Step | Hyp | Ref
| Expression |
1 | | iccssxr 12256 |
. . . . . . 7
⊢
(0[,]+∞) ⊆ ℝ* |
2 | | xrge0gsumle.g |
. . . . . . . . . 10
⊢ 𝐺 =
(ℝ*𝑠 ↾s
(0[,]+∞)) |
3 | | xrsbas 19762 |
. . . . . . . . . 10
⊢
ℝ* =
(Base‘ℝ*𝑠) |
4 | 2, 3 | ressbas2 15931 |
. . . . . . . . 9
⊢
((0[,]+∞) ⊆ ℝ* → (0[,]+∞) =
(Base‘𝐺)) |
5 | 1, 4 | ax-mp 5 |
. . . . . . . 8
⊢
(0[,]+∞) = (Base‘𝐺) |
6 | | eqid 2622 |
. . . . . . . . . 10
⊢
(ℝ*𝑠 ↾s
(ℝ* ∖ {-∞})) =
(ℝ*𝑠 ↾s
(ℝ* ∖ {-∞})) |
7 | 6 | xrge0subm 19787 |
. . . . . . . . 9
⊢
(0[,]+∞) ∈
(SubMnd‘(ℝ*𝑠 ↾s
(ℝ* ∖ {-∞}))) |
8 | | xrex 11829 |
. . . . . . . . . . . . 13
⊢
ℝ* ∈ V |
9 | | difexg 4808 |
. . . . . . . . . . . . 13
⊢
(ℝ* ∈ V → (ℝ* ∖
{-∞}) ∈ V) |
10 | 8, 9 | ax-mp 5 |
. . . . . . . . . . . 12
⊢
(ℝ* ∖ {-∞}) ∈ V |
11 | | simpl 473 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ*
∧ 0 ≤ 𝑥) →
𝑥 ∈
ℝ*) |
12 | | ge0nemnf 12004 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ*
∧ 0 ≤ 𝑥) →
𝑥 ≠
-∞) |
13 | 11, 12 | jca 554 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ*
∧ 0 ≤ 𝑥) →
(𝑥 ∈
ℝ* ∧ 𝑥
≠ -∞)) |
14 | | elxrge0 12281 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (0[,]+∞) ↔
(𝑥 ∈
ℝ* ∧ 0 ≤ 𝑥)) |
15 | | eldifsn 4317 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (ℝ*
∖ {-∞}) ↔ (𝑥 ∈ ℝ* ∧ 𝑥 ≠
-∞)) |
16 | 13, 14, 15 | 3imtr4i 281 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (0[,]+∞) →
𝑥 ∈
(ℝ* ∖ {-∞})) |
17 | 16 | ssriv 3607 |
. . . . . . . . . . . 12
⊢
(0[,]+∞) ⊆ (ℝ* ∖
{-∞}) |
18 | | ressabs 15939 |
. . . . . . . . . . . 12
⊢
(((ℝ* ∖ {-∞}) ∈ V ∧ (0[,]+∞)
⊆ (ℝ* ∖ {-∞})) →
((ℝ*𝑠 ↾s
(ℝ* ∖ {-∞})) ↾s (0[,]+∞)) =
(ℝ*𝑠 ↾s
(0[,]+∞))) |
19 | 10, 17, 18 | mp2an 708 |
. . . . . . . . . . 11
⊢
((ℝ*𝑠 ↾s
(ℝ* ∖ {-∞})) ↾s (0[,]+∞)) =
(ℝ*𝑠 ↾s
(0[,]+∞)) |
20 | 2, 19 | eqtr4i 2647 |
. . . . . . . . . 10
⊢ 𝐺 =
((ℝ*𝑠 ↾s
(ℝ* ∖ {-∞})) ↾s
(0[,]+∞)) |
21 | 6 | xrs10 19785 |
. . . . . . . . . 10
⊢ 0 =
(0g‘(ℝ*𝑠
↾s (ℝ* ∖ {-∞}))) |
22 | 20, 21 | subm0 17356 |
. . . . . . . . 9
⊢
((0[,]+∞) ∈
(SubMnd‘(ℝ*𝑠 ↾s
(ℝ* ∖ {-∞}))) → 0 =
(0g‘𝐺)) |
23 | 7, 22 | ax-mp 5 |
. . . . . . . 8
⊢ 0 =
(0g‘𝐺) |
24 | | xrge0cmn 19788 |
. . . . . . . . . 10
⊢
(ℝ*𝑠 ↾s
(0[,]+∞)) ∈ CMnd |
25 | 2, 24 | eqeltri 2697 |
. . . . . . . . 9
⊢ 𝐺 ∈ CMnd |
26 | 25 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐺 ∈ CMnd) |
27 | | elfpw 8268 |
. . . . . . . . . 10
⊢ (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑠 ⊆ 𝐴 ∧ 𝑠 ∈ Fin)) |
28 | 27 | simprbi 480 |
. . . . . . . . 9
⊢ (𝑠 ∈ (𝒫 𝐴 ∩ Fin) → 𝑠 ∈ Fin) |
29 | 28 | adantl 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑠 ∈ Fin) |
30 | | xrge0gsumle.f |
. . . . . . . . 9
⊢ (𝜑 → 𝐹:𝐴⟶(0[,]+∞)) |
31 | 27 | simplbi 476 |
. . . . . . . . 9
⊢ (𝑠 ∈ (𝒫 𝐴 ∩ Fin) → 𝑠 ⊆ 𝐴) |
32 | | fssres 6070 |
. . . . . . . . 9
⊢ ((𝐹:𝐴⟶(0[,]+∞) ∧ 𝑠 ⊆ 𝐴) → (𝐹 ↾ 𝑠):𝑠⟶(0[,]+∞)) |
33 | 30, 31, 32 | syl2an 494 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ 𝑠):𝑠⟶(0[,]+∞)) |
34 | | c0ex 10034 |
. . . . . . . . . 10
⊢ 0 ∈
V |
35 | 34 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ (𝒫 𝐴 ∩ Fin)) → 0 ∈
V) |
36 | 33, 29, 35 | fdmfifsupp 8285 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ 𝑠) finSupp 0) |
37 | 5, 23, 26, 29, 33, 36 | gsumcl 18316 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ 𝑠)) ∈ (0[,]+∞)) |
38 | 1, 37 | sseldi 3601 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ 𝑠)) ∈
ℝ*) |
39 | | eqid 2622 |
. . . . . 6
⊢ (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg
(𝐹 ↾ 𝑠))) = (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑠))) |
40 | 38, 39 | fmptd 6385 |
. . . . 5
⊢ (𝜑 → (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑠))):(𝒫 𝐴 ∩
Fin)⟶ℝ*) |
41 | | frn 6053 |
. . . . 5
⊢ ((𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg
(𝐹 ↾ 𝑠))):(𝒫 𝐴 ∩ Fin)⟶ℝ*
→ ran (𝑠 ∈
(𝒫 𝐴 ∩ Fin)
↦ (𝐺
Σg (𝐹 ↾ 𝑠))) ⊆
ℝ*) |
42 | 40, 41 | syl 17 |
. . . 4
⊢ (𝜑 → ran (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑠))) ⊆
ℝ*) |
43 | | 0ss 3972 |
. . . . . . 7
⊢ ∅
⊆ 𝐴 |
44 | | 0fin 8188 |
. . . . . . 7
⊢ ∅
∈ Fin |
45 | | elfpw 8268 |
. . . . . . 7
⊢ (∅
∈ (𝒫 𝐴 ∩
Fin) ↔ (∅ ⊆ 𝐴 ∧ ∅ ∈ Fin)) |
46 | 43, 44, 45 | mpbir2an 955 |
. . . . . 6
⊢ ∅
∈ (𝒫 𝐴 ∩
Fin) |
47 | | 0cn 10032 |
. . . . . 6
⊢ 0 ∈
ℂ |
48 | | reseq2 5391 |
. . . . . . . . . 10
⊢ (𝑠 = ∅ → (𝐹 ↾ 𝑠) = (𝐹 ↾ ∅)) |
49 | | res0 5400 |
. . . . . . . . . 10
⊢ (𝐹 ↾ ∅) =
∅ |
50 | 48, 49 | syl6eq 2672 |
. . . . . . . . 9
⊢ (𝑠 = ∅ → (𝐹 ↾ 𝑠) = ∅) |
51 | 50 | oveq2d 6666 |
. . . . . . . 8
⊢ (𝑠 = ∅ → (𝐺 Σg
(𝐹 ↾ 𝑠)) = (𝐺 Σg
∅)) |
52 | 23 | gsum0 17278 |
. . . . . . . 8
⊢ (𝐺 Σg
∅) = 0 |
53 | 51, 52 | syl6eq 2672 |
. . . . . . 7
⊢ (𝑠 = ∅ → (𝐺 Σg
(𝐹 ↾ 𝑠)) = 0) |
54 | 39, 53 | elrnmpt1s 5373 |
. . . . . 6
⊢ ((∅
∈ (𝒫 𝐴 ∩
Fin) ∧ 0 ∈ ℂ) → 0 ∈ ran (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑠)))) |
55 | 46, 47, 54 | mp2an 708 |
. . . . 5
⊢ 0 ∈
ran (𝑠 ∈ (𝒫
𝐴 ∩ Fin) ↦ (𝐺 Σg
(𝐹 ↾ 𝑠))) |
56 | 55 | a1i 11 |
. . . 4
⊢ (𝜑 → 0 ∈ ran (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg
(𝐹 ↾ 𝑠)))) |
57 | 42, 56 | sseldd 3604 |
. . 3
⊢ (𝜑 → 0 ∈
ℝ*) |
58 | 25 | a1i 11 |
. . . . 5
⊢ (𝜑 → 𝐺 ∈ CMnd) |
59 | | xrge0gsumle.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ (𝒫 𝐴 ∩ Fin)) |
60 | | elfpw 8268 |
. . . . . . . 8
⊢ (𝐵 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) |
61 | 60 | simprbi 480 |
. . . . . . 7
⊢ (𝐵 ∈ (𝒫 𝐴 ∩ Fin) → 𝐵 ∈ Fin) |
62 | 59, 61 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ Fin) |
63 | | diffi 8192 |
. . . . . 6
⊢ (𝐵 ∈ Fin → (𝐵 ∖ 𝐶) ∈ Fin) |
64 | 62, 63 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝐵 ∖ 𝐶) ∈ Fin) |
65 | 60 | simplbi 476 |
. . . . . . . 8
⊢ (𝐵 ∈ (𝒫 𝐴 ∩ Fin) → 𝐵 ⊆ 𝐴) |
66 | 59, 65 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
67 | 66 | ssdifssd 3748 |
. . . . . 6
⊢ (𝜑 → (𝐵 ∖ 𝐶) ⊆ 𝐴) |
68 | 30, 67 | fssresd 6071 |
. . . . 5
⊢ (𝜑 → (𝐹 ↾ (𝐵 ∖ 𝐶)):(𝐵 ∖ 𝐶)⟶(0[,]+∞)) |
69 | 34 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 0 ∈
V) |
70 | 68, 64, 69 | fdmfifsupp 8285 |
. . . . 5
⊢ (𝜑 → (𝐹 ↾ (𝐵 ∖ 𝐶)) finSupp 0) |
71 | 5, 23, 58, 64, 68, 70 | gsumcl 18316 |
. . . 4
⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐵 ∖ 𝐶))) ∈ (0[,]+∞)) |
72 | 1, 71 | sseldi 3601 |
. . 3
⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐵 ∖ 𝐶))) ∈
ℝ*) |
73 | | xrge0gsumle.c |
. . . . . 6
⊢ (𝜑 → 𝐶 ⊆ 𝐵) |
74 | | ssfi 8180 |
. . . . . 6
⊢ ((𝐵 ∈ Fin ∧ 𝐶 ⊆ 𝐵) → 𝐶 ∈ Fin) |
75 | 62, 73, 74 | syl2anc 693 |
. . . . 5
⊢ (𝜑 → 𝐶 ∈ Fin) |
76 | 73, 66 | sstrd 3613 |
. . . . . 6
⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
77 | 30, 76 | fssresd 6071 |
. . . . 5
⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶(0[,]+∞)) |
78 | 77, 75, 69 | fdmfifsupp 8285 |
. . . . 5
⊢ (𝜑 → (𝐹 ↾ 𝐶) finSupp 0) |
79 | 5, 23, 58, 75, 77, 78 | gsumcl 18316 |
. . . 4
⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐶)) ∈ (0[,]+∞)) |
80 | 1, 79 | sseldi 3601 |
. . 3
⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐶)) ∈
ℝ*) |
81 | | elxrge0 12281 |
. . . . 5
⊢ ((𝐺 Σg
(𝐹 ↾ (𝐵 ∖ 𝐶))) ∈ (0[,]+∞) ↔ ((𝐺 Σg
(𝐹 ↾ (𝐵 ∖ 𝐶))) ∈ ℝ* ∧ 0 ≤
(𝐺
Σg (𝐹 ↾ (𝐵 ∖ 𝐶))))) |
82 | 81 | simprbi 480 |
. . . 4
⊢ ((𝐺 Σg
(𝐹 ↾ (𝐵 ∖ 𝐶))) ∈ (0[,]+∞) → 0 ≤
(𝐺
Σg (𝐹 ↾ (𝐵 ∖ 𝐶)))) |
83 | 71, 82 | syl 17 |
. . 3
⊢ (𝜑 → 0 ≤ (𝐺 Σg (𝐹 ↾ (𝐵 ∖ 𝐶)))) |
84 | | xleadd2a 12084 |
. . 3
⊢ (((0
∈ ℝ* ∧ (𝐺 Σg (𝐹 ↾ (𝐵 ∖ 𝐶))) ∈ ℝ* ∧ (𝐺 Σg
(𝐹 ↾ 𝐶)) ∈ ℝ*)
∧ 0 ≤ (𝐺
Σg (𝐹 ↾ (𝐵 ∖ 𝐶)))) → ((𝐺 Σg (𝐹 ↾ 𝐶)) +𝑒 0) ≤ ((𝐺 Σg
(𝐹 ↾ 𝐶)) +𝑒 (𝐺 Σg
(𝐹 ↾ (𝐵 ∖ 𝐶))))) |
85 | 57, 72, 80, 83, 84 | syl31anc 1329 |
. 2
⊢ (𝜑 → ((𝐺 Σg (𝐹 ↾ 𝐶)) +𝑒 0) ≤ ((𝐺 Σg
(𝐹 ↾ 𝐶)) +𝑒 (𝐺 Σg
(𝐹 ↾ (𝐵 ∖ 𝐶))))) |
86 | | xaddid1 12072 |
. . 3
⊢ ((𝐺 Σg
(𝐹 ↾ 𝐶)) ∈ ℝ*
→ ((𝐺
Σg (𝐹 ↾ 𝐶)) +𝑒 0) = (𝐺 Σg
(𝐹 ↾ 𝐶))) |
87 | 80, 86 | syl 17 |
. 2
⊢ (𝜑 → ((𝐺 Σg (𝐹 ↾ 𝐶)) +𝑒 0) = (𝐺 Σg
(𝐹 ↾ 𝐶))) |
88 | | ovex 6678 |
. . . . 5
⊢
(0[,]+∞) ∈ V |
89 | | xrsadd 19763 |
. . . . . 6
⊢
+𝑒 =
(+g‘ℝ*𝑠) |
90 | 2, 89 | ressplusg 15993 |
. . . . 5
⊢
((0[,]+∞) ∈ V → +𝑒 =
(+g‘𝐺)) |
91 | 88, 90 | ax-mp 5 |
. . . 4
⊢
+𝑒 = (+g‘𝐺) |
92 | 30, 66 | fssresd 6071 |
. . . 4
⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶(0[,]+∞)) |
93 | 92, 62, 69 | fdmfifsupp 8285 |
. . . 4
⊢ (𝜑 → (𝐹 ↾ 𝐵) finSupp 0) |
94 | | disjdif 4040 |
. . . . 5
⊢ (𝐶 ∩ (𝐵 ∖ 𝐶)) = ∅ |
95 | 94 | a1i 11 |
. . . 4
⊢ (𝜑 → (𝐶 ∩ (𝐵 ∖ 𝐶)) = ∅) |
96 | | undif2 4044 |
. . . . 5
⊢ (𝐶 ∪ (𝐵 ∖ 𝐶)) = (𝐶 ∪ 𝐵) |
97 | | ssequn1 3783 |
. . . . . 6
⊢ (𝐶 ⊆ 𝐵 ↔ (𝐶 ∪ 𝐵) = 𝐵) |
98 | 73, 97 | sylib 208 |
. . . . 5
⊢ (𝜑 → (𝐶 ∪ 𝐵) = 𝐵) |
99 | 96, 98 | syl5req 2669 |
. . . 4
⊢ (𝜑 → 𝐵 = (𝐶 ∪ (𝐵 ∖ 𝐶))) |
100 | 5, 23, 91, 58, 59, 92, 93, 95, 99 | gsumsplit 18328 |
. . 3
⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐵)) = ((𝐺 Σg ((𝐹 ↾ 𝐵) ↾ 𝐶)) +𝑒 (𝐺 Σg ((𝐹 ↾ 𝐵) ↾ (𝐵 ∖ 𝐶))))) |
101 | 73 | resabs1d 5428 |
. . . . 5
⊢ (𝜑 → ((𝐹 ↾ 𝐵) ↾ 𝐶) = (𝐹 ↾ 𝐶)) |
102 | 101 | oveq2d 6666 |
. . . 4
⊢ (𝜑 → (𝐺 Σg ((𝐹 ↾ 𝐵) ↾ 𝐶)) = (𝐺 Σg (𝐹 ↾ 𝐶))) |
103 | | difss 3737 |
. . . . . 6
⊢ (𝐵 ∖ 𝐶) ⊆ 𝐵 |
104 | | resabs1 5427 |
. . . . . 6
⊢ ((𝐵 ∖ 𝐶) ⊆ 𝐵 → ((𝐹 ↾ 𝐵) ↾ (𝐵 ∖ 𝐶)) = (𝐹 ↾ (𝐵 ∖ 𝐶))) |
105 | 103, 104 | mp1i 13 |
. . . . 5
⊢ (𝜑 → ((𝐹 ↾ 𝐵) ↾ (𝐵 ∖ 𝐶)) = (𝐹 ↾ (𝐵 ∖ 𝐶))) |
106 | 105 | oveq2d 6666 |
. . . 4
⊢ (𝜑 → (𝐺 Σg ((𝐹 ↾ 𝐵) ↾ (𝐵 ∖ 𝐶))) = (𝐺 Σg (𝐹 ↾ (𝐵 ∖ 𝐶)))) |
107 | 102, 106 | oveq12d 6668 |
. . 3
⊢ (𝜑 → ((𝐺 Σg ((𝐹 ↾ 𝐵) ↾ 𝐶)) +𝑒 (𝐺 Σg ((𝐹 ↾ 𝐵) ↾ (𝐵 ∖ 𝐶)))) = ((𝐺 Σg (𝐹 ↾ 𝐶)) +𝑒 (𝐺 Σg (𝐹 ↾ (𝐵 ∖ 𝐶))))) |
108 | 100, 107 | eqtr2d 2657 |
. 2
⊢ (𝜑 → ((𝐺 Σg (𝐹 ↾ 𝐶)) +𝑒 (𝐺 Σg (𝐹 ↾ (𝐵 ∖ 𝐶)))) = (𝐺 Σg (𝐹 ↾ 𝐵))) |
109 | 85, 87, 108 | 3brtr3d 4684 |
1
⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐶)) ≤ (𝐺 Σg (𝐹 ↾ 𝐵))) |