Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sge0tsms Structured version   Visualization version   GIF version

Theorem sge0tsms 39273
 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*𝑠s (0[,]+∞))
sge0tsms.x (𝜑𝑋𝑉)
sge0tsms.f (𝜑𝐹: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.
StepHypRef Expression
1 eqid 2610 . . . 4 sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )
21a1i 11 . . 3 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
3 xrltso 11850 . . . . . 6 < Or ℝ*
43supex 8252 . . . . 5 sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ V
54a1i 11 . . . 4 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ V)
6 elsng 4139 . . . 4 (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ V → (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )} ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )))
75, 6syl 17 . . 3 (𝜑 → (sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )} ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )))
82, 7mpbird 246 . 2 (𝜑 → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )})
9 sge0tsms.x . . . . . . 7 (𝜑𝑋𝑉)
109adantr 480 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝑋𝑉)
11 sge0tsms.f . . . . . . 7 (𝜑𝐹:𝑋⟶(0[,]+∞))
1211adantr 480 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
13 simpr 476 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → +∞ ∈ ran 𝐹)
1410, 12, 13sge0pnfval 39266 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = +∞)
15 ffn 5958 . . . . . . . . . 10 (𝐹:𝑋⟶(0[,]+∞) → 𝐹 Fn 𝑋)
1611, 15syl 17 . . . . . . . . 9 (𝜑𝐹 Fn 𝑋)
1716adantr 480 . . . . . . . 8 ((𝜑 ∧ +∞ ∈ ran 𝐹) → 𝐹 Fn 𝑋)
18 fvelrnb 6153 . . . . . . . 8 (𝐹 Fn 𝑋 → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
1917, 18syl 17 . . . . . . 7 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
2013, 19mpbid 221 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∃𝑦𝑋 (𝐹𝑦) = +∞)
21 iccssxr 12127 . . . . . . . . . . . . . 14 (0[,]+∞) ⊆ ℝ*
22 sge0tsms.g . . . . . . . . . . . . . . 15 𝐺 = (ℝ*𝑠s (0[,]+∞))
23 simpr 476 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ (𝒫 𝑋 ∩ Fin))
2411adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐹:𝑋⟶(0[,]+∞))
25 elinel1 3761 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
26 elpwi 4117 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
2725, 26syl 17 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥𝑋)
2827adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥𝑋)
29 fssres 5983 . . . . . . . . . . . . . . . 16 ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥𝑋) → (𝐹𝑥):𝑥⟶(0[,]+∞))
3024, 28, 29syl2anc 691 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,]+∞))
31 elinel2 3762 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
3231adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
33 0red 9920 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 0 ∈ ℝ)
3430, 32, 33fdmfifsupp 8168 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥) finSupp 0)
3522, 23, 30, 34gsumge0cl 39264 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹𝑥)) ∈ (0[,]+∞))
3621, 35sseldi 3566 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹𝑥)) ∈ ℝ*)
3736ralrimiva 2949 . . . . . . . . . . . 12 (𝜑 → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹𝑥)) ∈ ℝ*)
38373ad2ant1 1075 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹𝑥)) ∈ ℝ*)
39 eqid 2610 . . . . . . . . . . . 12 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥)))
4039rnmptss 6299 . . . . . . . . . . 11 (∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹𝑥)) ∈ ℝ* → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) ⊆ ℝ*)
4138, 40syl 17 . . . . . . . . . 10 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) ⊆ ℝ*)
42 snelpwi 4839 . . . . . . . . . . . . . 14 (𝑦𝑋 → {𝑦} ∈ 𝒫 𝑋)
43 snfi 7923 . . . . . . . . . . . . . . 15 {𝑦} ∈ Fin
4443a1i 11 . . . . . . . . . . . . . 14 (𝑦𝑋 → {𝑦} ∈ Fin)
4542, 44elind 3760 . . . . . . . . . . . . 13 (𝑦𝑋 → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
46453ad2ant2 1076 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
4711adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦𝑋) → 𝐹:𝑋⟶(0[,]+∞))
48 snssi 4280 . . . . . . . . . . . . . . . . . . 19 (𝑦𝑋 → {𝑦} ⊆ 𝑋)
4948adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦𝑋) → {𝑦} ⊆ 𝑋)
5047, 49fssresd 5984 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦𝑋) → (𝐹 ↾ {𝑦}):{𝑦}⟶(0[,]+∞))
5150feqmptd 6159 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝑋) → (𝐹 ↾ {𝑦}) = (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)))
52 fvres 6117 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ {𝑦} → ((𝐹 ↾ {𝑦})‘𝑥) = (𝐹𝑥))
5352mpteq2ia 4668 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)) = (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))
5453a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝑋) → (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)) = (𝑥 ∈ {𝑦} ↦ (𝐹𝑥)))
5551, 54eqtrd 2644 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝑋) → (𝐹 ↾ {𝑦}) = (𝑥 ∈ {𝑦} ↦ (𝐹𝑥)))
5655oveq2d 6565 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑋) → (𝐺 Σg (𝐹 ↾ {𝑦})) = (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))))
57563adant3 1074 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐺 Σg (𝐹 ↾ {𝑦})) = (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))))
58 xrge0cmn 19607 . . . . . . . . . . . . . . . . 17 (ℝ*𝑠s (0[,]+∞)) ∈ CMnd
5922, 58eqeltri 2684 . . . . . . . . . . . . . . . 16 𝐺 ∈ CMnd
60 cmnmnd 18031 . . . . . . . . . . . . . . . 16 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
6159, 60ax-mp 5 . . . . . . . . . . . . . . 15 𝐺 ∈ Mnd
6261a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → 𝐺 ∈ Mnd)
63 simp2 1055 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → 𝑦𝑋)
6411ffvelrnda 6267 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝑋) → (𝐹𝑦) ∈ (0[,]+∞))
65643adant3 1074 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐹𝑦) ∈ (0[,]+∞))
66 df-ss 3554 . . . . . . . . . . . . . . . . . 18 ((0[,]+∞) ⊆ ℝ* ↔ ((0[,]+∞) ∩ ℝ*) = (0[,]+∞))
6721, 66mpbi 219 . . . . . . . . . . . . . . . . 17 ((0[,]+∞) ∩ ℝ*) = (0[,]+∞)
6867eqcomi 2619 . . . . . . . . . . . . . . . 16 (0[,]+∞) = ((0[,]+∞) ∩ ℝ*)
69 ovex 6577 . . . . . . . . . . . . . . . . 17 (0[,]+∞) ∈ V
70 xrsbas 19581 . . . . . . . . . . . . . . . . . 18 * = (Base‘ℝ*𝑠)
7122, 70ressbas 15757 . . . . . . . . . . . . . . . . 17 ((0[,]+∞) ∈ V → ((0[,]+∞) ∩ ℝ*) = (Base‘𝐺))
7269, 71ax-mp 5 . . . . . . . . . . . . . . . 16 ((0[,]+∞) ∩ ℝ*) = (Base‘𝐺)
7368, 72eqtri 2632 . . . . . . . . . . . . . . 15 (0[,]+∞) = (Base‘𝐺)
74 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
7573, 74gsumsn 18177 . . . . . . . . . . . . . 14 ((𝐺 ∈ Mnd ∧ 𝑦𝑋 ∧ (𝐹𝑦) ∈ (0[,]+∞)) → (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))) = (𝐹𝑦))
7662, 63, 65, 75syl3anc 1318 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))) = (𝐹𝑦))
77 simp3 1056 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐹𝑦) = +∞)
7857, 76, 773eqtrrd 2649 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ = (𝐺 Σg (𝐹 ↾ {𝑦})))
79 reseq2 5312 . . . . . . . . . . . . . . 15 (𝑥 = {𝑦} → (𝐹𝑥) = (𝐹 ↾ {𝑦}))
8079oveq2d 6565 . . . . . . . . . . . . . 14 (𝑥 = {𝑦} → (𝐺 Σg (𝐹𝑥)) = (𝐺 Σg (𝐹 ↾ {𝑦})))
8180eqeq2d 2620 . . . . . . . . . . . . 13 (𝑥 = {𝑦} → (+∞ = (𝐺 Σg (𝐹𝑥)) ↔ +∞ = (𝐺 Σg (𝐹 ↾ {𝑦}))))
8281rspcev 3282 . . . . . . . . . . . 12 (({𝑦} ∈ (𝒫 𝑋 ∩ Fin) ∧ +∞ = (𝐺 Σg (𝐹 ↾ {𝑦}))) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹𝑥)))
8346, 78, 82syl2anc 691 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹𝑥)))
84 pnfxr 9971 . . . . . . . . . . . . 13 +∞ ∈ ℝ*
8584a1i 11 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ ∈ ℝ*)
8639elrnmpt 5293 . . . . . . . . . . . 12 (+∞ ∈ ℝ* → (+∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹𝑥))))
8785, 86syl 17 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (+∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) ↔ ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹𝑥))))
8883, 87mpbird 246 . . . . . . . . . 10 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))))
89 supxrpnf 12020 . . . . . . . . . 10 ((ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) ⊆ ℝ* ∧ +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥)))) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞)
9041, 88, 89syl2anc 691 . . . . . . . . 9 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞)
91903exp 1256 . . . . . . . 8 (𝜑 → (𝑦𝑋 → ((𝐹𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞)))
9291adantr 480 . . . . . . 7 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (𝑦𝑋 → ((𝐹𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞)))
9392rexlimdv 3012 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (∃𝑦𝑋 (𝐹𝑦) = +∞ → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞))
9420, 93mpd 15 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞)
9514, 94eqtr4d 2647 . . . 4 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
969adantr 480 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝑋𝑉)
9711adantr 480 . . . . . . 7 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,]+∞))
98 simpr 476 . . . . . . 7 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ¬ +∞ ∈ ran 𝐹)
9997, 98fge0iccico 39263 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
10096, 99sge0reval 39265 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
10124, 28feqresmpt 6160 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥) = (𝑦𝑥 ↦ (𝐹𝑦)))
102101adantlr 747 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥) = (𝑦𝑥 ↦ (𝐹𝑦)))
103102oveq2d 6565 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹𝑥)) = (𝐺 Σg (𝑦𝑥 ↦ (𝐹𝑦))))
10422fveq2i 6106 . . . . . . . . . . 11 (+g𝐺) = (+g‘(ℝ*𝑠s (0[,]+∞)))
105 eqid 2610 . . . . . . . . . . . . . 14 (ℝ*𝑠s (0[,]+∞)) = (ℝ*𝑠s (0[,]+∞))
106 xrsadd 19582 . . . . . . . . . . . . . 14 +𝑒 = (+g‘ℝ*𝑠)
107105, 106ressplusg 15818 . . . . . . . . . . . . 13 ((0[,]+∞) ∈ V → +𝑒 = (+g‘(ℝ*𝑠s (0[,]+∞))))
10869, 107ax-mp 5 . . . . . . . . . . . 12 +𝑒 = (+g‘(ℝ*𝑠s (0[,]+∞)))
109108eqcomi 2619 . . . . . . . . . . 11 (+g‘(ℝ*𝑠s (0[,]+∞))) = +𝑒
110104, 109eqtr2i 2633 . . . . . . . . . 10 +𝑒 = (+g𝐺)
11122oveq1i 6559 . . . . . . . . . . 11 (𝐺s (0[,)+∞)) = ((ℝ*𝑠s (0[,]+∞)) ↾s (0[,)+∞))
112 icossicc 12131 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ (0[,]+∞)
11369, 112pm3.2i 470 . . . . . . . . . . . 12 ((0[,]+∞) ∈ V ∧ (0[,)+∞) ⊆ (0[,]+∞))
114 ressabs 15766 . . . . . . . . . . . 12 (((0[,]+∞) ∈ V ∧ (0[,)+∞) ⊆ (0[,]+∞)) → ((ℝ*𝑠s (0[,]+∞)) ↾s (0[,)+∞)) = (ℝ*𝑠s (0[,)+∞)))
115113, 114ax-mp 5 . . . . . . . . . . 11 ((ℝ*𝑠s (0[,]+∞)) ↾s (0[,)+∞)) = (ℝ*𝑠s (0[,)+∞))
116111, 115eqtr2i 2633 . . . . . . . . . 10 (ℝ*𝑠s (0[,)+∞)) = (𝐺s (0[,)+∞))
11759elexi 3186 . . . . . . . . . . 11 𝐺 ∈ V
118117a1i 11 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝐺 ∈ V)
119 simpr 476 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ (𝒫 𝑋 ∩ Fin))
120112a1i 11 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (0[,)+∞) ⊆ (0[,]+∞))
121 0xr 9965 . . . . . . . . . . . . 13 0 ∈ ℝ*
122121a1i 11 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 0 ∈ ℝ*)
12384a1i 11 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → +∞ ∈ ℝ*)
12497ad2antrr 758 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝐹:𝑋⟶(0[,]+∞))
12527sselda 3568 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → 𝑦𝑋)
126125adantll 746 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑋)
127124, 126ffvelrnd 6268 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,]+∞))
12821, 127sseldi 3566 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ*)
129 iccgelb 12101 . . . . . . . . . . . . 13 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝐹𝑦) ∈ (0[,]+∞)) → 0 ≤ (𝐹𝑦))
130122, 123, 127, 129syl3anc 1318 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 0 ≤ (𝐹𝑦))
131 id 22 . . . . . . . . . . . . . . . . . . . 20 ((𝐹𝑦) = +∞ → (𝐹𝑦) = +∞)
132131eqcomd 2616 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑦) = +∞ → +∞ = (𝐹𝑦))
133132adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → +∞ = (𝐹𝑦))
134 ffun 5961 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹:𝑋⟶(0[,]+∞) → Fun 𝐹)
13511, 134syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → Fun 𝐹)
136135ad2antrr 758 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → Fun 𝐹)
13723, 125sylan 487 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑋)
138 fdm 5964 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹:𝑋⟶(0[,]+∞) → dom 𝐹 = 𝑋)
13911, 138syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → dom 𝐹 = 𝑋)
140139eqcomd 2616 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑋 = dom 𝐹)
141140ad2antrr 758 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑋 = dom 𝐹)
142137, 141eleqtrd 2690 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦 ∈ dom 𝐹)
143 fvelrn 6260 . . . . . . . . . . . . . . . . . . . 20 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝐹𝑦) ∈ ran 𝐹)
144136, 142, 143syl2anc 691 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ran 𝐹)
145144adantr 480 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → (𝐹𝑦) ∈ ran 𝐹)
146133, 145eqeltrd 2688 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → +∞ ∈ ran 𝐹)
147146adantlllr 38222 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → +∞ ∈ ran 𝐹)
14898ad3antrrr 762 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → ¬ +∞ ∈ ran 𝐹)
149147, 148pm2.65da 598 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → ¬ (𝐹𝑦) = +∞)
150149neqned 2789 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ≠ +∞)
151 ge0xrre 38605 . . . . . . . . . . . . . 14 (((𝐹𝑦) ∈ (0[,]+∞) ∧ (𝐹𝑦) ≠ +∞) → (𝐹𝑦) ∈ ℝ)
152127, 150, 151syl2anc 691 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ)
153152ltpnfd 11831 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) < +∞)
154122, 123, 128, 130, 153elicod 12095 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,)+∞))
155 eqid 2610 . . . . . . . . . . 11 (𝑦𝑥 ↦ (𝐹𝑦)) = (𝑦𝑥 ↦ (𝐹𝑦))
156154, 155fmptd 6292 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝑦𝑥 ↦ (𝐹𝑦)):𝑥⟶(0[,)+∞))
157 0e0icopnf 12153 . . . . . . . . . . 11 0 ∈ (0[,)+∞)
158157a1i 11 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 0 ∈ (0[,)+∞))
15921sseli 3564 . . . . . . . . . . . 12 (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*)
160 xaddid2 11947 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → (0 +𝑒 𝑦) = 𝑦)
161 xaddid1 11946 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → (𝑦 +𝑒 0) = 𝑦)
162160, 161jca 553 . . . . . . . . . . . 12 (𝑦 ∈ ℝ* → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
163159, 162syl 17 . . . . . . . . . . 11 (𝑦 ∈ (0[,]+∞) → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
164163adantl 481 . . . . . . . . . 10 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦 ∈ (0[,]+∞)) → ((0 +𝑒 𝑦) = 𝑦 ∧ (𝑦 +𝑒 0) = 𝑦))
16573, 110, 116, 118, 119, 120, 156, 158, 164gsumress 17099 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℝ*𝑠s (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
166 rege0subm 19621 . . . . . . . . . . . . 13 (0[,)+∞) ∈ (SubMnd‘ℂfld)
167166a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (0[,)+∞) ∈ (SubMnd‘ℂfld))
168 eqid 2610 . . . . . . . . . . . 12 (ℂflds (0[,)+∞)) = (ℂflds (0[,)+∞))
169119, 167, 156, 168gsumsubm 17196 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℂflds (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
170 eqidd 2611 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℂflds (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℂflds (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
171 vex 3176 . . . . . . . . . . . . . 14 𝑥 ∈ V
172171mptex 6390 . . . . . . . . . . . . 13 (𝑦𝑥 ↦ (𝐹𝑦)) ∈ V
173172a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝑦𝑥 ↦ (𝐹𝑦)) ∈ V)
174 ovex 6577 . . . . . . . . . . . . 13 (ℂflds (0[,)+∞)) ∈ V
175174a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂflds (0[,)+∞)) ∈ V)
176 ovex 6577 . . . . . . . . . . . . 13 (ℝ*𝑠s (0[,)+∞)) ∈ V
177176a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℝ*𝑠s (0[,)+∞)) ∈ V)
178 rge0ssre 12151 . . . . . . . . . . . . . . . . 17 (0[,)+∞) ⊆ ℝ
179 ax-resscn 9872 . . . . . . . . . . . . . . . . 17 ℝ ⊆ ℂ
180178, 179sstri 3577 . . . . . . . . . . . . . . . 16 (0[,)+∞) ⊆ ℂ
181 cnfldbas 19571 . . . . . . . . . . . . . . . . 17 ℂ = (Base‘ℂfld)
182168, 181ressbas2 15758 . . . . . . . . . . . . . . . 16 ((0[,)+∞) ⊆ ℂ → (0[,)+∞) = (Base‘(ℂflds (0[,)+∞))))
183180, 182ax-mp 5 . . . . . . . . . . . . . . 15 (0[,)+∞) = (Base‘(ℂflds (0[,)+∞)))
184183eqcomi 2619 . . . . . . . . . . . . . 14 (Base‘(ℂflds (0[,)+∞))) = (0[,)+∞)
185112, 21sstri 3577 . . . . . . . . . . . . . . 15 (0[,)+∞) ⊆ ℝ*
186 eqid 2610 . . . . . . . . . . . . . . . 16 (ℝ*𝑠s (0[,)+∞)) = (ℝ*𝑠s (0[,)+∞))
187186, 70ressbas2 15758 . . . . . . . . . . . . . . 15 ((0[,)+∞) ⊆ ℝ* → (0[,)+∞) = (Base‘(ℝ*𝑠s (0[,)+∞))))
188185, 187ax-mp 5 . . . . . . . . . . . . . 14 (0[,)+∞) = (Base‘(ℝ*𝑠s (0[,)+∞)))
189184, 188eqtri 2632 . . . . . . . . . . . . 13 (Base‘(ℂflds (0[,)+∞))) = (Base‘(ℝ*𝑠s (0[,)+∞)))
190189a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Base‘(ℂflds (0[,)+∞))) = (Base‘(ℝ*𝑠s (0[,)+∞))))
191 rge0srg 19636 . . . . . . . . . . . . . . 15 (ℂflds (0[,)+∞)) ∈ SRing
192191a1i 11 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → (ℂflds (0[,)+∞)) ∈ SRing)
193 simpl 472 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → 𝑠 ∈ (Base‘(ℂflds (0[,)+∞))))
194 simpr 476 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → 𝑡 ∈ (Base‘(ℂflds (0[,)+∞))))
195 eqid 2610 . . . . . . . . . . . . . . 15 (Base‘(ℂflds (0[,)+∞))) = (Base‘(ℂflds (0[,)+∞)))
196 eqid 2610 . . . . . . . . . . . . . . 15 (+g‘(ℂflds (0[,)+∞))) = (+g‘(ℂflds (0[,)+∞)))
197195, 196srgacl 18347 . . . . . . . . . . . . . 14 (((ℂflds (0[,)+∞)) ∈ SRing ∧ 𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) ∈ (Base‘(ℂflds (0[,)+∞))))
198192, 193, 194, 197syl3anc 1318 . . . . . . . . . . . . 13 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) ∈ (Base‘(ℂflds (0[,)+∞))))
199198adantl 481 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞))))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) ∈ (Base‘(ℂflds (0[,)+∞))))
200178a1i 11 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) → (0[,)+∞) ⊆ ℝ)
201 id 22 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑠 ∈ (Base‘(ℂflds (0[,)+∞))))
202201, 184syl6eleq 2698 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑠 ∈ (0[,)+∞))
203200, 202sseldd 3569 . . . . . . . . . . . . . . 15 (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑠 ∈ ℝ)
204203adantr 480 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → 𝑠 ∈ ℝ)
205178a1i 11 . . . . . . . . . . . . . . . 16 (𝑡 ∈ (Base‘(ℂflds (0[,)+∞))) → (0[,)+∞) ⊆ ℝ)
206 id 22 . . . . . . . . . . . . . . . . 17 (𝑡 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑡 ∈ (Base‘(ℂflds (0[,)+∞))))
207206, 184syl6eleq 2698 . . . . . . . . . . . . . . . 16 (𝑡 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑡 ∈ (0[,)+∞))
208205, 207sseldd 3569 . . . . . . . . . . . . . . 15 (𝑡 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑡 ∈ ℝ)
209208adantl 481 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → 𝑡 ∈ ℝ)
210 rexadd 11937 . . . . . . . . . . . . . . . 16 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠 +𝑒 𝑡) = (𝑠 + 𝑡))
211210eqcomd 2616 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠 + 𝑡) = (𝑠 +𝑒 𝑡))
212166elexi 3186 . . . . . . . . . . . . . . . . . . . 20 (0[,)+∞) ∈ V
213 cnfldadd 19572 . . . . . . . . . . . . . . . . . . . . 21 + = (+g‘ℂfld)
214168, 213ressplusg 15818 . . . . . . . . . . . . . . . . . . . 20 ((0[,)+∞) ∈ V → + = (+g‘(ℂflds (0[,)+∞))))
215212, 214ax-mp 5 . . . . . . . . . . . . . . . . . . 19 + = (+g‘(ℂflds (0[,)+∞)))
216215, 213eqtr3i 2634 . . . . . . . . . . . . . . . . . 18 (+g‘(ℂflds (0[,)+∞))) = (+g‘ℂfld)
217216, 213eqtr4i 2635 . . . . . . . . . . . . . . . . 17 (+g‘(ℂflds (0[,)+∞))) = +
218217oveqi 6562 . . . . . . . . . . . . . . . 16 (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠 + 𝑡)
219218a1i 11 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠 + 𝑡))
220186, 106ressplusg 15818 . . . . . . . . . . . . . . . . . . 19 ((0[,)+∞) ∈ V → +𝑒 = (+g‘(ℝ*𝑠s (0[,)+∞))))
221212, 220ax-mp 5 . . . . . . . . . . . . . . . . . 18 +𝑒 = (+g‘(ℝ*𝑠s (0[,)+∞)))
222221eqcomi 2619 . . . . . . . . . . . . . . . . 17 (+g‘(ℝ*𝑠s (0[,)+∞))) = +𝑒
223222oveqi 6562 . . . . . . . . . . . . . . . 16 (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡) = (𝑠 +𝑒 𝑡)
224223a1i 11 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡) = (𝑠 +𝑒 𝑡))
225211, 219, 2243eqtr4d 2654 . . . . . . . . . . . . . 14 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡))
226204, 209, 225syl2anc 691 . . . . . . . . . . . . 13 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡))
227226adantl 481 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞))))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡))
228 funmpt 5840 . . . . . . . . . . . . 13 Fun (𝑦𝑥 ↦ (𝐹𝑦))
229228a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Fun (𝑦𝑥 ↦ (𝐹𝑦)))
230154, 183syl6eleq 2698 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (Base‘(ℂflds (0[,)+∞))))
231230ralrimiva 2949 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ∀𝑦𝑥 (𝐹𝑦) ∈ (Base‘(ℂflds (0[,)+∞))))
232155rnmptss 6299 . . . . . . . . . . . . 13 (∀𝑦𝑥 (𝐹𝑦) ∈ (Base‘(ℂflds (0[,)+∞))) → ran (𝑦𝑥 ↦ (𝐹𝑦)) ⊆ (Base‘(ℂflds (0[,)+∞))))
233231, 232syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ran (𝑦𝑥 ↦ (𝐹𝑦)) ⊆ (Base‘(ℂflds (0[,)+∞))))
234173, 175, 177, 190, 199, 227, 229, 233gsumpropd2 17097 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℂflds (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℝ*𝑠s (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
235169, 170, 2343eqtrd 2648 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℝ*𝑠s (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
23631adantl 481 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
237152recnd 9947 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℂ)
238236, 237gsumfsum 19632 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦𝑥 ↦ (𝐹𝑦))) = Σ𝑦𝑥 (𝐹𝑦))
239235, 238eqtr3d 2646 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℝ*𝑠s (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))) = Σ𝑦𝑥 (𝐹𝑦))
240103, 165, 2393eqtrrd 2649 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) = (𝐺 Σg (𝐹𝑥)))
241240mpteq2dva 4672 . . . . . . 7 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))))
242241rneqd 5274 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))))
243242supeq1d 8235 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
244100, 243eqtrd 2644 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
24595, 244pm2.61dan 828 . . 3 (𝜑 → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
24622, 9, 11, 1xrge0tsms 22445 . . 3 (𝜑 → (𝐺 tsums 𝐹) = {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )})
247245, 246eleq12d 2682 . 2 (𝜑 → ((Σ^𝐹) ∈ (𝐺 tsums 𝐹) ↔ sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) ∈ {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )}))
2488, 247mpbird 246 1 (𝜑 → (Σ^𝐹) ∈ (𝐺 tsums 𝐹))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125   class class class wbr 4583   ↦ cmpt 4643  dom cdm 5038  ran crn 5039   ↾ cres 5040  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  supcsup 8229  ℂcc 9813  ℝcr 9814  0cc0 9815   + caddc 9818  +∞cpnf 9950  ℝ*cxr 9952   < clt 9953   ≤ cle 9954   +𝑒 cxad 11820  [,)cico 12048  [,]cicc 12049  Σcsu 14264  Basecbs 15695   ↾s cress 15696  +gcplusg 15768   Σg cgsu 15924  ℝ*𝑠cxrs 15983  Mndcmnd 17117  SubMndcsubmnd 17157  CMndccmn 18016  SRingcsrg 18328  ℂfldccnfld 19567   tsums ctsu 21739  Σ^csumge0 39255 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-fi 8200  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xadd 11823  df-ioo 12050  df-ioc 12051  df-ico 12052  df-icc 12053  df-fz 12198  df-fzo 12335  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-starv 15783  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-rest 15906  df-topn 15907  df-0g 15925  df-gsum 15926  df-topgen 15927  df-ordt 15984  df-xrs 15985  df-mre 16069  df-mrc 16070  df-acs 16072  df-ps 17023  df-tsr 17024  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-mulg 17364  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-srg 18329  df-ring 18372  df-cring 18373  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-ntr 20634  df-nei 20712  df-cn 20841  df-haus 20929  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-tsms 21740  df-sumge0 39256 This theorem is referenced by: (None)
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