Step | Hyp | Ref
| Expression |
1 | | iccssxr 12545 |
. . . . . . 7
⊢
(0[,]+∞) ⊆ ℝ* |
2 | | xrge0gsumle.g |
. . . . . . . . . 10
⊢ 𝐺 =
(ℝ*𝑠 ↾s
(0[,]+∞)) |
3 | | xrsbas 20123 |
. . . . . . . . . 10
⊢
ℝ* =
(Base‘ℝ*𝑠) |
4 | 2, 3 | ressbas2 16295 |
. . . . . . . . 9
⊢
((0[,]+∞) ⊆ ℝ* → (0[,]+∞) =
(Base‘𝐺)) |
5 | 1, 4 | ax-mp 5 |
. . . . . . . 8
⊢
(0[,]+∞) = (Base‘𝐺) |
6 | | eqid 2826 |
. . . . . . . . . 10
⊢
(ℝ*𝑠 ↾s
(ℝ* ∖ {-∞})) =
(ℝ*𝑠 ↾s
(ℝ* ∖ {-∞})) |
7 | 6 | xrge0subm 20148 |
. . . . . . . . 9
⊢
(0[,]+∞) ∈
(SubMnd‘(ℝ*𝑠 ↾s
(ℝ* ∖ {-∞}))) |
8 | | xrex 12110 |
. . . . . . . . . . . . 13
⊢
ℝ* ∈ V |
9 | | difexg 5034 |
. . . . . . . . . . . . 13
⊢
(ℝ* ∈ V → (ℝ* ∖
{-∞}) ∈ V) |
10 | 8, 9 | ax-mp 5 |
. . . . . . . . . . . 12
⊢
(ℝ* ∖ {-∞}) ∈ V |
11 | | simpl 476 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ*
∧ 0 ≤ 𝑥) →
𝑥 ∈
ℝ*) |
12 | | ge0nemnf 12293 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ ℝ*
∧ 0 ≤ 𝑥) →
𝑥 ≠
-∞) |
13 | 11, 12 | jca 509 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ ℝ*
∧ 0 ≤ 𝑥) →
(𝑥 ∈
ℝ* ∧ 𝑥
≠ -∞)) |
14 | | elxrge0 12572 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (0[,]+∞) ↔
(𝑥 ∈
ℝ* ∧ 0 ≤ 𝑥)) |
15 | | eldifsn 4537 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ (ℝ*
∖ {-∞}) ↔ (𝑥 ∈ ℝ* ∧ 𝑥 ≠
-∞)) |
16 | 13, 14, 15 | 3imtr4i 284 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ (0[,]+∞) →
𝑥 ∈
(ℝ* ∖ {-∞})) |
17 | 16 | ssriv 3832 |
. . . . . . . . . . . 12
⊢
(0[,]+∞) ⊆ (ℝ* ∖
{-∞}) |
18 | | ressabs 16304 |
. . . . . . . . . . . 12
⊢
(((ℝ* ∖ {-∞}) ∈ V ∧ (0[,]+∞)
⊆ (ℝ* ∖ {-∞})) →
((ℝ*𝑠 ↾s
(ℝ* ∖ {-∞})) ↾s (0[,]+∞)) =
(ℝ*𝑠 ↾s
(0[,]+∞))) |
19 | 10, 17, 18 | mp2an 685 |
. . . . . . . . . . 11
⊢
((ℝ*𝑠 ↾s
(ℝ* ∖ {-∞})) ↾s (0[,]+∞)) =
(ℝ*𝑠 ↾s
(0[,]+∞)) |
20 | 2, 19 | eqtr4i 2853 |
. . . . . . . . . 10
⊢ 𝐺 =
((ℝ*𝑠 ↾s
(ℝ* ∖ {-∞})) ↾s
(0[,]+∞)) |
21 | 6 | xrs10 20146 |
. . . . . . . . . 10
⊢ 0 =
(0g‘(ℝ*𝑠
↾s (ℝ* ∖ {-∞}))) |
22 | 20, 21 | subm0 17710 |
. . . . . . . . 9
⊢
((0[,]+∞) ∈
(SubMnd‘(ℝ*𝑠 ↾s
(ℝ* ∖ {-∞}))) → 0 =
(0g‘𝐺)) |
23 | 7, 22 | ax-mp 5 |
. . . . . . . 8
⊢ 0 =
(0g‘𝐺) |
24 | | xrge0cmn 20149 |
. . . . . . . . . 10
⊢
(ℝ*𝑠 ↾s
(0[,]+∞)) ∈ CMnd |
25 | 2, 24 | eqeltri 2903 |
. . . . . . . . 9
⊢ 𝐺 ∈ CMnd |
26 | 25 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (𝒫 𝐴 ∩ Fin)) → 𝐺 ∈ CMnd) |
27 | | elfpw 8538 |
. . . . . . . . . 10
⊢ (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝑠 ⊆ 𝐴 ∧ 𝑠 ∈ Fin)) |
28 | 27 | simprbi 492 |
. . . . . . . . 9
⊢ (𝑠 ∈ (𝒫 𝐴 ∩ Fin) → 𝑠 ∈ Fin) |
29 | 28 | adantl 475 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (𝒫 𝐴 ∩ Fin)) → 𝑠 ∈ Fin) |
30 | | xrge0gsumle.f |
. . . . . . . . 9
⊢ (𝜑 → 𝐹:𝐴⟶(0[,]+∞)) |
31 | 27 | simplbi 493 |
. . . . . . . . 9
⊢ (𝑠 ∈ (𝒫 𝐴 ∩ Fin) → 𝑠 ⊆ 𝐴) |
32 | | fssres 6308 |
. . . . . . . . 9
⊢ ((𝐹:𝐴⟶(0[,]+∞) ∧ 𝑠 ⊆ 𝐴) → (𝐹 ↾ 𝑠):𝑠⟶(0[,]+∞)) |
33 | 30, 31, 32 | syl2an 591 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ 𝑠):𝑠⟶(0[,]+∞)) |
34 | | c0ex 10351 |
. . . . . . . . . 10
⊢ 0 ∈
V |
35 | 34 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑠 ∈ (𝒫 𝐴 ∩ Fin)) → 0 ∈
V) |
36 | 33, 29, 35 | fdmfifsupp 8555 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑠 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐹 ↾ 𝑠) finSupp 0) |
37 | 5, 23, 26, 29, 33, 36 | gsumcl 18670 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑠 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ 𝑠)) ∈ (0[,]+∞)) |
38 | 1, 37 | sseldi 3826 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑠 ∈ (𝒫 𝐴 ∩ Fin)) → (𝐺 Σg (𝐹 ↾ 𝑠)) ∈
ℝ*) |
39 | 38 | fmpttd 6635 |
. . . . 5
⊢ (𝜑 → (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑠))):(𝒫 𝐴 ∩
Fin)⟶ℝ*) |
40 | 39 | frnd 6286 |
. . . 4
⊢ (𝜑 → ran (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑠))) ⊆
ℝ*) |
41 | | 0ss 4198 |
. . . . . . 7
⊢ ∅
⊆ 𝐴 |
42 | | 0fin 8458 |
. . . . . . 7
⊢ ∅
∈ Fin |
43 | | elfpw 8538 |
. . . . . . 7
⊢ (∅
∈ (𝒫 𝐴 ∩
Fin) ↔ (∅ ⊆ 𝐴 ∧ ∅ ∈ Fin)) |
44 | 41, 42, 43 | mpbir2an 704 |
. . . . . 6
⊢ ∅
∈ (𝒫 𝐴 ∩
Fin) |
45 | | 0cn 10349 |
. . . . . 6
⊢ 0 ∈
ℂ |
46 | | eqid 2826 |
. . . . . . 7
⊢ (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg
(𝐹 ↾ 𝑠))) = (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑠))) |
47 | | reseq2 5625 |
. . . . . . . . . 10
⊢ (𝑠 = ∅ → (𝐹 ↾ 𝑠) = (𝐹 ↾ ∅)) |
48 | | res0 5634 |
. . . . . . . . . 10
⊢ (𝐹 ↾ ∅) =
∅ |
49 | 47, 48 | syl6eq 2878 |
. . . . . . . . 9
⊢ (𝑠 = ∅ → (𝐹 ↾ 𝑠) = ∅) |
50 | 49 | oveq2d 6922 |
. . . . . . . 8
⊢ (𝑠 = ∅ → (𝐺 Σg
(𝐹 ↾ 𝑠)) = (𝐺 Σg
∅)) |
51 | 23 | gsum0 17632 |
. . . . . . . 8
⊢ (𝐺 Σg
∅) = 0 |
52 | 50, 51 | syl6eq 2878 |
. . . . . . 7
⊢ (𝑠 = ∅ → (𝐺 Σg
(𝐹 ↾ 𝑠)) = 0) |
53 | 46, 52 | elrnmpt1s 5607 |
. . . . . 6
⊢ ((∅
∈ (𝒫 𝐴 ∩
Fin) ∧ 0 ∈ ℂ) → 0 ∈ ran (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg (𝐹 ↾ 𝑠)))) |
54 | 44, 45, 53 | mp2an 685 |
. . . . 5
⊢ 0 ∈
ran (𝑠 ∈ (𝒫
𝐴 ∩ Fin) ↦ (𝐺 Σg
(𝐹 ↾ 𝑠))) |
55 | 54 | a1i 11 |
. . . 4
⊢ (𝜑 → 0 ∈ ran (𝑠 ∈ (𝒫 𝐴 ∩ Fin) ↦ (𝐺 Σg
(𝐹 ↾ 𝑠)))) |
56 | 40, 55 | sseldd 3829 |
. . 3
⊢ (𝜑 → 0 ∈
ℝ*) |
57 | 25 | a1i 11 |
. . . . 5
⊢ (𝜑 → 𝐺 ∈ CMnd) |
58 | | xrge0gsumle.b |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ (𝒫 𝐴 ∩ Fin)) |
59 | | elfpw 8538 |
. . . . . . . 8
⊢ (𝐵 ∈ (𝒫 𝐴 ∩ Fin) ↔ (𝐵 ⊆ 𝐴 ∧ 𝐵 ∈ Fin)) |
60 | 59 | simprbi 492 |
. . . . . . 7
⊢ (𝐵 ∈ (𝒫 𝐴 ∩ Fin) → 𝐵 ∈ Fin) |
61 | 58, 60 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈ Fin) |
62 | | diffi 8462 |
. . . . . 6
⊢ (𝐵 ∈ Fin → (𝐵 ∖ 𝐶) ∈ Fin) |
63 | 61, 62 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝐵 ∖ 𝐶) ∈ Fin) |
64 | 59 | simplbi 493 |
. . . . . . . 8
⊢ (𝐵 ∈ (𝒫 𝐴 ∩ Fin) → 𝐵 ⊆ 𝐴) |
65 | 58, 64 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
66 | 65 | ssdifssd 3976 |
. . . . . 6
⊢ (𝜑 → (𝐵 ∖ 𝐶) ⊆ 𝐴) |
67 | 30, 66 | fssresd 6309 |
. . . . 5
⊢ (𝜑 → (𝐹 ↾ (𝐵 ∖ 𝐶)):(𝐵 ∖ 𝐶)⟶(0[,]+∞)) |
68 | 34 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 0 ∈
V) |
69 | 67, 63, 68 | fdmfifsupp 8555 |
. . . . 5
⊢ (𝜑 → (𝐹 ↾ (𝐵 ∖ 𝐶)) finSupp 0) |
70 | 5, 23, 57, 63, 67, 69 | gsumcl 18670 |
. . . 4
⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐵 ∖ 𝐶))) ∈ (0[,]+∞)) |
71 | 1, 70 | sseldi 3826 |
. . 3
⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐵 ∖ 𝐶))) ∈
ℝ*) |
72 | | xrge0gsumle.c |
. . . . . 6
⊢ (𝜑 → 𝐶 ⊆ 𝐵) |
73 | | ssfi 8450 |
. . . . . 6
⊢ ((𝐵 ∈ Fin ∧ 𝐶 ⊆ 𝐵) → 𝐶 ∈ Fin) |
74 | 61, 72, 73 | syl2anc 581 |
. . . . 5
⊢ (𝜑 → 𝐶 ∈ Fin) |
75 | 72, 65 | sstrd 3838 |
. . . . . 6
⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
76 | 30, 75 | fssresd 6309 |
. . . . 5
⊢ (𝜑 → (𝐹 ↾ 𝐶):𝐶⟶(0[,]+∞)) |
77 | 76, 74, 68 | fdmfifsupp 8555 |
. . . . 5
⊢ (𝜑 → (𝐹 ↾ 𝐶) finSupp 0) |
78 | 5, 23, 57, 74, 76, 77 | gsumcl 18670 |
. . . 4
⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐶)) ∈ (0[,]+∞)) |
79 | 1, 78 | sseldi 3826 |
. . 3
⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐶)) ∈
ℝ*) |
80 | | elxrge0 12572 |
. . . . 5
⊢ ((𝐺 Σg
(𝐹 ↾ (𝐵 ∖ 𝐶))) ∈ (0[,]+∞) ↔ ((𝐺 Σg
(𝐹 ↾ (𝐵 ∖ 𝐶))) ∈ ℝ* ∧ 0 ≤
(𝐺
Σg (𝐹 ↾ (𝐵 ∖ 𝐶))))) |
81 | 80 | simprbi 492 |
. . . 4
⊢ ((𝐺 Σg
(𝐹 ↾ (𝐵 ∖ 𝐶))) ∈ (0[,]+∞) → 0 ≤
(𝐺
Σg (𝐹 ↾ (𝐵 ∖ 𝐶)))) |
82 | 70, 81 | syl 17 |
. . 3
⊢ (𝜑 → 0 ≤ (𝐺 Σg (𝐹 ↾ (𝐵 ∖ 𝐶)))) |
83 | | xleadd2a 12373 |
. . 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))))) |
84 | 56, 71, 79, 82, 83 | syl31anc 1498 |
. 2
⊢ (𝜑 → ((𝐺 Σg (𝐹 ↾ 𝐶)) +𝑒 0) ≤ ((𝐺 Σg
(𝐹 ↾ 𝐶)) +𝑒 (𝐺 Σg
(𝐹 ↾ (𝐵 ∖ 𝐶))))) |
85 | | xaddid1 12361 |
. . 3
⊢ ((𝐺 Σg
(𝐹 ↾ 𝐶)) ∈ ℝ*
→ ((𝐺
Σg (𝐹 ↾ 𝐶)) +𝑒 0) = (𝐺 Σg
(𝐹 ↾ 𝐶))) |
86 | 79, 85 | syl 17 |
. 2
⊢ (𝜑 → ((𝐺 Σg (𝐹 ↾ 𝐶)) +𝑒 0) = (𝐺 Σg
(𝐹 ↾ 𝐶))) |
87 | | ovex 6938 |
. . . . 5
⊢
(0[,]+∞) ∈ V |
88 | | xrsadd 20124 |
. . . . . 6
⊢
+𝑒 =
(+g‘ℝ*𝑠) |
89 | 2, 88 | ressplusg 16353 |
. . . . 5
⊢
((0[,]+∞) ∈ V → +𝑒 =
(+g‘𝐺)) |
90 | 87, 89 | ax-mp 5 |
. . . 4
⊢
+𝑒 = (+g‘𝐺) |
91 | 30, 65 | fssresd 6309 |
. . . 4
⊢ (𝜑 → (𝐹 ↾ 𝐵):𝐵⟶(0[,]+∞)) |
92 | 91, 61, 68 | fdmfifsupp 8555 |
. . . 4
⊢ (𝜑 → (𝐹 ↾ 𝐵) finSupp 0) |
93 | | disjdif 4264 |
. . . . 5
⊢ (𝐶 ∩ (𝐵 ∖ 𝐶)) = ∅ |
94 | 93 | a1i 11 |
. . . 4
⊢ (𝜑 → (𝐶 ∩ (𝐵 ∖ 𝐶)) = ∅) |
95 | | undif2 4268 |
. . . . 5
⊢ (𝐶 ∪ (𝐵 ∖ 𝐶)) = (𝐶 ∪ 𝐵) |
96 | | ssequn1 4011 |
. . . . . 6
⊢ (𝐶 ⊆ 𝐵 ↔ (𝐶 ∪ 𝐵) = 𝐵) |
97 | 72, 96 | sylib 210 |
. . . . 5
⊢ (𝜑 → (𝐶 ∪ 𝐵) = 𝐵) |
98 | 95, 97 | syl5req 2875 |
. . . 4
⊢ (𝜑 → 𝐵 = (𝐶 ∪ (𝐵 ∖ 𝐶))) |
99 | 5, 23, 90, 57, 58, 91, 92, 94, 98 | gsumsplit 18682 |
. . 3
⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐵)) = ((𝐺 Σg ((𝐹 ↾ 𝐵) ↾ 𝐶)) +𝑒 (𝐺 Σg ((𝐹 ↾ 𝐵) ↾ (𝐵 ∖ 𝐶))))) |
100 | 72 | resabs1d 5665 |
. . . . 5
⊢ (𝜑 → ((𝐹 ↾ 𝐵) ↾ 𝐶) = (𝐹 ↾ 𝐶)) |
101 | 100 | oveq2d 6922 |
. . . 4
⊢ (𝜑 → (𝐺 Σg ((𝐹 ↾ 𝐵) ↾ 𝐶)) = (𝐺 Σg (𝐹 ↾ 𝐶))) |
102 | | difss 3965 |
. . . . . 6
⊢ (𝐵 ∖ 𝐶) ⊆ 𝐵 |
103 | | resabs1 5664 |
. . . . . 6
⊢ ((𝐵 ∖ 𝐶) ⊆ 𝐵 → ((𝐹 ↾ 𝐵) ↾ (𝐵 ∖ 𝐶)) = (𝐹 ↾ (𝐵 ∖ 𝐶))) |
104 | 102, 103 | mp1i 13 |
. . . . 5
⊢ (𝜑 → ((𝐹 ↾ 𝐵) ↾ (𝐵 ∖ 𝐶)) = (𝐹 ↾ (𝐵 ∖ 𝐶))) |
105 | 104 | oveq2d 6922 |
. . . 4
⊢ (𝜑 → (𝐺 Σg ((𝐹 ↾ 𝐵) ↾ (𝐵 ∖ 𝐶))) = (𝐺 Σg (𝐹 ↾ (𝐵 ∖ 𝐶)))) |
106 | 101, 105 | oveq12d 6924 |
. . 3
⊢ (𝜑 → ((𝐺 Σg ((𝐹 ↾ 𝐵) ↾ 𝐶)) +𝑒 (𝐺 Σg ((𝐹 ↾ 𝐵) ↾ (𝐵 ∖ 𝐶)))) = ((𝐺 Σg (𝐹 ↾ 𝐶)) +𝑒 (𝐺 Σg (𝐹 ↾ (𝐵 ∖ 𝐶))))) |
107 | 99, 106 | eqtr2d 2863 |
. 2
⊢ (𝜑 → ((𝐺 Σg (𝐹 ↾ 𝐶)) +𝑒 (𝐺 Σg (𝐹 ↾ (𝐵 ∖ 𝐶)))) = (𝐺 Σg (𝐹 ↾ 𝐵))) |
108 | 84, 86, 107 | 3brtr3d 4905 |
1
⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ 𝐶)) ≤ (𝐺 Σg (𝐹 ↾ 𝐵))) |