Theorem sge0tsms 42711

Description: Σ^{^} applied to a nonnegative function (its meaningful domain) is the same as the infinite group sum (that's always convergent, in this case). (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
sge0tsms.g	⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞))
sge0tsms.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
sge0tsms.f	⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))

Assertion

Ref	Expression
sge0tsms	⊢ (𝜑 → (Σ^‘𝐹) ∈ (𝐺 tsums 𝐹))

Proof of Theorem sge0tsms

Dummy variables 𝑠 𝑡 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2821	. . . 4 ⊢ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )
2	1	a1i 11	. . 3 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ))
3		xrltso 12535	. . . . . 6 ⊢ < Or ℝ*
4	3	supex 8927	. . . . 5 ⊢ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ V)
6		elsng 4581	. . . 4 ⊢ (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ V → (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )} ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )))
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )} ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )))
8	2, 7	mpbird 259	. 2 ⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )})
9		sge0tsms.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
10	9	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
11		sge0tsms.f	. . . . . . 7 ⊢ (𝜑 → 𝐹:𝑋⟶(0[,]+∞))
12	11	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
13		simpr 487	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
14	10, 12, 13	sge0pnfval 42704	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = +∞)
15	11	ffnd 6515	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn 𝑋)
16	15	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹 Fn 𝑋)
17		fvelrnb 6726	. . . . . . . 8 ⊢ (𝐹 Fn 𝑋 → (+∞ ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞))
18	16, 17	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (+∞ ∈ ran 𝐹 ↔ ∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞))
19	13, 18	mpbid 234	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞)
20		iccssxr 12820	. . . . . . . . . . . . . 14 ⊢ (0[,]+∞) ⊆ ℝ*
21		sge0tsms.g	. . . . . . . . . . . . . . 15 ⊢ 𝐺 = (ℝ*𝑠 ↾s (0[,]+∞))
22		simpr 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ (𝒫 𝑋 ∩ Fin))
23	11	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,]+∞))
24		elinel1 4172	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
25		elpwi 4548	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ 𝒫 𝑋 → 𝑥 ⊆ 𝑋)
26	24, 25	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ⊆ 𝑋)
27	26	adantl 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ⊆ 𝑋)
28		fssres 6544	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥 ⊆ 𝑋) → (𝐹 ↾ 𝑥):𝑥⟶(0[,]+∞))
29	23, 27, 28	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥):𝑥⟶(0[,]+∞))
30		elinel2 4173	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
31	30	adantl 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
32		0red 10644	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 0 ∈ ℝ)
33	29, 31, 32	fdmfifsupp 8843	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥) finSupp 0)
34	21, 22, 29, 33	gsumge0cl 42702	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ 𝑥)) ∈ (0[,]+∞))
35	20, 34	sseldi 3965	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ 𝑥)) ∈ ℝ*)
36	35	ralrimiva 3182	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹 ↾ 𝑥)) ∈ ℝ*)
37	36	3ad2ant1 1129	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹 ↾ 𝑥)) ∈ ℝ*)
38		eqid 2821	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥)))
39	38	rnmptss 6886	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹 ↾ 𝑥)) ∈ ℝ* → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) ⊆ ℝ*)
40	37, 39	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) ⊆ ℝ*)
41		snelpwi 5337	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝑋 → {𝑦} ∈ 𝒫 𝑋)
42		snfi 8594	. . . . . . . . . . . . . . 15 ⊢ {𝑦} ∈ Fin
43	42	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ 𝑋 → {𝑦} ∈ Fin)
44	41, 43	elind 4171	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑋 → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
45	44	3ad2ant2 1130	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
46	11	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → 𝐹:𝑋⟶(0[,]+∞))
47		snssi 4741	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ 𝑋 → {𝑦} ⊆ 𝑋)
48	47	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → {𝑦} ⊆ 𝑋)
49	46, 48	fssresd 6545	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹 ↾ {𝑦}):{𝑦}⟶(0[,]+∞))
50	49	feqmptd 6733	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹 ↾ {𝑦}) = (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)))
51		fvres 6689	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ {𝑦} → ((𝐹 ↾ {𝑦})‘𝑥) = (𝐹‘𝑥))
52	51	mpteq2ia 5157	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)) = (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥))
53	52	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)) = (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥)))
54	50, 53	eqtrd 2856	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹 ↾ {𝑦}) = (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥)))
55	54	oveq2d 7172	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐺 Σg (𝐹 ↾ {𝑦})) = (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥))))
56	55	3adant3 1128	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (𝐺 Σg (𝐹 ↾ {𝑦})) = (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥))))
57		xrge0cmn 20587	. . . . . . . . . . . . . . . . 17 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) ∈ CMnd
58	21, 57	eqeltri 2909	. . . . . . . . . . . . . . . 16 ⊢ 𝐺 ∈ CMnd
59		cmnmnd 18922	. . . . . . . . . . . . . . . 16 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
60	58, 59	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ 𝐺 ∈ Mnd
61	60	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → 𝐺 ∈ Mnd)
62		simp2 1133	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → 𝑦 ∈ 𝑋)
63	11	ffvelrnda 6851	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋) → (𝐹‘𝑦) ∈ (0[,]+∞))
64	63	3adant3 1128	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (𝐹‘𝑦) ∈ (0[,]+∞))
65		df-ss 3952	. . . . . . . . . . . . . . . . . 18 ⊢ ((0[,]+∞) ⊆ ℝ* ↔ ((0[,]+∞) ∩ ℝ*) = (0[,]+∞))
66	20, 65	mpbi 232	. . . . . . . . . . . . . . . . 17 ⊢ ((0[,]+∞) ∩ ℝ*) = (0[,]+∞)
67	66	eqcomi 2830	. . . . . . . . . . . . . . . 16 ⊢ (0[,]+∞) = ((0[,]+∞) ∩ ℝ*)
68		ovex 7189	. . . . . . . . . . . . . . . . 17 ⊢ (0[,]+∞) ∈ V
69		xrsbas 20561	. . . . . . . . . . . . . . . . . 18 ⊢ ℝ* = (Base‘ℝ*𝑠)
70	21, 69	ressbas 16554	. . . . . . . . . . . . . . . . 17 ⊢ ((0[,]+∞) ∈ V → ((0[,]+∞) ∩ ℝ*) = (Base‘𝐺))
71	68, 70	ax-mp 5	. . . . . . . . . . . . . . . 16 ⊢ ((0[,]+∞) ∩ ℝ*) = (Base‘𝐺)
72	67, 71	eqtri 2844	. . . . . . . . . . . . . . 15 ⊢ (0[,]+∞) = (Base‘𝐺)
73		fveq2 6670	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
74	72, 73	gsumsn 19074	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) ∈ (0[,]+∞)) → (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥))) = (𝐹‘𝑦))
75	61, 62, 64, 74	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹‘𝑥))) = (𝐹‘𝑦))
76		simp3 1134	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (𝐹‘𝑦) = +∞)
77	56, 75, 76	3eqtrrd 2861	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → +∞ = (𝐺 Σg (𝐹 ↾ {𝑦})))
78		reseq2 5848	. . . . . . . . . . . . . 14 ⊢ (𝑥 = {𝑦} → (𝐹 ↾ 𝑥) = (𝐹 ↾ {𝑦}))
79	78	oveq2d 7172	. . . . . . . . . . . . 13 ⊢ (𝑥 = {𝑦} → (𝐺 Σg (𝐹 ↾ 𝑥)) = (𝐺 Σg (𝐹 ↾ {𝑦})))
80	79	rspceeqv 3638	. . . . . . . . . . . 12 ⊢ (({𝑦} ∈ (𝒫 𝑋 ∩ Fin) ∧ +∞ = (𝐺 Σg (𝐹 ↾ {𝑦}))) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹 ↾ 𝑥)))
81	45, 77, 80	syl2anc 586	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹 ↾ 𝑥)))
82		pnfxr 10695	. . . . . . . . . . . . 13 ⊢ +∞ ∈ ℝ*
83	82	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → +∞ ∈ ℝ*)
84	38	elrnmpt 5828	. . . . . . . . . . . 12 ⊢ (+∞ ∈ ℝ* → (+∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹 ↾ 𝑥))))
85	83, 84	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → (+∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹 ↾ 𝑥))))
86	81, 85	mpbird 259	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))))
87		supxrpnf 12712	. . . . . . . . . 10 ⊢ ((ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))) ⊆ ℝ* ∧ +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥)))) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)
88	40, 86, 87	syl2anc 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑋 ∧ (𝐹‘𝑦) = +∞) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)
89	88	3exp 1115	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ 𝑋 → ((𝐹‘𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)))
90	89	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (𝑦 ∈ 𝑋 → ((𝐹‘𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)))
91	90	rexlimdv 3283	. . . . . 6 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (∃𝑦 ∈ 𝑋 (𝐹‘𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞))
92	19, 91	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) = +∞)
93	14, 92	eqtr4d 2859	. . . 4 ⊢ ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ))
94	9	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝑋 ∈ 𝑉)
95	11	adantr 483	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
96		simpr 487	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
97	95, 96	fge0iccico 42701	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
98	94, 97	sge0reval 42703	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ))
99	23, 27	feqresmpt 6734	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥) = (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)))
100	99	adantlr 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹 ↾ 𝑥) = (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)))
101	100	oveq2d 7172	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ 𝑥)) = (𝐺 Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
102	21	fveq2i 6673	. . . . . . . . . . 11 ⊢ (+g‘𝐺) = (+g‘(ℝ*𝑠 ↾s (0[,]+∞)))
103		eqid 2821	. . . . . . . . . . . . . 14 ⊢ (ℝ*𝑠 ↾s (0[,]+∞)) = (ℝ*𝑠 ↾s (0[,]+∞))
104		xrsadd 20562	. . . . . . . . . . . . . 14 ⊢ +𝑒 = (+g‘ℝ*𝑠)
105	103, 104	ressplusg 16612	. . . . . . . . . . . . 13 ⊢ ((0[,]+∞) ∈ V → +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞))))
106	68, 105	ax-mp 5	. . . . . . . . . . . 12 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,]+∞)))
107	106	eqcomi 2830	. . . . . . . . . . 11 ⊢ (+g‘(ℝ*𝑠 ↾s (0[,]+∞))) = +𝑒
108	102, 107	eqtr2i 2845	. . . . . . . . . 10 ⊢ +𝑒 = (+g‘𝐺)
109	21	oveq1i 7166	. . . . . . . . . . 11 ⊢ (𝐺 ↾s (0[,)+∞)) = ((ℝ*𝑠 ↾s (0[,]+∞)) ↾s (0[,)+∞))
110		icossicc 12825	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
111	68, 110	pm3.2i 473	. . . . . . . . . . . 12 ⊢ ((0[,]+∞) ∈ V ∧ (0[,)+∞) ⊆ (0[,]+∞))
112		ressabs 16563	. . . . . . . . . . . 12 ⊢ (((0[,]+∞) ∈ V ∧ (0[,)+∞) ⊆ (0[,]+∞)) → ((ℝ*𝑠 ↾s (0[,]+∞)) ↾s (0[,)+∞)) = (ℝ*𝑠 ↾s (0[,)+∞)))
113	111, 112	ax-mp 5	. . . . . . . . . . 11 ⊢ ((ℝ*𝑠 ↾s (0[,]+∞)) ↾s (0[,)+∞)) = (ℝ*𝑠 ↾s (0[,)+∞))
114	109, 113	eqtr2i 2845	. . . . . . . . . 10 ⊢ (ℝ*𝑠 ↾s (0[,)+∞)) = (𝐺 ↾s (0[,)+∞))
115	58	elexi 3513	. . . . . . . . . . 11 ⊢ 𝐺 ∈ V
116	115	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐺 ∈ V)
117		simpr 487	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ (𝒫 𝑋 ∩ Fin))
118	110	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (0[,)+∞) ⊆ (0[,]+∞))
119		0xr 10688	. . . . . . . . . . . . 13 ⊢ 0 ∈ ℝ*
120	119	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 0 ∈ ℝ*)
121	82	a1i 11	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → +∞ ∈ ℝ*)
122	95	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝐹:𝑋⟶(0[,]+∞))
123	26	sselda 3967	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋)
124	123	adantll 712	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋)
125	122, 124	ffvelrnd 6852	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (0[,]+∞))
126	20, 125	sseldi 3965	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℝ*)
127		iccgelb 12794	. . . . . . . . . . . . 13 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝐹‘𝑦) ∈ (0[,]+∞)) → 0 ≤ (𝐹‘𝑦))
128	120, 121, 125, 127	syl3anc 1367	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 0 ≤ (𝐹‘𝑦))
129		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹‘𝑦) = +∞ → (𝐹‘𝑦) = +∞)
130	129	eqcomd 2827	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑦) = +∞ → +∞ = (𝐹‘𝑦))
131	130	adantl 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ (𝐹‘𝑦) = +∞) → +∞ = (𝐹‘𝑦))
132	11	ffund 6518	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → Fun 𝐹)
133	132	ad2antrr 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → Fun 𝐹)
134	22, 123	sylan 582	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑋)
135	11	fdmd 6523	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → dom 𝐹 = 𝑋)
136	135	eqcomd 2827	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝑋 = dom 𝐹)
137	136	ad2antrr 724	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑋 = dom 𝐹)
138	134, 137	eleqtrd 2915	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ dom 𝐹)
139		fvelrn 6844	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ dom 𝐹) → (𝐹‘𝑦) ∈ ran 𝐹)
140	133, 138, 139	syl2anc 586	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ran 𝐹)
141	140	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ (𝐹‘𝑦) = +∞) → (𝐹‘𝑦) ∈ ran 𝐹)
142	131, 141	eqeltrd 2913	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ (𝐹‘𝑦) = +∞) → +∞ ∈ ran 𝐹)
143	142	adantl3r 748	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ (𝐹‘𝑦) = +∞) → +∞ ∈ ran 𝐹)
144	96	ad3antrrr 728	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) ∧ (𝐹‘𝑦) = +∞) → ¬ +∞ ∈ ran 𝐹)
145	143, 144	pm2.65da 815	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → ¬ (𝐹‘𝑦) = +∞)
146	145	neqned 3023	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ≠ +∞)
147		ge0xrre 41856	. . . . . . . . . . . . . 14 ⊢ (((𝐹‘𝑦) ∈ (0[,]+∞) ∧ (𝐹‘𝑦) ≠ +∞) → (𝐹‘𝑦) ∈ ℝ)
148	125, 146, 147	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℝ)
149	148	ltpnfd 12517	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) < +∞)
150	120, 121, 126, 128, 149	elicod 12788	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (0[,)+∞))
151		eqid 2821	. . . . . . . . . . 11 ⊢ (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)) = (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))
152	150, 151	fmptd 6878	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)):𝑥⟶(0[,)+∞))
153		0e0icopnf 12847	. . . . . . . . . . 11 ⊢ 0 ∈ (0[,)+∞)
154	153	a1i 11	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 0 ∈ (0[,)+∞))
155		eliccxr 12824	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*)
156		xaddid2 12636	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ* → (0 +𝑒 𝑦) = 𝑦)
157		xaddid1 12635	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℝ* → (𝑦 +𝑒 0) = 𝑦)
158	156, 157	jca 514	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℝ* → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
159	155, 158	syl 17	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (0[,]+∞) → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
160	159	adantl 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ (0[,]+∞)) → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
161	72, 108, 114, 116, 117, 118, 152, 154, 160	gsumress 17892	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = ((ℝ*𝑠 ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
162		rege0subm 20601	. . . . . . . . . . . . 13 ⊢ (0[,)+∞) ∈ (SubMnd‘ℂfld)
163	162	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (0[,)+∞) ∈ (SubMnd‘ℂfld))
164		eqid 2821	. . . . . . . . . . . 12 ⊢ (ℂfld ↾s (0[,)+∞)) = (ℂfld ↾s (0[,)+∞))
165	117, 163, 152, 164	gsumsubm 17999	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = ((ℂfld ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
166		eqidd 2822	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℂfld ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = ((ℂfld ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
167		vex 3497	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
168	167	mptex 6986	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)) ∈ V
169	168	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)) ∈ V)
170		ovexd 7191	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld ↾s (0[,)+∞)) ∈ V)
171		ovexd 7191	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℝ*𝑠 ↾s (0[,)+∞)) ∈ V)
172		rge0ssre 12845	. . . . . . . . . . . . . . . . 17 ⊢ (0[,)+∞) ⊆ ℝ
173		ax-resscn 10594	. . . . . . . . . . . . . . . . 17 ⊢ ℝ ⊆ ℂ
174	172, 173	sstri 3976	. . . . . . . . . . . . . . . 16 ⊢ (0[,)+∞) ⊆ ℂ
175		cnfldbas 20549	. . . . . . . . . . . . . . . . 17 ⊢ ℂ = (Base‘ℂfld)
176	164, 175	ressbas2 16555	. . . . . . . . . . . . . . . 16 ⊢ ((0[,)+∞) ⊆ ℂ → (0[,)+∞) = (Base‘(ℂfld ↾s (0[,)+∞))))
177	174, 176	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (0[,)+∞) = (Base‘(ℂfld ↾s (0[,)+∞)))
178	177	eqcomi 2830	. . . . . . . . . . . . . 14 ⊢ (Base‘(ℂfld ↾s (0[,)+∞))) = (0[,)+∞)
179	110, 20	sstri 3976	. . . . . . . . . . . . . . 15 ⊢ (0[,)+∞) ⊆ ℝ*
180		eqid 2821	. . . . . . . . . . . . . . . 16 ⊢ (ℝ*𝑠 ↾s (0[,)+∞)) = (ℝ*𝑠 ↾s (0[,)+∞))
181	180, 69	ressbas2 16555	. . . . . . . . . . . . . . 15 ⊢ ((0[,)+∞) ⊆ ℝ* → (0[,)+∞) = (Base‘(ℝ*𝑠 ↾s (0[,)+∞))))
182	179, 181	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (0[,)+∞) = (Base‘(ℝ*𝑠 ↾s (0[,)+∞)))
183	178, 182	eqtri 2844	. . . . . . . . . . . . 13 ⊢ (Base‘(ℂfld ↾s (0[,)+∞))) = (Base‘(ℝ*𝑠 ↾s (0[,)+∞)))
184	183	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Base‘(ℂfld ↾s (0[,)+∞))) = (Base‘(ℝ*𝑠 ↾s (0[,)+∞))))
185		rge0srg 20616	. . . . . . . . . . . . . . 15 ⊢ (ℂfld ↾s (0[,)+∞)) ∈ SRing
186	185	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → (ℂfld ↾s (0[,)+∞)) ∈ SRing)
187		simpl 485	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → 𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
188		simpr 487	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
189		eqid 2821	. . . . . . . . . . . . . . 15 ⊢ (Base‘(ℂfld ↾s (0[,)+∞))) = (Base‘(ℂfld ↾s (0[,)+∞)))
190		eqid 2821	. . . . . . . . . . . . . . 15 ⊢ (+g‘(ℂfld ↾s (0[,)+∞))) = (+g‘(ℂfld ↾s (0[,)+∞)))
191	189, 190	srgacl 19274	. . . . . . . . . . . . . 14 ⊢ (((ℂfld ↾s (0[,)+∞)) ∈ SRing ∧ 𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
192	186, 187, 188, 191	syl3anc 1367	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
193	192	adantl 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
194	172	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → (0[,)+∞) ⊆ ℝ)
195		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
196	195, 178	eleqtrdi 2923	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑠 ∈ (0[,)+∞))
197	194, 196	sseldd 3968	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑠 ∈ ℝ)
198	197	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → 𝑠 ∈ ℝ)
199	172	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → (0[,)+∞) ⊆ ℝ)
200		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
201	200, 178	eleqtrdi 2923	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑡 ∈ (0[,)+∞))
202	199, 201	sseldd 3968	. . . . . . . . . . . . . . 15 ⊢ (𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → 𝑡 ∈ ℝ)
203	202	adantl 484	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → 𝑡 ∈ ℝ)
204		rexadd 12626	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠 +𝑒 𝑡) = (𝑠 + 𝑡))
205	204	eqcomd 2827	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠 + 𝑡) = (𝑠 +𝑒 𝑡))
206	162	elexi 3513	. . . . . . . . . . . . . . . . . . . 20 ⊢ (0[,)+∞) ∈ V
207		cnfldadd 20550	. . . . . . . . . . . . . . . . . . . . 21 ⊢ + = (+g‘ℂfld)
208	164, 207	ressplusg 16612	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0[,)+∞) ∈ V → + = (+g‘(ℂfld ↾s (0[,)+∞))))
209	206, 208	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ + = (+g‘(ℂfld ↾s (0[,)+∞)))
210	209, 207	eqtr3i 2846	. . . . . . . . . . . . . . . . . 18 ⊢ (+g‘(ℂfld ↾s (0[,)+∞))) = (+g‘ℂfld)
211	210, 207	eqtr4i 2847	. . . . . . . . . . . . . . . . 17 ⊢ (+g‘(ℂfld ↾s (0[,)+∞))) = +
212	211	oveqi 7169	. . . . . . . . . . . . . . . 16 ⊢ (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) = (𝑠 + 𝑡)
213	212	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) = (𝑠 + 𝑡))
214	180, 104	ressplusg 16612	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0[,)+∞) ∈ V → +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,)+∞))))
215	206, 214	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ +𝑒 = (+g‘(ℝ*𝑠 ↾s (0[,)+∞)))
216	215	eqcomi 2830	. . . . . . . . . . . . . . . . 17 ⊢ (+g‘(ℝ*𝑠 ↾s (0[,)+∞))) = +𝑒
217	216	oveqi 7169	. . . . . . . . . . . . . . . 16 ⊢ (𝑠(+g‘(ℝ*𝑠 ↾s (0[,)+∞)))𝑡) = (𝑠 +𝑒 𝑡)
218	217	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℝ*𝑠 ↾s (0[,)+∞)))𝑡) = (𝑠 +𝑒 𝑡))
219	205, 213, 218	3eqtr4d 2866	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠 ↾s (0[,)+∞)))𝑡))
220	198, 203, 219	syl2anc 586	. . . . . . . . . . . . 13 ⊢ ((𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞)))) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠 ↾s (0[,)+∞)))𝑡))
221	220	adantl 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (𝑠 ∈ (Base‘(ℂfld ↾s (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂfld ↾s (0[,)+∞))))) → (𝑠(+g‘(ℂfld ↾s (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠 ↾s (0[,)+∞)))𝑡))
222		funmpt 6393	. . . . . . . . . . . . 13 ⊢ Fun (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))
223	222	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Fun (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)))
224	150, 177	eleqtrdi 2923	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
225	224	ralrimiva 3182	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (Base‘(ℂfld ↾s (0[,)+∞))))
226	151	rnmptss 6886	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ 𝑥 (𝐹‘𝑦) ∈ (Base‘(ℂfld ↾s (0[,)+∞))) → ran (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)) ⊆ (Base‘(ℂfld ↾s (0[,)+∞))))
227	225, 226	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ran (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦)) ⊆ (Base‘(ℂfld ↾s (0[,)+∞))))
228	169, 170, 171, 184, 193, 221, 223, 227	gsumpropd2 17890	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℂfld ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = ((ℝ*𝑠 ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
229	165, 166, 228	3eqtrd 2860	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = ((ℝ*𝑠 ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))))
230	30	adantl 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
231	148	recnd 10669	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ 𝑥) → (𝐹‘𝑦) ∈ ℂ)
232	230, 231	gsumfsum 20612	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
233	229, 232	eqtr3d 2858	. . . . . . . . 9 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℝ*𝑠 ↾s (0[,)+∞)) Σg (𝑦 ∈ 𝑥 ↦ (𝐹‘𝑦))) = Σ𝑦 ∈ 𝑥 (𝐹‘𝑦))
234	101, 161, 233	3eqtrrd 2861	. . . . . . . 8 ⊢ (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦 ∈ 𝑥 (𝐹‘𝑦) = (𝐺 Σg (𝐹 ↾ 𝑥)))
235	234	mpteq2dva 5161	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))))
236	235	rneqd 5808	. . . . . 6 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)) = ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))))
237	236	supeq1d 8910	. . . . 5 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦 ∈ 𝑥 (𝐹‘𝑦)), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ))
238	98, 237	eqtrd 2856	. . . 4 ⊢ ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ))
239	93, 238	pm2.61dan 811	. . 3 ⊢ (𝜑 → (Σ^‘𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ))
240	21, 9, 11, 1	xrge0tsms 23442	. . 3 ⊢ (𝜑 → (𝐺 tsums 𝐹) = {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )})
241	239, 240	eleq12d 2907	. 2 ⊢ (𝜑 → ((Σ^‘𝐹) ∈ (𝐺 tsums 𝐹) ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑥))), ℝ*, < )}))
242	8, 241	mpbird 259	1 ⊢ (𝜑 → (Σ^‘𝐹) ∈ (𝐺 tsums 𝐹))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ∩ cin 3935   ⊆ wss 3936  𝒫 cpw 4539  {csn 4567   class class class wbr 5066   ↦ cmpt 5146  dom cdm 5555  ran crn 5556   ↾ cres 5557  Fun wfun 6349   Fn wfn 6350  ⟶wf 6351  ‘cfv 6355  (class class class)co 7156  Fincfn 8509  supcsup 8904  ℂcc 10535  ℝcr 10536  0cc0 10537   + caddc 10540  +∞cpnf 10672  ℝ^*cxr 10674   < clt 10675   ≤ cle 10676   +_𝑒 cxad 12506  [,)cico 12741  [,]cicc 12742  Σcsu 15042  Basecbs 16483   ↾_s cress 16484  +_gcplusg 16565   Σ_g cgsu 16714  ℝ^*_𝑠cxrs 16773  Mndcmnd 17911  SubMndcsubmnd 17955  CMndccmn 18906  SRingcsrg 19255  ℂ_fldccnfld 20545   tsums ctsu 22734  Σ^{^}csumge0 42693

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-inf2 9104  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615  ax-addf 10616  ax-mulf 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-iin 4922  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-isom 6364  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7409  df-om 7581  df-1st 7689  df-2nd 7690  df-supp 7831  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-fsupp 8834  df-fi 8875  df-sup 8906  df-inf 8907  df-oi 8974  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-z 11983  df-dec 12100  df-uz 12245  df-q 12350  df-rp 12391  df-xadd 12509  df-ioo 12743  df-ioc 12744  df-ico 12745  df-icc 12746  df-fz 12894  df-fzo 13035  df-seq 13371  df-exp 13431  df-hash 13692  df-cj 14458  df-re 14459  df-im 14460  df-sqrt 14594  df-abs 14595  df-clim 14845  df-sum 15043  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-mulr 16579  df-starv 16580  df-tset 16584  df-ple 16585  df-ds 16587  df-unif 16588  df-rest 16696  df-topn 16697  df-0g 16715  df-gsum 16716  df-topgen 16717  df-ordt 16774  df-xrs 16775  df-mre 16857  df-mrc 16858  df-acs 16860  df-ps 17810  df-tsr 17811  df-mgm 17852  df-sgrp 17901  df-mnd 17912  df-submnd 17957  df-grp 18106  df-minusg 18107  df-mulg 18225  df-cntz 18447  df-cmn 18908  df-abl 18909  df-mgp 19240  df-ur 19252  df-srg 19256  df-ring 19299  df-cring 19300  df-fbas 20542  df-fg 20543  df-cnfld 20546  df-top 21502  df-topon 21519  df-topsp 21541  df-bases 21554  df-ntr 21628  df-nei 21706  df-cn 21835  df-haus 21923  df-fil 22454  df-fm 22546  df-flim 22547  df-flf 22548  df-tsms 22735  df-sumge0 42694

This theorem is referenced by: (None)