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Theorem sge0tsms 40915
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 2651 . . . 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 12012 . . . . . 6 < Or ℝ*
43supex 8410 . . . . 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 4224 . . . 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 247 . 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 40908 . . . . 5 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (Σ^𝐹) = +∞)
15 ffn 6083 . . . . . . . . . 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 6282 . . . . . . . 8 (𝐹 Fn 𝑋 → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
1917, 18syl 17 . . . . . . 7 ((𝜑 ∧ +∞ ∈ ran 𝐹) → (+∞ ∈ ran 𝐹 ↔ ∃𝑦𝑋 (𝐹𝑦) = +∞))
2013, 19mpbid 222 . . . . . 6 ((𝜑 ∧ +∞ ∈ ran 𝐹) → ∃𝑦𝑋 (𝐹𝑦) = +∞)
21 iccssxr 12294 . . . . . . . . . . . . . 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 3832 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ 𝒫 𝑋)
26 elpwi 4201 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ 𝒫 𝑋𝑥𝑋)
2725, 26syl 17 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥𝑋)
2827adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥𝑋)
29 fssres 6108 . . . . . . . . . . . . . . . 16 ((𝐹:𝑋⟶(0[,]+∞) ∧ 𝑥𝑋) → (𝐹𝑥):𝑥⟶(0[,]+∞))
3024, 28, 29syl2anc 694 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥):𝑥⟶(0[,]+∞))
31 elinel2 3833 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) → 𝑥 ∈ Fin)
3231adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
33 0red 10079 . . . . . . . . . . . . . . . 16 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 0 ∈ ℝ)
3430, 32, 33fdmfifsupp 8326 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥) finSupp 0)
3522, 23, 30, 34gsumge0cl 40906 . . . . . . . . . . . . . 14 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹𝑥)) ∈ (0[,]+∞))
3621, 35sseldi 3634 . . . . . . . . . . . . 13 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹𝑥)) ∈ ℝ*)
3736ralrimiva 2995 . . . . . . . . . . . 12 (𝜑 → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹𝑥)) ∈ ℝ*)
38373ad2ant1 1102 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → ∀𝑥 ∈ (𝒫 𝑋 ∩ Fin)(𝐺 Σg (𝐹𝑥)) ∈ ℝ*)
39 eqid 2651 . . . . . . . . . . . 12 (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥)))
4039rnmptss 6432 . . . . . . . . . . 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 4942 . . . . . . . . . . . . . 14 (𝑦𝑋 → {𝑦} ∈ 𝒫 𝑋)
43 snfi 8079 . . . . . . . . . . . . . . 15 {𝑦} ∈ Fin
4443a1i 11 . . . . . . . . . . . . . 14 (𝑦𝑋 → {𝑦} ∈ Fin)
4542, 44elind 3831 . . . . . . . . . . . . 13 (𝑦𝑋 → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
46453ad2ant2 1103 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → {𝑦} ∈ (𝒫 𝑋 ∩ Fin))
4711adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦𝑋) → 𝐹:𝑋⟶(0[,]+∞))
48 snssi 4371 . . . . . . . . . . . . . . . . . . 19 (𝑦𝑋 → {𝑦} ⊆ 𝑋)
4948adantl 481 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑦𝑋) → {𝑦} ⊆ 𝑋)
5047, 49fssresd 6109 . . . . . . . . . . . . . . . . 17 ((𝜑𝑦𝑋) → (𝐹 ↾ {𝑦}):{𝑦}⟶(0[,]+∞))
5150feqmptd 6288 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝑋) → (𝐹 ↾ {𝑦}) = (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)))
52 fvres 6245 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ {𝑦} → ((𝐹 ↾ {𝑦})‘𝑥) = (𝐹𝑥))
5352mpteq2ia 4773 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)) = (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))
5453a1i 11 . . . . . . . . . . . . . . . 16 ((𝜑𝑦𝑋) → (𝑥 ∈ {𝑦} ↦ ((𝐹 ↾ {𝑦})‘𝑥)) = (𝑥 ∈ {𝑦} ↦ (𝐹𝑥)))
5551, 54eqtrd 2685 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝑋) → (𝐹 ↾ {𝑦}) = (𝑥 ∈ {𝑦} ↦ (𝐹𝑥)))
5655oveq2d 6706 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑋) → (𝐺 Σg (𝐹 ↾ {𝑦})) = (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))))
57563adant3 1101 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐺 Σg (𝐹 ↾ {𝑦})) = (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))))
58 xrge0cmn 19836 . . . . . . . . . . . . . . . . 17 (ℝ*𝑠s (0[,]+∞)) ∈ CMnd
5922, 58eqeltri 2726 . . . . . . . . . . . . . . . 16 𝐺 ∈ CMnd
60 cmnmnd 18254 . . . . . . . . . . . . . . . 16 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
6159, 60ax-mp 5 . . . . . . . . . . . . . . 15 𝐺 ∈ Mnd
6261a1i 11 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → 𝐺 ∈ Mnd)
63 simp2 1082 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → 𝑦𝑋)
6411ffvelrnda 6399 . . . . . . . . . . . . . . 15 ((𝜑𝑦𝑋) → (𝐹𝑦) ∈ (0[,]+∞))
65643adant3 1101 . . . . . . . . . . . . . 14 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐹𝑦) ∈ (0[,]+∞))
66 df-ss 3621 . . . . . . . . . . . . . . . . . 18 ((0[,]+∞) ⊆ ℝ* ↔ ((0[,]+∞) ∩ ℝ*) = (0[,]+∞))
6721, 66mpbi 220 . . . . . . . . . . . . . . . . 17 ((0[,]+∞) ∩ ℝ*) = (0[,]+∞)
6867eqcomi 2660 . . . . . . . . . . . . . . . 16 (0[,]+∞) = ((0[,]+∞) ∩ ℝ*)
69 ovex 6718 . . . . . . . . . . . . . . . . 17 (0[,]+∞) ∈ V
70 xrsbas 19810 . . . . . . . . . . . . . . . . . 18 * = (Base‘ℝ*𝑠)
7122, 70ressbas 15977 . . . . . . . . . . . . . . . . 17 ((0[,]+∞) ∈ V → ((0[,]+∞) ∩ ℝ*) = (Base‘𝐺))
7269, 71ax-mp 5 . . . . . . . . . . . . . . . 16 ((0[,]+∞) ∩ ℝ*) = (Base‘𝐺)
7368, 72eqtri 2673 . . . . . . . . . . . . . . 15 (0[,]+∞) = (Base‘𝐺)
74 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
7573, 74gsumsn 18400 . . . . . . . . . . . . . 14 ((𝐺 ∈ Mnd ∧ 𝑦𝑋 ∧ (𝐹𝑦) ∈ (0[,]+∞)) → (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))) = (𝐹𝑦))
7662, 63, 65, 75syl3anc 1366 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐺 Σg (𝑥 ∈ {𝑦} ↦ (𝐹𝑥))) = (𝐹𝑦))
77 simp3 1083 . . . . . . . . . . . . 13 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → (𝐹𝑦) = +∞)
7857, 76, 773eqtrrd 2690 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ = (𝐺 Σg (𝐹 ↾ {𝑦})))
79 reseq2 5423 . . . . . . . . . . . . . . 15 (𝑥 = {𝑦} → (𝐹𝑥) = (𝐹 ↾ {𝑦}))
8079oveq2d 6706 . . . . . . . . . . . . . 14 (𝑥 = {𝑦} → (𝐺 Σg (𝐹𝑥)) = (𝐺 Σg (𝐹 ↾ {𝑦})))
8180eqeq2d 2661 . . . . . . . . . . . . 13 (𝑥 = {𝑦} → (+∞ = (𝐺 Σg (𝐹𝑥)) ↔ +∞ = (𝐺 Σg (𝐹 ↾ {𝑦}))))
8281rspcev 3340 . . . . . . . . . . . 12 (({𝑦} ∈ (𝒫 𝑋 ∩ Fin) ∧ +∞ = (𝐺 Σg (𝐹 ↾ {𝑦}))) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹𝑥)))
8346, 78, 82syl2anc 694 . . . . . . . . . . 11 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → ∃𝑥 ∈ (𝒫 𝑋 ∩ Fin)+∞ = (𝐺 Σg (𝐹𝑥)))
84 pnfxr 10130 . . . . . . . . . . . . 13 +∞ ∈ ℝ*
8584a1i 11 . . . . . . . . . . . 12 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ ∈ ℝ*)
8639elrnmpt 5404 . . . . . . . . . . . 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 247 . . . . . . . . . 10 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → +∞ ∈ ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))))
89 supxrpnf 12186 . . . . . . . . . 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 694 . . . . . . . . 9 ((𝜑𝑦𝑋 ∧ (𝐹𝑦) = +∞) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ) = +∞)
91903exp 1283 . . . . . . . 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 3059 . . . . . 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 2688 . . . 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 40905 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → 𝐹:𝑋⟶(0[,)+∞))
10096, 99sge0reval 40907 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ))
10124, 28feqresmpt 6289 . . . . . . . . . . 11 ((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥) = (𝑦𝑥 ↦ (𝐹𝑦)))
102101adantlr 751 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐹𝑥) = (𝑦𝑥 ↦ (𝐹𝑦)))
103102oveq2d 6706 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝐹𝑥)) = (𝐺 Σg (𝑦𝑥 ↦ (𝐹𝑦))))
10422fveq2i 6232 . . . . . . . . . . 11 (+g𝐺) = (+g‘(ℝ*𝑠s (0[,]+∞)))
105 eqid 2651 . . . . . . . . . . . . . 14 (ℝ*𝑠s (0[,]+∞)) = (ℝ*𝑠s (0[,]+∞))
106 xrsadd 19811 . . . . . . . . . . . . . 14 +𝑒 = (+g‘ℝ*𝑠)
107105, 106ressplusg 16040 . . . . . . . . . . . . 13 ((0[,]+∞) ∈ V → +𝑒 = (+g‘(ℝ*𝑠s (0[,]+∞))))
10869, 107ax-mp 5 . . . . . . . . . . . 12 +𝑒 = (+g‘(ℝ*𝑠s (0[,]+∞)))
109108eqcomi 2660 . . . . . . . . . . 11 (+g‘(ℝ*𝑠s (0[,]+∞))) = +𝑒
110104, 109eqtr2i 2674 . . . . . . . . . 10 +𝑒 = (+g𝐺)
11122oveq1i 6700 . . . . . . . . . . 11 (𝐺s (0[,)+∞)) = ((ℝ*𝑠s (0[,]+∞)) ↾s (0[,)+∞))
112 icossicc 12298 . . . . . . . . . . . . 13 (0[,)+∞) ⊆ (0[,]+∞)
11369, 112pm3.2i 470 . . . . . . . . . . . 12 ((0[,]+∞) ∈ V ∧ (0[,)+∞) ⊆ (0[,]+∞))
114 ressabs 15986 . . . . . . . . . . . 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 2674 . . . . . . . . . 10 (ℝ*𝑠s (0[,)+∞)) = (𝐺s (0[,)+∞))
11759elexi 3244 . . . . . . . . . . 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 10124 . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝐹:𝑋⟶(0[,]+∞))
12527sselda 3636 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑦𝑥) → 𝑦𝑋)
126125adantll 750 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑋)
127124, 126ffvelrnd 6400 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,]+∞))
12821, 127sseldi 3634 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ*)
129 iccgelb 12268 . . . . . . . . . . . . 13 ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (𝐹𝑦) ∈ (0[,]+∞)) → 0 ≤ (𝐹𝑦))
130122, 123, 127, 129syl3anc 1366 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 0 ≤ (𝐹𝑦))
131 id 22 . . . . . . . . . . . . . . . . . . . 20 ((𝐹𝑦) = +∞ → (𝐹𝑦) = +∞)
132131eqcomd 2657 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝑦) = +∞ → +∞ = (𝐹𝑦))
133132adantl 481 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → +∞ = (𝐹𝑦))
134 ffun 6086 . . . . . . . . . . . . . . . . . . . . . 22 (𝐹:𝑋⟶(0[,]+∞) → Fun 𝐹)
13511, 134syl 17 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → Fun 𝐹)
136135ad2antrr 762 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → Fun 𝐹)
13723, 125sylan 487 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦𝑋)
138 fdm 6089 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐹:𝑋⟶(0[,]+∞) → dom 𝐹 = 𝑋)
13911, 138syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → dom 𝐹 = 𝑋)
140139eqcomd 2657 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑𝑋 = dom 𝐹)
141140ad2antrr 762 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑋 = dom 𝐹)
142137, 141eleqtrd 2732 . . . . . . . . . . . . . . . . . . . 20 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → 𝑦 ∈ dom 𝐹)
143 fvelrn 6392 . . . . . . . . . . . . . . . . . . . 20 ((Fun 𝐹𝑦 ∈ dom 𝐹) → (𝐹𝑦) ∈ ran 𝐹)
144136, 142, 143syl2anc 694 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ran 𝐹)
145144adantr 480 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → (𝐹𝑦) ∈ ran 𝐹)
146133, 145eqeltrd 2730 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → +∞ ∈ ran 𝐹)
147146adantlllr 39513 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → +∞ ∈ ran 𝐹)
14898ad3antrrr 766 . . . . . . . . . . . . . . . 16 (((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) ∧ (𝐹𝑦) = +∞) → ¬ +∞ ∈ ran 𝐹)
149147, 148pm2.65da 599 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → ¬ (𝐹𝑦) = +∞)
150149neqned 2830 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ≠ +∞)
151 ge0xrre 40076 . . . . . . . . . . . . . 14 (((𝐹𝑦) ∈ (0[,]+∞) ∧ (𝐹𝑦) ≠ +∞) → (𝐹𝑦) ∈ ℝ)
152127, 150, 151syl2anc 694 . . . . . . . . . . . . 13 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℝ)
153152ltpnfd 11993 . . . . . . . . . . . 12 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) < +∞)
154122, 123, 128, 130, 153elicod 12262 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (0[,)+∞))
155 eqid 2651 . . . . . . . . . . 11 (𝑦𝑥 ↦ (𝐹𝑦)) = (𝑦𝑥 ↦ (𝐹𝑦))
156154, 155fmptd 6425 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝑦𝑥 ↦ (𝐹𝑦)):𝑥⟶(0[,)+∞))
157 0e0icopnf 12320 . . . . . . . . . . 11 0 ∈ (0[,)+∞)
158157a1i 11 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 0 ∈ (0[,)+∞))
15921sseli 3632 . . . . . . . . . . . 12 (𝑦 ∈ (0[,]+∞) → 𝑦 ∈ ℝ*)
160 xaddid2 12111 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ* → (0 +𝑒 𝑦) = 𝑦)
161 xaddid1 12110 . . . . . . . . . . . . 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 17323 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝐺 Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℝ*𝑠s (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
166 rege0subm 19850 . . . . . . . . . . . . 13 (0[,)+∞) ∈ (SubMnd‘ℂfld)
167166a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (0[,)+∞) ∈ (SubMnd‘ℂfld))
168 eqid 2651 . . . . . . . . . . . 12 (ℂflds (0[,)+∞)) = (ℂflds (0[,)+∞))
169119, 167, 156, 168gsumsubm 17420 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℂflds (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
170 eqidd 2652 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℂflds (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℂflds (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
171 vex 3234 . . . . . . . . . . . . . 14 𝑥 ∈ V
172171mptex 6527 . . . . . . . . . . . . 13 (𝑦𝑥 ↦ (𝐹𝑦)) ∈ V
173172a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (𝑦𝑥 ↦ (𝐹𝑦)) ∈ V)
174 ovexd 6720 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂflds (0[,)+∞)) ∈ V)
175 ovexd 6720 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℝ*𝑠s (0[,)+∞)) ∈ V)
176 rge0ssre 12318 . . . . . . . . . . . . . . . . 17 (0[,)+∞) ⊆ ℝ
177 ax-resscn 10031 . . . . . . . . . . . . . . . . 17 ℝ ⊆ ℂ
178176, 177sstri 3645 . . . . . . . . . . . . . . . 16 (0[,)+∞) ⊆ ℂ
179 cnfldbas 19798 . . . . . . . . . . . . . . . . 17 ℂ = (Base‘ℂfld)
180168, 179ressbas2 15978 . . . . . . . . . . . . . . . 16 ((0[,)+∞) ⊆ ℂ → (0[,)+∞) = (Base‘(ℂflds (0[,)+∞))))
181178, 180ax-mp 5 . . . . . . . . . . . . . . 15 (0[,)+∞) = (Base‘(ℂflds (0[,)+∞)))
182181eqcomi 2660 . . . . . . . . . . . . . 14 (Base‘(ℂflds (0[,)+∞))) = (0[,)+∞)
183112, 21sstri 3645 . . . . . . . . . . . . . . 15 (0[,)+∞) ⊆ ℝ*
184 eqid 2651 . . . . . . . . . . . . . . . 16 (ℝ*𝑠s (0[,)+∞)) = (ℝ*𝑠s (0[,)+∞))
185184, 70ressbas2 15978 . . . . . . . . . . . . . . 15 ((0[,)+∞) ⊆ ℝ* → (0[,)+∞) = (Base‘(ℝ*𝑠s (0[,)+∞))))
186183, 185ax-mp 5 . . . . . . . . . . . . . 14 (0[,)+∞) = (Base‘(ℝ*𝑠s (0[,)+∞)))
187182, 186eqtri 2673 . . . . . . . . . . . . 13 (Base‘(ℂflds (0[,)+∞))) = (Base‘(ℝ*𝑠s (0[,)+∞)))
188187a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (Base‘(ℂflds (0[,)+∞))) = (Base‘(ℝ*𝑠s (0[,)+∞))))
189 rge0srg 19865 . . . . . . . . . . . . . . 15 (ℂflds (0[,)+∞)) ∈ SRing
190189a1i 11 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → (ℂflds (0[,)+∞)) ∈ SRing)
191 simpl 472 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → 𝑠 ∈ (Base‘(ℂflds (0[,)+∞))))
192 simpr 476 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → 𝑡 ∈ (Base‘(ℂflds (0[,)+∞))))
193 eqid 2651 . . . . . . . . . . . . . . 15 (Base‘(ℂflds (0[,)+∞))) = (Base‘(ℂflds (0[,)+∞)))
194 eqid 2651 . . . . . . . . . . . . . . 15 (+g‘(ℂflds (0[,)+∞))) = (+g‘(ℂflds (0[,)+∞)))
195193, 194srgacl 18570 . . . . . . . . . . . . . 14 (((ℂflds (0[,)+∞)) ∈ SRing ∧ 𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) ∈ (Base‘(ℂflds (0[,)+∞))))
196190, 191, 192, 195syl3anc 1366 . . . . . . . . . . . . 13 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) ∈ (Base‘(ℂflds (0[,)+∞))))
197196adantl 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[,)+∞))))
198176a1i 11 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) → (0[,)+∞) ⊆ ℝ)
199 id 22 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑠 ∈ (Base‘(ℂflds (0[,)+∞))))
200199, 182syl6eleq 2740 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑠 ∈ (0[,)+∞))
201198, 200sseldd 3637 . . . . . . . . . . . . . . 15 (𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑠 ∈ ℝ)
202201adantr 480 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → 𝑠 ∈ ℝ)
203176a1i 11 . . . . . . . . . . . . . . . 16 (𝑡 ∈ (Base‘(ℂflds (0[,)+∞))) → (0[,)+∞) ⊆ ℝ)
204 id 22 . . . . . . . . . . . . . . . . 17 (𝑡 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑡 ∈ (Base‘(ℂflds (0[,)+∞))))
205204, 182syl6eleq 2740 . . . . . . . . . . . . . . . 16 (𝑡 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑡 ∈ (0[,)+∞))
206203, 205sseldd 3637 . . . . . . . . . . . . . . 15 (𝑡 ∈ (Base‘(ℂflds (0[,)+∞))) → 𝑡 ∈ ℝ)
207206adantl 481 . . . . . . . . . . . . . 14 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → 𝑡 ∈ ℝ)
208 rexadd 12101 . . . . . . . . . . . . . . . 16 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠 +𝑒 𝑡) = (𝑠 + 𝑡))
209208eqcomd 2657 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠 + 𝑡) = (𝑠 +𝑒 𝑡))
210166elexi 3244 . . . . . . . . . . . . . . . . . . . 20 (0[,)+∞) ∈ V
211 cnfldadd 19799 . . . . . . . . . . . . . . . . . . . . 21 + = (+g‘ℂfld)
212168, 211ressplusg 16040 . . . . . . . . . . . . . . . . . . . 20 ((0[,)+∞) ∈ V → + = (+g‘(ℂflds (0[,)+∞))))
213210, 212ax-mp 5 . . . . . . . . . . . . . . . . . . 19 + = (+g‘(ℂflds (0[,)+∞)))
214213, 211eqtr3i 2675 . . . . . . . . . . . . . . . . . 18 (+g‘(ℂflds (0[,)+∞))) = (+g‘ℂfld)
215214, 211eqtr4i 2676 . . . . . . . . . . . . . . . . 17 (+g‘(ℂflds (0[,)+∞))) = +
216215oveqi 6703 . . . . . . . . . . . . . . . 16 (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠 + 𝑡)
217216a1i 11 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠 + 𝑡))
218184, 106ressplusg 16040 . . . . . . . . . . . . . . . . . . 19 ((0[,)+∞) ∈ V → +𝑒 = (+g‘(ℝ*𝑠s (0[,)+∞))))
219210, 218ax-mp 5 . . . . . . . . . . . . . . . . . 18 +𝑒 = (+g‘(ℝ*𝑠s (0[,)+∞)))
220219eqcomi 2660 . . . . . . . . . . . . . . . . 17 (+g‘(ℝ*𝑠s (0[,)+∞))) = +𝑒
221220oveqi 6703 . . . . . . . . . . . . . . . 16 (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡) = (𝑠 +𝑒 𝑡)
222221a1i 11 . . . . . . . . . . . . . . 15 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡) = (𝑠 +𝑒 𝑡))
223209, 217, 2223eqtr4d 2695 . . . . . . . . . . . . . 14 ((𝑠 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡))
224202, 207, 223syl2anc 694 . . . . . . . . . . . . 13 ((𝑠 ∈ (Base‘(ℂflds (0[,)+∞))) ∧ 𝑡 ∈ (Base‘(ℂflds (0[,)+∞)))) → (𝑠(+g‘(ℂflds (0[,)+∞)))𝑡) = (𝑠(+g‘(ℝ*𝑠s (0[,)+∞)))𝑡))
225224adantl 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))
226 funmpt 5964 . . . . . . . . . . . . 13 Fun (𝑦𝑥 ↦ (𝐹𝑦))
227226a1i 11 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Fun (𝑦𝑥 ↦ (𝐹𝑦)))
228154, 181syl6eleq 2740 . . . . . . . . . . . . . 14 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ (Base‘(ℂflds (0[,)+∞))))
229228ralrimiva 2995 . . . . . . . . . . . . 13 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ∀𝑦𝑥 (𝐹𝑦) ∈ (Base‘(ℂflds (0[,)+∞))))
230155rnmptss 6432 . . . . . . . . . . . . 13 (∀𝑦𝑥 (𝐹𝑦) ∈ (Base‘(ℂflds (0[,)+∞))) → ran (𝑦𝑥 ↦ (𝐹𝑦)) ⊆ (Base‘(ℂflds (0[,)+∞))))
231229, 230syl 17 . . . . . . . . . . . 12 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ran (𝑦𝑥 ↦ (𝐹𝑦)) ⊆ (Base‘(ℂflds (0[,)+∞))))
232173, 174, 175, 188, 197, 225, 227, 231gsumpropd2 17321 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℂflds (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℝ*𝑠s (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
233169, 170, 2323eqtrd 2689 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦𝑥 ↦ (𝐹𝑦))) = ((ℝ*𝑠s (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))))
23431adantl 481 . . . . . . . . . . 11 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → 𝑥 ∈ Fin)
235152recnd 10106 . . . . . . . . . . 11 ((((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) ∧ 𝑦𝑥) → (𝐹𝑦) ∈ ℂ)
236234, 235gsumfsum 19861 . . . . . . . . . 10 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → (ℂfld Σg (𝑦𝑥 ↦ (𝐹𝑦))) = Σ𝑦𝑥 (𝐹𝑦))
237233, 236eqtr3d 2687 . . . . . . . . 9 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → ((ℝ*𝑠s (0[,)+∞)) Σg (𝑦𝑥 ↦ (𝐹𝑦))) = Σ𝑦𝑥 (𝐹𝑦))
238103, 165, 2373eqtrrd 2690 . . . . . . . 8 (((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) ∧ 𝑥 ∈ (𝒫 𝑋 ∩ Fin)) → Σ𝑦𝑥 (𝐹𝑦) = (𝐺 Σg (𝐹𝑥)))
239238mpteq2dva 4777 . . . . . . 7 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))))
240239rneqd 5385 . . . . . 6 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)) = ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))))
241240supeq1d 8393 . . . . 5 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ Σ𝑦𝑥 (𝐹𝑦)), ℝ*, < ) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
242100, 241eqtrd 2685 . . . 4 ((𝜑 ∧ ¬ +∞ ∈ ran 𝐹) → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
24395, 242pm2.61dan 849 . . 3 (𝜑 → (Σ^𝐹) = sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < ))
24422, 9, 11, 1xrge0tsms 22684 . . 3 (𝜑 → (𝐺 tsums 𝐹) = {sup(ran (𝑥 ∈ (𝒫 𝑋 ∩ Fin) ↦ (𝐺 Σg (𝐹𝑥))), ℝ*, < )})
245243, 244eleq12d 2724 . 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*, < )}))
2468, 245mpbird 247 1 (𝜑 → (Σ^𝐹) ∈ (𝐺 tsums 𝐹))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  Vcvv 3231  cin 3606  wss 3607  𝒫 cpw 4191  {csn 4210   class class class wbr 4685  cmpt 4762  dom cdm 5143  ran crn 5144  cres 5145  Fun wfun 5920   Fn wfn 5921  wf 5922  cfv 5926  (class class class)co 6690  Fincfn 7997  supcsup 8387  cc 9972  cr 9973  0cc0 9974   + caddc 9977  +∞cpnf 10109  *cxr 10111   < clt 10112  cle 10113   +𝑒 cxad 11982  [,)cico 12215  [,]cicc 12216  Σcsu 14460  Basecbs 15904  s cress 15905  +gcplusg 15988   Σg cgsu 16148  *𝑠cxrs 16207  Mndcmnd 17341  SubMndcsubmnd 17381  CMndccmn 18239  SRingcsrg 18551  fldccnfld 19794   tsums ctsu 21976  Σ^csumge0 40897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052  ax-addf 10053  ax-mulf 10054
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-fi 8358  df-sup 8389  df-inf 8390  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xadd 11985  df-ioo 12217  df-ioc 12218  df-ico 12219  df-icc 12220  df-fz 12365  df-fzo 12505  df-seq 12842  df-exp 12901  df-hash 13158  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-clim 14263  df-sum 14461  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-starv 16003  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-rest 16130  df-topn 16131  df-0g 16149  df-gsum 16150  df-topgen 16151  df-ordt 16208  df-xrs 16209  df-mre 16293  df-mrc 16294  df-acs 16296  df-ps 17247  df-tsr 17248  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-grp 17472  df-minusg 17473  df-mulg 17588  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-srg 18552  df-ring 18595  df-cring 18596  df-fbas 19791  df-fg 19792  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-ntr 20872  df-nei 20950  df-cn 21079  df-haus 21167  df-fil 21697  df-fm 21789  df-flim 21790  df-flf 21791  df-tsms 21977  df-sumge0 40898
This theorem is referenced by: (None)
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