Theorem jensen 25726

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 6505	. . . . . . . 8 ⊢ (𝜑 → 𝑇 Fn 𝐴)
4		fnresdm 6455	. . . . . . . 8 ⊢ (𝑇 Fn 𝐴 → (𝑇 ↾ 𝐴) = 𝑇)
5	3, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑇 ↾ 𝐴) = 𝑇)
6	5	oveq2d 7186	. . . . . 6 ⊢ (𝜑 → (ℂfld Σg (𝑇 ↾ 𝐴)) = (ℂfld Σg 𝑇))
7	1, 6	breqtrrd 5058	. . . . 5 ⊢ (𝜑 → 0 < (ℂfld Σg (𝑇 ↾ 𝐴)))
8		ssid 3899	. . . . 5 ⊢ 𝐴 ⊆ 𝐴
9	7, 8	jctil 523	. . . 4 ⊢ (𝜑 → (𝐴 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝐴))))
10		jensen.4	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin)
11		sseq1 3902	. . . . . . . . 9 ⊢ (𝑎 = ∅ → (𝑎 ⊆ 𝐴 ↔ ∅ ⊆ 𝐴))
12		reseq2 5820	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∅ → (𝑇 ↾ 𝑎) = (𝑇 ↾ ∅))
13		res0 5829	. . . . . . . . . . . . 13 ⊢ (𝑇 ↾ ∅) = ∅
14	12, 13	eqtrdi 2789	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → (𝑇 ↾ 𝑎) = ∅)
15	14	oveq2d 7186	. . . . . . . . . . 11 ⊢ (𝑎 = ∅ → (ℂfld Σg (𝑇 ↾ 𝑎)) = (ℂfld Σg ∅))
16		cnfld0 20241	. . . . . . . . . . . 12 ⊢ 0 = (0g‘ℂfld)
17	16	gsum0 18010	. . . . . . . . . . 11 ⊢ (ℂfld Σg ∅) = 0
18	15, 17	eqtrdi 2789	. . . . . . . . . 10 ⊢ (𝑎 = ∅ → (ℂfld Σg (𝑇 ↾ 𝑎)) = 0)
19	18	breq2d 5042	. . . . . . . . 9 ⊢ (𝑎 = ∅ → (0 < (ℂfld Σg (𝑇 ↾ 𝑎)) ↔ 0 < 0))
20	11, 19	anbi12d 634	. . . . . . . 8 ⊢ (𝑎 = ∅ → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑎))) ↔ (∅ ⊆ 𝐴 ∧ 0 < 0)))
21		reseq2 5820	. . . . . . . . . . 11 ⊢ (𝑎 = ∅ → ((𝑇 ∘f · 𝑋) ↾ 𝑎) = ((𝑇 ∘f · 𝑋) ↾ ∅))
22	21	oveq2d 7186	. . . . . . . . . 10 ⊢ (𝑎 = ∅ → (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ ∅)))
23	22, 18	oveq12d 7188	. . . . . . . . 9 ⊢ (𝑎 = ∅ → ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) = ((ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ ∅)) / 0))
24		reseq2 5820	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∅ → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎) = ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅))
25	24	oveq2d 7186	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)))
26	25, 18	oveq12d 7188	. . . . . . . . . . 11 ⊢ (𝑎 = ∅ → ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) / (ℂfld Σg (𝑇 ↾ 𝑎))) = ((ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ ∅)) / 0))
27	26	breq2d 5042	. . . . . . . . . 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 3381	. . . . . . . . 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 2827	. . . . . . . 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 348	. . . . . . 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 344	. . . . . 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 3902	. . . . . . . . 9 ⊢ (𝑎 = 𝑘 → (𝑎 ⊆ 𝐴 ↔ 𝑘 ⊆ 𝐴))
33		reseq2 5820	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑘 → (𝑇 ↾ 𝑎) = (𝑇 ↾ 𝑘))
34	33	oveq2d 7186	. . . . . . . . . 10 ⊢ (𝑎 = 𝑘 → (ℂfld Σg (𝑇 ↾ 𝑎)) = (ℂfld Σg (𝑇 ↾ 𝑘)))
35	34	breq2d 5042	. . . . . . . . 9 ⊢ (𝑎 = 𝑘 → (0 < (ℂfld Σg (𝑇 ↾ 𝑎)) ↔ 0 < (ℂfld Σg (𝑇 ↾ 𝑘))))
36	32, 35	anbi12d 634	. . . . . . . 8 ⊢ (𝑎 = 𝑘 → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑎))) ↔ (𝑘 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑘)))))
37		reseq2 5820	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑘 → ((𝑇 ∘f · 𝑋) ↾ 𝑎) = ((𝑇 ∘f · 𝑋) ↾ 𝑘))
38	37	oveq2d 7186	. . . . . . . . . 10 ⊢ (𝑎 = 𝑘 → (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑘)))
39	38, 34	oveq12d 7188	. . . . . . . . 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 5820	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑘 → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎) = ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘))
41	40	oveq2d 7186	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑘 → (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑘)))
42	41, 34	oveq12d 7188	. . . . . . . . . . 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 5042	. . . . . . . . . 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 3381	. . . . . . . . 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 2827	. . . . . . . 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 348	. . . . . . 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 344	. . . . . 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 3902	. . . . . . . . 9 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (𝑎 ⊆ 𝐴 ↔ (𝑘 ∪ {𝑐}) ⊆ 𝐴))
49		reseq2 5820	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (𝑇 ↾ 𝑎) = (𝑇 ↾ (𝑘 ∪ {𝑐})))
50	49	oveq2d 7186	. . . . . . . . . 10 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld Σg (𝑇 ↾ 𝑎)) = (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))
51	50	breq2d 5042	. . . . . . . . 9 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (0 < (ℂfld Σg (𝑇 ↾ 𝑎)) ↔ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐})))))
52	48, 51	anbi12d 634	. . . . . . . 8 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑎))) ↔ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))))
53		reseq2 5820	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑇 ∘f · 𝑋) ↾ 𝑎) = ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐})))
54	53	oveq2d 7186	. . . . . . . . . 10 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐}))))
55	54, 50	oveq12d 7188	. . . . . . . . 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 5820	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎) = ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐})))
57	56	oveq2d 7186	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑘 ∪ {𝑐}) → (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ (𝑘 ∪ {𝑐}))))
58	57, 50	oveq12d 7188	. . . . . . . . . . 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 5042	. . . . . . . . . 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 3381	. . . . . . . . 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 2827	. . . . . . . 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 348	. . . . . . 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 344	. . . . . 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 3902	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (𝑎 ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
65		reseq2 5820	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑇 ↾ 𝑎) = (𝑇 ↾ 𝐴))
66	65	oveq2d 7186	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (ℂfld Σg (𝑇 ↾ 𝑎)) = (ℂfld Σg (𝑇 ↾ 𝐴)))
67	66	breq2d 5042	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (0 < (ℂfld Σg (𝑇 ↾ 𝑎)) ↔ 0 < (ℂfld Σg (𝑇 ↾ 𝐴))))
68	64, 67	anbi12d 634	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → ((𝑎 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝑎))) ↔ (𝐴 ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ 𝐴)))))
69		reseq2 5820	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → ((𝑇 ∘f · 𝑋) ↾ 𝑎) = ((𝑇 ∘f · 𝑋) ↾ 𝐴))
70	69	oveq2d 7186	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)))
71	70, 66	oveq12d 7188	. . . . . . . . 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 5820	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐴 → ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎) = ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴))
73	72	oveq2d 7186	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐴 → (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝑎)) = (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)))
74	73, 66	oveq12d 7188	. . . . . . . . . . 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 5042	. . . . . . . . . 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 3381	. . . . . . . . 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 2827	. . . . . . . 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 348	. . . . . . 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 344	. . . . . 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 10721	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
81	80	ltnri 10827	. . . . . . . . 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 485	. . . . . . 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 454	. . . . . . . . . . . 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 771	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑘 ∪ {𝑐}) ⊆ 𝐴)
87	86	unssad 4077	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑘 ⊆ 𝐴)
88		simpr 488	. . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . . 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 776	. . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . 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 20239	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ℂfld ∈ Ring
108		ringcmn 19453	. . . . . . . . . . . . . . . . . . . . . . 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 8819	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → 𝑘 ∈ Fin)
112		rege0subm 20273	. . . . . . . . . . . . . . . . . . . . . . 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 6545	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑇 ↾ 𝑘):𝑘⟶(0[,)+∞))
116		c0ex 10713	. . . . . . . . . . . . . . . . . . . . . . . 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 8916	. . . . . . . . . . . . . . . . . . . . . 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 19158	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (ℂfld Σg (𝑇 ↾ 𝑘)) ∈ (0[,)+∞))
120		elrege0 12928	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((ℂfld Σg (𝑇 ↾ 𝑘)) ∈ (0[,)+∞) ↔ ((ℂfld Σg (𝑇 ↾ 𝑘)) ∈ ℝ ∧ 0 ≤ (ℂfld Σg (𝑇 ↾ 𝑘))))
121	120	simplbi 501	. . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . . . 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 12511	. . . . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . . . 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 6674	. . . . . . . . . . . . . . . . . . . . . 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 5040	. . . . . . . . . . . . . . . . . . . . 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 3588	. . . . . . . . . . . . . . . . . . . 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 221	. . . . . . . . . . . . . . . . . . 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 498	. . . . . . . . . . . . . . . . . 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 499	. . . . . . . . . . . . . . . . . 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 25725	. . . . . . . . . . . . . . . . 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 6674	. . . . . . . . . . . . . . . . . . 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 5040	. . . . . . . . . . . . . . . . . 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 3588	. . . . . . . . . . . . . . . . 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 237	. . . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . . 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 20221	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ℂ = (Base‘ℂfld)
141		ringmnd 19426	. . . . . . . . . . . . . . . . . . . . . 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 8819	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) → (𝑘 ∪ {𝑐}) ∈ Fin)
144	143	adantr 484	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝑘 ∪ {𝑐}) ∈ Fin)
145		ssun2 4063	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ {𝑐} ⊆ (𝑘 ∪ {𝑐})
146		vsnid 4553	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ 𝑐 ∈ {𝑐}
147	145, 146	sselii 3874	. . . . . . . . . . . . . . . . . . . . . 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 10700	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑥 · 𝑦) ∈ ℝ)
150	149	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ)) → (𝑥 · 𝑦) ∈ ℝ)
151		rge0ssre 12930	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (0[,)+∞) ⊆ ℝ
152		fss 6521	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑇:𝐴⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℝ) → 𝑇:𝐴⟶ℝ)
153	2, 151, 152	sylancl 589	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑇:𝐴⟶ℝ)
154	98, 89	fssd 6522	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑋:𝐴⟶ℝ)
155		inidm 4109	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝐴 ∩ 𝐴) = 𝐴
156	150, 153, 154, 10, 10, 155	off 7442	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑇 ∘f · 𝑋):𝐴⟶ℝ)
157		ax-resscn 10672	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ℝ ⊆ ℂ
158		fss 6521	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑇 ∘f · 𝑋):𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → (𝑇 ∘f · 𝑋):𝐴⟶ℂ)
159	156, 157, 158	sylancl 589	. . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . 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 6545	. . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝐴 ∈ Fin)
165	163, 164	fexd 7000	. . . . . . . . . . . . . . . . . . . . . . . 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 7000	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑋 ∈ V)
168		offres 7709	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑇 ∈ V ∧ 𝑋 ∈ V) → ((𝑇 ∘f · 𝑋) ↾ (𝑘 ∪ {𝑐})) = ((𝑇 ↾ (𝑘 ∪ {𝑐})) ∘f · (𝑋 ↾ (𝑘 ∪ {𝑐}))))
169	165, 167, 168	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . 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 7185	. . . . . . . . . . . . . . . . . . . . . 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 3886	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (0[,)+∞) ⊆ ℂ
172		fss 6521	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑇:𝐴⟶(0[,)+∞) ∧ (0[,)+∞) ⊆ ℂ) → 𝑇:𝐴⟶ℂ)
173	163, 171, 172	sylancl 589	. . . . . . . . . . . . . . . . . . . . . . . . 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 6545	. . . . . . . . . . . . . . . . . . . . . . . 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 4017	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) → 𝑥 ∈ (𝑘 ∪ {𝑐}))
176	175	adantl 485	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6694	. . . . . . . . . . . . . . . . . . . . . . . . 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 4370	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) = (𝑘 ∖ {𝑐})
179		difss 4022	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑘 ∖ {𝑐}) ⊆ 𝑘
180	178, 179	eqsstri 3911	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) ⊆ 𝑘
181	180	sseli 3873	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 ∈ ((𝑘 ∪ {𝑐}) ∖ {𝑐}) → 𝑥 ∈ 𝑘)
182		simpr 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑘 ⊆ 𝐴)
184	163, 183	feqresmpt 6738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7186	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑘 ∈ Fin)
187	183	sselda 3877	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6861	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 594	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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	sseldi 3875	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 20284	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12928	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑇‘𝑥) ∈ (0[,)+∞) ↔ ((𝑇‘𝑥) ∈ ℝ ∧ 0 ≤ (𝑇‘𝑥)))
194	189, 193	sylib 221	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 498	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 499	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 15246	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 3121	. . . . . . . . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . . . . . . . . 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 7889	. . . . . . . . . . . . . . . . . . . . . . 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 10896	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ ℂ → (0 · 𝑥) = 0)
204	203	adantl 485	. . . . . . . . . . . . . . . . . . . . . . 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 3889	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝐷 ⊆ ℂ)
207	166, 206	fssd 6522	. . . . . . . . . . . . . . . . . . . . . . . 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 6545	. . . . . . . . . . . . . . . . . . . . . . 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 7894	. . . . . . . . . . . . . . . . . . . . . 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 3915	. . . . . . . . . . . . . . . . . . . . 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 19201	. . . . . . . . . . . . . . . . . . . 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 6694	. . . . . . . . . . . . . . . . . . . 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 6505	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑇 Fn 𝐴)
215	166	ffnd 6505	. . . . . . . . . . . . . . . . . . . . 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 3878	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 𝑐 ∈ 𝐴)
217		fnfvof 7441	. . . . . . . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . . . . . 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 19201	. . . . . . . . . . . . . . . . . . . 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 6694	. . . . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . . . 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 7188	. . . . . . . . . . . . . . . . . 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 6862	. . . . . . . . . . . . . . . . . . 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 6862	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝑇‘𝑐) ∈ ℂ)
226		simplrr 778	. . . . . . . . . . . . . . . . . . . . 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 5056	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → 0 < (𝑇‘𝑐))
228	227	gt0ne0d 11282	. . . . . . . . . . . . . . . . . . 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 11499	. . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . 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 6862	. . . . . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . . . . 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	ffvelrnd 6862	. . . . . . . . . . . . . . . . . 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 11284	. . . . . . . . . . . . . . . . 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 6678	. . . . . . . . . . . . . . . . 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 6528	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐹:𝐷⟶ℝ ∧ 𝑋:𝐴⟶𝐷) → (𝐹 ∘ 𝑋):𝐴⟶ℝ)
238	91, 98, 237	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (𝐹 ∘ 𝑋):𝐴⟶ℝ)
239	150, 153, 238, 10, 10, 155	off 7442	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → (𝑇 ∘f · (𝐹 ∘ 𝑋)):𝐴⟶ℝ)
240		fss 6521	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑇 ∘f · (𝐹 ∘ 𝑋)):𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → (𝑇 ∘f · (𝐹 ∘ 𝑋)):𝐴⟶ℂ)
241	239, 157, 240	sylancl 589	. . . . . . . . . . . . . . . . . . . . . . 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 6545	. . . . . . . . . . . . . . . . . . . . 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 7000	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝜑 ∧ ¬ 𝑐 ∈ 𝑘) ∧ ((𝑘 ∪ {𝑐}) ⊆ 𝐴 ∧ 0 < (ℂfld Σg (𝑇 ↾ (𝑘 ∪ {𝑐}))))) ∧ 0 = (ℂfld Σg (𝑇 ↾ 𝑘))) → (𝐹 ∘ 𝑋) ∈ V)
246		offres 7709	. . . . . . . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . . . . . . 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 7185	. . . . . . . . . . . . . . . . . . . . . 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 6521	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝐹 ∘ 𝑋):𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → (𝐹 ∘ 𝑋):𝐴⟶ℂ)
250	244, 157, 249	sylancl 589	. . . . . . . . . . . . . . . . . . . . . . . 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 6545	. . . . . . . . . . . . . . . . . . . . . . 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 7894	. . . . . . . . . . . . . . . . . . . . . 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 3915	. . . . . . . . . . . . . . . . . . . . 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 19201	. . . . . . . . . . . . . . . . . . . 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 6694	. . . . . . . . . . . . . . . . . . . 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 6505	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐹 Fn 𝐷)
257		fnfco 6543	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐹 Fn 𝐷 ∧ 𝑋:𝐴⟶𝐷) → (𝐹 ∘ 𝑋) Fn 𝐴)
258	256, 98, 257	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . 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 7441	. . . . . . . . . . . . . . . . . . . . . 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 6767	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑋:𝐴⟶𝐷 ∧ 𝑐 ∈ 𝐴) → ((𝐹 ∘ 𝑋)‘𝑐) = (𝐹‘(𝑋‘𝑐)))
263	166, 216, 262	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . 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 7186	. . . . . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . . . . 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 2777	. . . . . . . . . . . . . . . . . . 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 7188	. . . . . . . . . . . . . . . . . 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 10747	. . . . . . . . . . . . . . . . . . 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 11499	. . . . . . . . . . . . . . . . . 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 2773	. . . . . . . . . . . . . . . . 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 5062	. . . . . . . . . . . . . . . 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 3590	. . . . . . . . . . . . . . 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 500	. . . . . . . . . . . . . . . 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 10805	. . . . . . . . . . . . . . . 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 590	. . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . 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 958	. . . . . . . . . . . . 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 245	. . . . . . . . . . 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 416	. . . . . . . . . 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 417	. . . . . . . 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 485	. . . . . . 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 8764	. . . . 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 6505	. . . . . 6 ⊢ (𝜑 → (𝑇 ∘f · 𝑋) Fn 𝐴)
291		fnresdm 6455	. . . . . 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 7186	. . . 4 ⊢ (𝜑 → (ℂfld Σg ((𝑇 ∘f · 𝑋) ↾ 𝐴)) = (ℂfld Σg (𝑇 ∘f · 𝑋)))
294	293, 6	oveq12d 7188	. . 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 7437	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ∘f · (𝐹 ∘ 𝑋)) Fn 𝐴)
296		fnresdm 6455	. . . . . . . 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 7186	. . . . . 6 ⊢ (𝜑 → (ℂfld Σg ((𝑇 ∘f · (𝐹 ∘ 𝑋)) ↾ 𝐴)) = (ℂfld Σg (𝑇 ∘f · (𝐹 ∘ 𝑋))))
299	298, 6	oveq12d 7188	. . . . 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 5042	. . . 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 3381	. . 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 2847	. 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 6674	. . . 4 ⊢ (𝑤 = ((ℂfld Σg (𝑇 ∘f · 𝑋)) / (ℂfld Σg 𝑇)) → (𝐹‘𝑤) = (𝐹‘((ℂfld Σg (𝑇 ∘f · 𝑋)) / (ℂfld Σg 𝑇))))
304	303	breq1d 5040	. . 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 3588	. 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 221	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 209   ∧ wa 399   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2114  ∀wral 3053  {crab 3057  Vcvv 3398   ∖ cdif 3840   ∪ cun 3841   ⊆ wss 3843  ∅c0 4211  {csn 4516   class class class wbr 5030   ↦ cmpt 5110   ↾ cres 5527   ∘ ccom 5529   Fn wfn 6334  ⟶wf 6335  ‘cfv 6339  (class class class)co 7170   ∘_f cof 7423   supp csupp 7856  Fincfn 8555  ℂcc 10613  ℝcr 10614  0cc0 10615  1c1 10616   + caddc 10618   · cmul 10620  +∞cpnf 10750   < clt 10753   ≤ cle 10754   − cmin 10948   / cdiv 11375  [,)cico 12823  [,]cicc 12824  Σcsu 15135   Σ_g cgsu 16817  Mndcmnd 18027  SubMndcsubmnd 18071  CMndccmn 19024  Ringcrg 19416  ℂ_fldccnfld 20217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-inf2 9177  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692  ax-pre-sup 10693  ax-addf 10694  ax-mulf 10695

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-int 4837  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-of 7425  df-om 7600  df-1st 7714  df-2nd 7715  df-supp 7857  df-tpos 7921  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-er 8320  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-fsupp 8907  df-sup 8979  df-oi 9047  df-card 9441  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-div 11376  df-nn 11717  df-2 11779  df-3 11780  df-4 11781  df-5 11782  df-6 11783  df-7 11784  df-8 11785  df-9 11786  df-n0 11977  df-z 12063  df-dec 12180  df-uz 12325  df-rp 12473  df-ico 12827  df-icc 12828  df-fz 12982  df-fzo 13125  df-seq 13461  df-exp 13522  df-hash 13783  df-cj 14548  df-re 14549  df-im 14550  df-sqrt 14684  df-abs 14685  df-clim 14935  df-sum 15136  df-struct 16588  df-ndx 16589  df-slot 16590  df-base 16592  df-sets 16593  df-ress 16594  df-plusg 16681  df-mulr 16682  df-starv 16683  df-tset 16687  df-ple 16688  df-ds 16690  df-unif 16691  df-0g 16818  df-gsum 16819  df-mre 16960  df-mrc 16961  df-acs 16963  df-mgm 17968  df-sgrp 18017  df-mnd 18028  df-submnd 18073  df-grp 18222  df-minusg 18223  df-mulg 18343  df-subg 18394  df-cntz 18565  df-cmn 19026  df-abl 19027  df-mgp 19359  df-ur 19371  df-ring 19418  df-cring 19419  df-oppr 19495  df-dvdsr 19513  df-unit 19514  df-invr 19544  df-dvr 19555  df-drng 19623  df-subrg 19652  df-cnfld 20218  df-refld 20421

This theorem is referenced by:  amgmlem  25727  amgmwlem  45959