3914  ∅c0 4296  {csn 4589   class class class wbr 5107   ↦ cmpt 5188   ↾ cres 5640   ∘ ccom 5642   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ∘_f cof 7651   supp csupp 8139  Fincfn 8918  ℂcc 11066  ℝcr 11067  0cc0 11068  1c1 11069   + caddc 11071   · cmul 11073  +∞cpnf 11205   < clt 11208   ≤ cle 11209   − cmin 11405   / cdiv 11835  [,)cico 13308  [,]cicc 13309  Σcsu 15652   Σ_g cgsu 17403  Mndcmnd 18661  SubMndcsubmnd 18709  CMndccmn 19710  Ringcrg 20142  ℂ_fldccnfld 21264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-inf2 9594  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146  ax-addf 11147

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4872  df-int 4911  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-se 5592  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-isom 6520  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-of 7653  df-om 7843  df-1st 7968  df-2nd 7969  df-supp 8140  df-tpos 8205  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-1o 8434  df-2o 8435  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-fsupp 9313  df-sup 9393  df-oi 9463  df-card 9892  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-nn 12187  df-2 12249  df-3 12250  df-4 12251  df-5 12252  df-6 12253  df-7 12254  df-8 12255  df-9 12256  df-n0 12443  df-z 12530  df-dec 12650  df-uz 12794  df-rp 12952  df-ico 13312  df-icc 13313  df-fz 13469  df-fzo 13616  df-seq 13967  df-exp 14027  df-hash 14296  df-cj 15065  df-re 15066  df-im 15067  df-sqrt 15201  df-abs 15202  df-clim 15454  df-sum 15653  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-starv 17235  df-tset 17239  df-ple 17240  df-ds 17242  df-unif 17243  df-0g 17404  df-gsum 17405  df-mre 17547  df-mrc 17548  df-acs 17550  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-submnd 18711  df-grp 18868  df-minusg 18869  df-mulg 19000  df-subg 19055  df-cntz 19249  df-cmn 19712  df-abl 19713  df-mgp 20050  df-rng 20062  df-ur 20091  df-ring 20144  df-cring 20145  df-oppr 20246  df-dvdsr 20266  df-unit 20267  df-invr 20297  df-dvr 20310  df-subrng 20455  df-subrg 20479  df-drng 20640  df-cnfld 21265  df-refld 21514

This theorem is referenced by:  amgmlem  26900  amgmwlem  49791