| Step | Hyp | Ref
| Expression |
| 1 | | df-esum 34029 |
. . 3
⊢
Σ*𝑘
∈ {𝑀}𝐴 = ∪
((ℝ*𝑠 ↾s (0[,]+∞))
tsums (𝑘 ∈ {𝑀} ↦ 𝐴)) |
| 2 | 1 | a1i 11 |
. 2
⊢ (𝜑 → Σ*𝑘 ∈ {𝑀}𝐴 = ∪
((ℝ*𝑠 ↾s (0[,]+∞))
tsums (𝑘 ∈ {𝑀} ↦ 𝐴))) |
| 3 | | eqid 2737 |
. . . 4
⊢
(ℝ*𝑠 ↾s
(0[,]+∞)) = (ℝ*𝑠 ↾s
(0[,]+∞)) |
| 4 | | snfi 9083 |
. . . . 5
⊢ {𝑀} ∈ Fin |
| 5 | 4 | a1i 11 |
. . . 4
⊢ (𝜑 → {𝑀} ∈ Fin) |
| 6 | | elsni 4643 |
. . . . . . . . 9
⊢ (𝑘 ∈ {𝑀} → 𝑘 = 𝑀) |
| 7 | | esumsnf.1 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 = 𝑀) → 𝐴 = 𝐵) |
| 8 | 6, 7 | sylan2 593 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ {𝑀}) → 𝐴 = 𝐵) |
| 9 | 8 | mpteq2dva 5242 |
. . . . . . 7
⊢ (𝜑 → (𝑘 ∈ {𝑀} ↦ 𝐴) = (𝑘 ∈ {𝑀} ↦ 𝐵)) |
| 10 | | esumsnf.2 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ 𝑉) |
| 11 | | esumsnf.3 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈ (0[,]+∞)) |
| 12 | | fmptsn 7187 |
. . . . . . . . 9
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ (0[,]+∞)) → {〈𝑀, 𝐵〉} = (𝑙 ∈ {𝑀} ↦ 𝐵)) |
| 13 | | nfcv 2905 |
. . . . . . . . . 10
⊢
Ⅎ𝑙𝐵 |
| 14 | | esumsnf.0 |
. . . . . . . . . 10
⊢
Ⅎ𝑘𝐵 |
| 15 | | eqidd 2738 |
. . . . . . . . . 10
⊢ (𝑘 = 𝑙 → 𝐵 = 𝐵) |
| 16 | 13, 14, 15 | cbvmpt 5253 |
. . . . . . . . 9
⊢ (𝑘 ∈ {𝑀} ↦ 𝐵) = (𝑙 ∈ {𝑀} ↦ 𝐵) |
| 17 | 12, 16 | eqtr4di 2795 |
. . . . . . . 8
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ (0[,]+∞)) → {〈𝑀, 𝐵〉} = (𝑘 ∈ {𝑀} ↦ 𝐵)) |
| 18 | 10, 11, 17 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → {〈𝑀, 𝐵〉} = (𝑘 ∈ {𝑀} ↦ 𝐵)) |
| 19 | 9, 18 | eqtr4d 2780 |
. . . . . 6
⊢ (𝜑 → (𝑘 ∈ {𝑀} ↦ 𝐴) = {〈𝑀, 𝐵〉}) |
| 20 | | fsng 7157 |
. . . . . . 7
⊢ ((𝑀 ∈ 𝑉 ∧ 𝐵 ∈ (0[,]+∞)) → ((𝑘 ∈ {𝑀} ↦ 𝐴):{𝑀}⟶{𝐵} ↔ (𝑘 ∈ {𝑀} ↦ 𝐴) = {〈𝑀, 𝐵〉})) |
| 21 | 10, 11, 20 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → ((𝑘 ∈ {𝑀} ↦ 𝐴):{𝑀}⟶{𝐵} ↔ (𝑘 ∈ {𝑀} ↦ 𝐴) = {〈𝑀, 𝐵〉})) |
| 22 | 19, 21 | mpbird 257 |
. . . . 5
⊢ (𝜑 → (𝑘 ∈ {𝑀} ↦ 𝐴):{𝑀}⟶{𝐵}) |
| 23 | 11 | snssd 4809 |
. . . . 5
⊢ (𝜑 → {𝐵} ⊆ (0[,]+∞)) |
| 24 | 22, 23 | fssd 6753 |
. . . 4
⊢ (𝜑 → (𝑘 ∈ {𝑀} ↦ 𝐴):{𝑀}⟶(0[,]+∞)) |
| 25 | | xrltso 13183 |
. . . . . . 7
⊢ < Or
ℝ* |
| 26 | 25 | a1i 11 |
. . . . . 6
⊢ (𝜑 → < Or
ℝ*) |
| 27 | | 0xr 11308 |
. . . . . . 7
⊢ 0 ∈
ℝ* |
| 28 | 27 | a1i 11 |
. . . . . 6
⊢ (𝜑 → 0 ∈
ℝ*) |
| 29 | | elxrge0 13497 |
. . . . . . . 8
⊢ (𝐵 ∈ (0[,]+∞) ↔
(𝐵 ∈
ℝ* ∧ 0 ≤ 𝐵)) |
| 30 | 11, 29 | sylib 218 |
. . . . . . 7
⊢ (𝜑 → (𝐵 ∈ ℝ* ∧ 0 ≤
𝐵)) |
| 31 | 30 | simpld 494 |
. . . . . 6
⊢ (𝜑 → 𝐵 ∈
ℝ*) |
| 32 | | suppr 9511 |
. . . . . 6
⊢ (( <
Or ℝ* ∧ 0 ∈ ℝ* ∧ 𝐵 ∈ ℝ*)
→ sup({0, 𝐵},
ℝ*, < ) = if(𝐵 < 0, 0, 𝐵)) |
| 33 | 26, 28, 31, 32 | syl3anc 1373 |
. . . . 5
⊢ (𝜑 → sup({0, 𝐵}, ℝ*, < ) = if(𝐵 < 0, 0, 𝐵)) |
| 34 | | 0fi 9082 |
. . . . . . . . . . 11
⊢ ∅
∈ Fin |
| 35 | 34 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → ∅ ∈
Fin) |
| 36 | | reseq2 5992 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = ∅ → ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ 𝑥) = ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ ∅)) |
| 37 | | res0 6001 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ ∅) =
∅ |
| 38 | 36, 37 | eqtrdi 2793 |
. . . . . . . . . . . . 13
⊢ (𝑥 = ∅ → ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ 𝑥) = ∅) |
| 39 | 38 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (𝑥 = ∅ →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ 𝑥)) =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg ∅)) |
| 40 | | xrge00 33017 |
. . . . . . . . . . . . 13
⊢ 0 =
(0g‘(ℝ*𝑠
↾s (0[,]+∞))) |
| 41 | 40 | gsum0 18697 |
. . . . . . . . . . . 12
⊢
((ℝ*𝑠 ↾s
(0[,]+∞)) Σg ∅) = 0 |
| 42 | 39, 41 | eqtrdi 2793 |
. . . . . . . . . . 11
⊢ (𝑥 = ∅ →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ 𝑥)) = 0) |
| 43 | 42 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 = ∅) →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ 𝑥)) = 0) |
| 44 | | reseq2 5992 |
. . . . . . . . . . . . 13
⊢ (𝑥 = {𝑀} → ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ 𝑥) = ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ {𝑀})) |
| 45 | | ssid 4006 |
. . . . . . . . . . . . . 14
⊢ {𝑀} ⊆ {𝑀} |
| 46 | | resmpt 6055 |
. . . . . . . . . . . . . 14
⊢ ({𝑀} ⊆ {𝑀} → ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ {𝑀}) = (𝑘 ∈ {𝑀} ↦ 𝐴)) |
| 47 | 45, 46 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ {𝑀}) = (𝑘 ∈ {𝑀} ↦ 𝐴) |
| 48 | 44, 47 | eqtrdi 2793 |
. . . . . . . . . . . 12
⊢ (𝑥 = {𝑀} → ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ 𝑥) = (𝑘 ∈ {𝑀} ↦ 𝐴)) |
| 49 | 48 | oveq2d 7447 |
. . . . . . . . . . 11
⊢ (𝑥 = {𝑀} →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ 𝑥)) =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ {𝑀} ↦ 𝐴))) |
| 50 | | xrge0base 33016 |
. . . . . . . . . . . 12
⊢
(0[,]+∞) = (Base‘(ℝ*𝑠
↾s (0[,]+∞))) |
| 51 | | xrge0cmn 21426 |
. . . . . . . . . . . . . 14
⊢
(ℝ*𝑠 ↾s
(0[,]+∞)) ∈ CMnd |
| 52 | | cmnmnd 19815 |
. . . . . . . . . . . . . 14
⊢
((ℝ*𝑠 ↾s
(0[,]+∞)) ∈ CMnd → (ℝ*𝑠
↾s (0[,]+∞)) ∈ Mnd) |
| 53 | 51, 52 | ax-mp 5 |
. . . . . . . . . . . . 13
⊢
(ℝ*𝑠 ↾s
(0[,]+∞)) ∈ Mnd |
| 54 | 53 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 →
(ℝ*𝑠 ↾s (0[,]+∞))
∈ Mnd) |
| 55 | | nfv 1914 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑘𝜑 |
| 56 | 50, 54, 10, 11, 7, 55, 14 | gsumsnfd 19969 |
. . . . . . . . . . 11
⊢ (𝜑 →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg (𝑘 ∈ {𝑀} ↦ 𝐴)) = 𝐵) |
| 57 | 49, 56 | sylan9eqr 2799 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 = {𝑀}) →
((ℝ*𝑠 ↾s (0[,]+∞))
Σg ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ 𝑥)) = 𝐵) |
| 58 | 35, 5, 28, 11, 43, 57 | fmptpr 7192 |
. . . . . . . . 9
⊢ (𝜑 → {〈∅, 0〉,
〈{𝑀}, 𝐵〉} = (𝑥 ∈ {∅, {𝑀}} ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ 𝑥)))) |
| 59 | | pwsn 4900 |
. . . . . . . . . . . . 13
⊢ 𝒫
{𝑀} = {∅, {𝑀}} |
| 60 | | prssi 4821 |
. . . . . . . . . . . . . 14
⊢ ((∅
∈ Fin ∧ {𝑀} ∈
Fin) → {∅, {𝑀}}
⊆ Fin) |
| 61 | 34, 4, 60 | mp2an 692 |
. . . . . . . . . . . . 13
⊢ {∅,
{𝑀}} ⊆
Fin |
| 62 | 59, 61 | eqsstri 4030 |
. . . . . . . . . . . 12
⊢ 𝒫
{𝑀} ⊆
Fin |
| 63 | | dfss2 3969 |
. . . . . . . . . . . 12
⊢
(𝒫 {𝑀}
⊆ Fin ↔ (𝒫 {𝑀} ∩ Fin) = 𝒫 {𝑀}) |
| 64 | 62, 63 | mpbi 230 |
. . . . . . . . . . 11
⊢
(𝒫 {𝑀} ∩
Fin) = 𝒫 {𝑀} |
| 65 | 64, 59 | eqtri 2765 |
. . . . . . . . . 10
⊢
(𝒫 {𝑀} ∩
Fin) = {∅, {𝑀}} |
| 66 | | eqid 2737 |
. . . . . . . . . 10
⊢
((ℝ*𝑠 ↾s
(0[,]+∞)) Σg ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ 𝑥)) =
((ℝ*𝑠 ↾s (0[,]+∞))
Σg ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ 𝑥)) |
| 67 | 65, 66 | mpteq12i 5248 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝒫 {𝑀} ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ 𝑥))) = (𝑥 ∈ {∅, {𝑀}} ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ 𝑥))) |
| 68 | 58, 67 | eqtr4di 2795 |
. . . . . . . 8
⊢ (𝜑 → {〈∅, 0〉,
〈{𝑀}, 𝐵〉} = (𝑥 ∈ (𝒫 {𝑀} ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ 𝑥)))) |
| 69 | 68 | rneqd 5949 |
. . . . . . 7
⊢ (𝜑 → ran {〈∅,
0〉, 〈{𝑀}, 𝐵〉} = ran (𝑥 ∈ (𝒫 {𝑀} ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ 𝑥)))) |
| 70 | | rnpropg 6242 |
. . . . . . . 8
⊢ ((∅
∈ Fin ∧ {𝑀} ∈
Fin) → ran {〈∅, 0〉, 〈{𝑀}, 𝐵〉} = {0, 𝐵}) |
| 71 | 35, 5, 70 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → ran {〈∅,
0〉, 〈{𝑀}, 𝐵〉} = {0, 𝐵}) |
| 72 | 69, 71 | eqtr3d 2779 |
. . . . . 6
⊢ (𝜑 → ran (𝑥 ∈ (𝒫 {𝑀} ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ 𝑥))) = {0, 𝐵}) |
| 73 | 72 | supeq1d 9486 |
. . . . 5
⊢ (𝜑 → sup(ran (𝑥 ∈ (𝒫 {𝑀} ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ 𝑥))), ℝ*, < ) = sup({0,
𝐵}, ℝ*,
< )) |
| 74 | 30 | simprd 495 |
. . . . . . . . 9
⊢ (𝜑 → 0 ≤ 𝐵) |
| 75 | | xrlenlt 11326 |
. . . . . . . . . 10
⊢ ((0
∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (0 ≤
𝐵 ↔ ¬ 𝐵 < 0)) |
| 76 | 28, 31, 75 | syl2anc 584 |
. . . . . . . . 9
⊢ (𝜑 → (0 ≤ 𝐵 ↔ ¬ 𝐵 < 0)) |
| 77 | 74, 76 | mpbid 232 |
. . . . . . . 8
⊢ (𝜑 → ¬ 𝐵 < 0) |
| 78 | | eqidd 2738 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 = 𝐵) |
| 79 | 77, 78 | jca 511 |
. . . . . . 7
⊢ (𝜑 → (¬ 𝐵 < 0 ∧ 𝐵 = 𝐵)) |
| 80 | 79 | olcd 875 |
. . . . . 6
⊢ (𝜑 → ((𝐵 < 0 ∧ 𝐵 = 0) ∨ (¬ 𝐵 < 0 ∧ 𝐵 = 𝐵))) |
| 81 | | eqif 4567 |
. . . . . 6
⊢ (𝐵 = if(𝐵 < 0, 0, 𝐵) ↔ ((𝐵 < 0 ∧ 𝐵 = 0) ∨ (¬ 𝐵 < 0 ∧ 𝐵 = 𝐵))) |
| 82 | 80, 81 | sylibr 234 |
. . . . 5
⊢ (𝜑 → 𝐵 = if(𝐵 < 0, 0, 𝐵)) |
| 83 | 33, 73, 82 | 3eqtr4rd 2788 |
. . . 4
⊢ (𝜑 → 𝐵 = sup(ran (𝑥 ∈ (𝒫 {𝑀} ∩ Fin) ↦
((ℝ*𝑠 ↾s (0[,]+∞))
Σg ((𝑘 ∈ {𝑀} ↦ 𝐴) ↾ 𝑥))), ℝ*, <
)) |
| 84 | 3, 5, 24, 83 | xrge0tsmsd 33065 |
. . 3
⊢ (𝜑 →
((ℝ*𝑠 ↾s (0[,]+∞))
tsums (𝑘 ∈ {𝑀} ↦ 𝐴)) = {𝐵}) |
| 85 | 84 | unieqd 4920 |
. 2
⊢ (𝜑 → ∪ ((ℝ*𝑠
↾s (0[,]+∞)) tsums (𝑘 ∈ {𝑀} ↦ 𝐴)) = ∪ {𝐵}) |
| 86 | | unisng 4925 |
. . 3
⊢ (𝐵 ∈ (0[,]+∞) →
∪ {𝐵} = 𝐵) |
| 87 | 11, 86 | syl 17 |
. 2
⊢ (𝜑 → ∪ {𝐵}
= 𝐵) |
| 88 | 2, 85, 87 | 3eqtrd 2781 |
1
⊢ (𝜑 → Σ*𝑘 ∈ {𝑀}𝐴 = 𝐵) |