Proof of Theorem sge0pr
| Step | Hyp | Ref
| Expression |
| 1 | | iccssxr 13470 |
. . . . . . 7
⊢
(0[,]+∞) ⊆ ℝ* |
| 2 | | sge0pr.e |
. . . . . . 7
⊢ (𝜑 → 𝐸 ∈ (0[,]+∞)) |
| 3 | 1, 2 | sselid 3981 |
. . . . . 6
⊢ (𝜑 → 𝐸 ∈
ℝ*) |
| 4 | | mnfxr 11318 |
. . . . . . . 8
⊢ -∞
∈ ℝ* |
| 5 | 4 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → -∞ ∈
ℝ*) |
| 6 | | 0xr 11308 |
. . . . . . . . 9
⊢ 0 ∈
ℝ* |
| 7 | 6 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 0 ∈
ℝ*) |
| 8 | | mnflt0 13167 |
. . . . . . . . 9
⊢ -∞
< 0 |
| 9 | 8 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → -∞ <
0) |
| 10 | | pnfxr 11315 |
. . . . . . . . . 10
⊢ +∞
∈ ℝ* |
| 11 | 10 | a1i 11 |
. . . . . . . . 9
⊢ (𝜑 → +∞ ∈
ℝ*) |
| 12 | | iccgelb 13443 |
. . . . . . . . 9
⊢ ((0
∈ ℝ* ∧ +∞ ∈ ℝ* ∧
𝐸 ∈ (0[,]+∞))
→ 0 ≤ 𝐸) |
| 13 | 7, 11, 2, 12 | syl3anc 1373 |
. . . . . . . 8
⊢ (𝜑 → 0 ≤ 𝐸) |
| 14 | 5, 7, 3, 9, 13 | xrltletrd 13203 |
. . . . . . 7
⊢ (𝜑 → -∞ < 𝐸) |
| 15 | 5, 3, 14 | xrgtned 45333 |
. . . . . 6
⊢ (𝜑 → 𝐸 ≠ -∞) |
| 16 | | xaddpnf2 13269 |
. . . . . 6
⊢ ((𝐸 ∈ ℝ*
∧ 𝐸 ≠ -∞)
→ (+∞ +𝑒 𝐸) = +∞) |
| 17 | 3, 15, 16 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → (+∞
+𝑒 𝐸) =
+∞) |
| 18 | 17 | eqcomd 2743 |
. . . 4
⊢ (𝜑 → +∞ = (+∞
+𝑒 𝐸)) |
| 19 | 18 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝐷 = +∞) → +∞ = (+∞
+𝑒 𝐸)) |
| 20 | | prex 5437 |
. . . . 5
⊢ {𝐴, 𝐵} ∈ V |
| 21 | 20 | a1i 11 |
. . . 4
⊢ ((𝜑 ∧ 𝐷 = +∞) → {𝐴, 𝐵} ∈ V) |
| 22 | | sge0pr.cd |
. . . . . . . . . 10
⊢ (𝑘 = 𝐴 → 𝐶 = 𝐷) |
| 23 | 22 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷) |
| 24 | | sge0pr.d |
. . . . . . . . . 10
⊢ (𝜑 → 𝐷 ∈ (0[,]+∞)) |
| 25 | 24 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐷 ∈ (0[,]+∞)) |
| 26 | 23, 25 | eqeltrd 2841 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,]+∞)) |
| 27 | 26 | adantlr 715 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,]+∞)) |
| 28 | | simpll 767 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝜑) |
| 29 | | simpl 482 |
. . . . . . . . . 10
⊢ ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘 ∈ {𝐴, 𝐵}) |
| 30 | | neqne 2948 |
. . . . . . . . . . 11
⊢ (¬
𝑘 = 𝐴 → 𝑘 ≠ 𝐴) |
| 31 | 30 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘 ≠ 𝐴) |
| 32 | | elprn1 45648 |
. . . . . . . . . 10
⊢ ((𝑘 ∈ {𝐴, 𝐵} ∧ 𝑘 ≠ 𝐴) → 𝑘 = 𝐵) |
| 33 | 29, 31, 32 | syl2anc 584 |
. . . . . . . . 9
⊢ ((𝑘 ∈ {𝐴, 𝐵} ∧ ¬ 𝑘 = 𝐴) → 𝑘 = 𝐵) |
| 34 | 33 | adantll 714 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝑘 = 𝐵) |
| 35 | | sge0pr.ce |
. . . . . . . . . 10
⊢ (𝑘 = 𝐵 → 𝐶 = 𝐸) |
| 36 | 35 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸) |
| 37 | 2 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐸 ∈ (0[,]+∞)) |
| 38 | 36, 37 | eqeltrd 2841 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 = 𝐵) → 𝐶 ∈ (0[,]+∞)) |
| 39 | 28, 34, 38 | syl2anc 584 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝐶 ∈ (0[,]+∞)) |
| 40 | 27, 39 | pm2.61dan 813 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝐶 ∈ (0[,]+∞)) |
| 41 | | eqid 2737 |
. . . . . 6
⊢ (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶) = (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶) |
| 42 | 40, 41 | fmptd 7134 |
. . . . 5
⊢ (𝜑 → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶):{𝐴, 𝐵}⟶(0[,]+∞)) |
| 43 | 42 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝐷 = +∞) → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶):{𝐴, 𝐵}⟶(0[,]+∞)) |
| 44 | | id 22 |
. . . . . . 7
⊢ (𝐷 = +∞ → 𝐷 = +∞) |
| 45 | 44 | eqcomd 2743 |
. . . . . 6
⊢ (𝐷 = +∞ → +∞ =
𝐷) |
| 46 | 45 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝐷 = +∞) → +∞ = 𝐷) |
| 47 | | prid1g 4760 |
. . . . . . . 8
⊢ (𝐷 ∈ (0[,]+∞) →
𝐷 ∈ {𝐷, 𝐸}) |
| 48 | 24, 47 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐷 ∈ {𝐷, 𝐸}) |
| 49 | | sge0pr.a |
. . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| 50 | | sge0pr.b |
. . . . . . . . 9
⊢ (𝜑 → 𝐵 ∈ 𝑊) |
| 51 | 49, 50, 41, 22, 35 | rnmptpr 45182 |
. . . . . . . 8
⊢ (𝜑 → ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶) = {𝐷, 𝐸}) |
| 52 | 51 | eqcomd 2743 |
. . . . . . 7
⊢ (𝜑 → {𝐷, 𝐸} = ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) |
| 53 | 48, 52 | eleqtrd 2843 |
. . . . . 6
⊢ (𝜑 → 𝐷 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) |
| 54 | 53 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝐷 = +∞) → 𝐷 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) |
| 55 | 46, 54 | eqeltrd 2841 |
. . . 4
⊢ ((𝜑 ∧ 𝐷 = +∞) → +∞ ∈ ran
(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) |
| 56 | 21, 43, 55 | sge0pnfval 46388 |
. . 3
⊢ ((𝜑 ∧ 𝐷 = +∞) →
(Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = +∞) |
| 57 | | oveq1 7438 |
. . . 4
⊢ (𝐷 = +∞ → (𝐷 +𝑒 𝐸) = (+∞
+𝑒 𝐸)) |
| 58 | 57 | adantl 481 |
. . 3
⊢ ((𝜑 ∧ 𝐷 = +∞) → (𝐷 +𝑒 𝐸) = (+∞ +𝑒 𝐸)) |
| 59 | 19, 56, 58 | 3eqtr4d 2787 |
. 2
⊢ ((𝜑 ∧ 𝐷 = +∞) →
(Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸)) |
| 60 | 1, 24 | sselid 3981 |
. . . . . . . 8
⊢ (𝜑 → 𝐷 ∈
ℝ*) |
| 61 | | iccgelb 13443 |
. . . . . . . . . . 11
⊢ ((0
∈ ℝ* ∧ +∞ ∈ ℝ* ∧
𝐷 ∈ (0[,]+∞))
→ 0 ≤ 𝐷) |
| 62 | 7, 11, 24, 61 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (𝜑 → 0 ≤ 𝐷) |
| 63 | 5, 7, 60, 9, 62 | xrltletrd 13203 |
. . . . . . . . 9
⊢ (𝜑 → -∞ < 𝐷) |
| 64 | 5, 60, 63 | xrgtned 45333 |
. . . . . . . 8
⊢ (𝜑 → 𝐷 ≠ -∞) |
| 65 | | xaddpnf1 13268 |
. . . . . . . 8
⊢ ((𝐷 ∈ ℝ*
∧ 𝐷 ≠ -∞)
→ (𝐷
+𝑒 +∞) = +∞) |
| 66 | 60, 64, 65 | syl2anc 584 |
. . . . . . 7
⊢ (𝜑 → (𝐷 +𝑒 +∞) =
+∞) |
| 67 | 66 | eqcomd 2743 |
. . . . . 6
⊢ (𝜑 → +∞ = (𝐷 +𝑒
+∞)) |
| 68 | 67 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝐸 = +∞) → +∞ = (𝐷 +𝑒
+∞)) |
| 69 | 20 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐸 = +∞) → {𝐴, 𝐵} ∈ V) |
| 70 | 42 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐸 = +∞) → (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶):{𝐴, 𝐵}⟶(0[,]+∞)) |
| 71 | | id 22 |
. . . . . . . . 9
⊢ (𝐸 = +∞ → 𝐸 = +∞) |
| 72 | 71 | eqcomd 2743 |
. . . . . . . 8
⊢ (𝐸 = +∞ → +∞ =
𝐸) |
| 73 | 72 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐸 = +∞) → +∞ = 𝐸) |
| 74 | | prid2g 4761 |
. . . . . . . . . 10
⊢ (𝐸 ∈ (0[,]+∞) →
𝐸 ∈ {𝐷, 𝐸}) |
| 75 | 2, 74 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐸 ∈ {𝐷, 𝐸}) |
| 76 | 75, 52 | eleqtrd 2843 |
. . . . . . . 8
⊢ (𝜑 → 𝐸 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) |
| 77 | 76 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐸 = +∞) → 𝐸 ∈ ran (𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) |
| 78 | 73, 77 | eqeltrd 2841 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐸 = +∞) → +∞ ∈ ran
(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) |
| 79 | 69, 70, 78 | sge0pnfval 46388 |
. . . . 5
⊢ ((𝜑 ∧ 𝐸 = +∞) →
(Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = +∞) |
| 80 | | oveq2 7439 |
. . . . . 6
⊢ (𝐸 = +∞ → (𝐷 +𝑒 𝐸) = (𝐷 +𝑒
+∞)) |
| 81 | 80 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝐸 = +∞) → (𝐷 +𝑒 𝐸) = (𝐷 +𝑒
+∞)) |
| 82 | 68, 79, 81 | 3eqtr4d 2787 |
. . . 4
⊢ ((𝜑 ∧ 𝐸 = +∞) →
(Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸)) |
| 83 | 82 | adantlr 715 |
. . 3
⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝐸 = +∞) →
(Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸)) |
| 84 | | rge0ssre 13496 |
. . . . . . . 8
⊢
(0[,)+∞) ⊆ ℝ |
| 85 | | ax-resscn 11212 |
. . . . . . . 8
⊢ ℝ
⊆ ℂ |
| 86 | 84, 85 | sstri 3993 |
. . . . . . 7
⊢
(0[,)+∞) ⊆ ℂ |
| 87 | 6 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 0 ∈
ℝ*) |
| 88 | 10 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → +∞ ∈
ℝ*) |
| 89 | 60 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ∈
ℝ*) |
| 90 | 62 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 0 ≤ 𝐷) |
| 91 | | pnfge 13172 |
. . . . . . . . . . . 12
⊢ (𝐷 ∈ ℝ*
→ 𝐷 ≤
+∞) |
| 92 | 60, 91 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐷 ≤ +∞) |
| 93 | 92 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ≤ +∞) |
| 94 | 44 | necon3bi 2967 |
. . . . . . . . . . 11
⊢ (¬
𝐷 = +∞ → 𝐷 ≠ +∞) |
| 95 | 94 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ≠ +∞) |
| 96 | 89, 88, 93, 95 | xrleneltd 45334 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 < +∞) |
| 97 | 87, 88, 89, 90, 96 | elicod 13437 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) → 𝐷 ∈ (0[,)+∞)) |
| 98 | 97 | adantr 480 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐷 ∈ (0[,)+∞)) |
| 99 | 86, 98 | sselid 3981 |
. . . . . 6
⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐷 ∈ ℂ) |
| 100 | 6 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 0 ∈
ℝ*) |
| 101 | 10 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → +∞ ∈
ℝ*) |
| 102 | 3 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈
ℝ*) |
| 103 | 13 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 0 ≤ 𝐸) |
| 104 | | pnfge 13172 |
. . . . . . . . . . . 12
⊢ (𝐸 ∈ ℝ*
→ 𝐸 ≤
+∞) |
| 105 | 3, 104 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐸 ≤ +∞) |
| 106 | 105 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ≤ +∞) |
| 107 | 71 | necon3bi 2967 |
. . . . . . . . . . 11
⊢ (¬
𝐸 = +∞ → 𝐸 ≠ +∞) |
| 108 | 107 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ≠ +∞) |
| 109 | 102, 101,
106, 108 | xrleneltd 45334 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 < +∞) |
| 110 | 100, 101,
102, 103, 109 | elicod 13437 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ (0[,)+∞)) |
| 111 | 86, 110 | sselid 3981 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℂ) |
| 112 | 111 | adantlr 715 |
. . . . . 6
⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℂ) |
| 113 | 99, 112 | jca 511 |
. . . . 5
⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (𝐷 ∈ ℂ ∧ 𝐸 ∈ ℂ)) |
| 114 | 49, 50 | jca 511 |
. . . . . 6
⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) |
| 115 | 114 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) |
| 116 | | sge0pr.ab |
. . . . . 6
⊢ (𝜑 → 𝐴 ≠ 𝐵) |
| 117 | 116 | ad2antrr 726 |
. . . . 5
⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐴 ≠ 𝐵) |
| 118 | 22, 35, 113, 115, 117 | sumpr 15784 |
. . . 4
⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → Σ𝑘 ∈ {𝐴, 𝐵}𝐶 = (𝐷 + 𝐸)) |
| 119 | | prfi 9363 |
. . . . . 6
⊢ {𝐴, 𝐵} ∈ Fin |
| 120 | 119 | a1i 11 |
. . . . 5
⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → {𝐴, 𝐵} ∈ Fin) |
| 121 | 22 | adantl 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝑘 = 𝐴) → 𝐶 = 𝐷) |
| 122 | 97 | adantr 480 |
. . . . . . . 8
⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝑘 = 𝐴) → 𝐷 ∈ (0[,)+∞)) |
| 123 | 121, 122 | eqeltrd 2841 |
. . . . . . 7
⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,)+∞)) |
| 124 | 123 | ad4ant14 752 |
. . . . . 6
⊢
(((((𝜑 ∧ ¬
𝐷 = +∞) ∧ ¬
𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ 𝑘 = 𝐴) → 𝐶 ∈ (0[,)+∞)) |
| 125 | | simp-4l 783 |
. . . . . . 7
⊢
(((((𝜑 ∧ ¬
𝐷 = +∞) ∧ ¬
𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝜑) |
| 126 | | simpllr 776 |
. . . . . . 7
⊢
(((((𝜑 ∧ ¬
𝐷 = +∞) ∧ ¬
𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → ¬ 𝐸 = +∞) |
| 127 | 33 | adantll 714 |
. . . . . . 7
⊢
(((((𝜑 ∧ ¬
𝐷 = +∞) ∧ ¬
𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝑘 = 𝐵) |
| 128 | 36 | 3adant2 1132 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵) → 𝐶 = 𝐸) |
| 129 | 110 | 3adant3 1133 |
. . . . . . . 8
⊢ ((𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵) → 𝐸 ∈ (0[,)+∞)) |
| 130 | 128, 129 | eqeltrd 2841 |
. . . . . . 7
⊢ ((𝜑 ∧ ¬ 𝐸 = +∞ ∧ 𝑘 = 𝐵) → 𝐶 ∈ (0[,)+∞)) |
| 131 | 125, 126,
127, 130 | syl3anc 1373 |
. . . . . 6
⊢
(((((𝜑 ∧ ¬
𝐷 = +∞) ∧ ¬
𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) ∧ ¬ 𝑘 = 𝐴) → 𝐶 ∈ (0[,)+∞)) |
| 132 | 124, 131 | pm2.61dan 813 |
. . . . 5
⊢ ((((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) ∧ 𝑘 ∈ {𝐴, 𝐵}) → 𝐶 ∈ (0[,)+∞)) |
| 133 | 120, 132 | sge0fsummpt 46405 |
. . . 4
⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) →
(Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = Σ𝑘 ∈ {𝐴, 𝐵}𝐶) |
| 134 | 84, 98 | sselid 3981 |
. . . . 5
⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐷 ∈ ℝ) |
| 135 | 84, 110 | sselid 3981 |
. . . . . 6
⊢ ((𝜑 ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℝ) |
| 136 | 135 | adantlr 715 |
. . . . 5
⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → 𝐸 ∈ ℝ) |
| 137 | | rexadd 13274 |
. . . . 5
⊢ ((𝐷 ∈ ℝ ∧ 𝐸 ∈ ℝ) → (𝐷 +𝑒 𝐸) = (𝐷 + 𝐸)) |
| 138 | 134, 136,
137 | syl2anc 584 |
. . . 4
⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) → (𝐷 +𝑒 𝐸) = (𝐷 + 𝐸)) |
| 139 | 118, 133,
138 | 3eqtr4d 2787 |
. . 3
⊢ (((𝜑 ∧ ¬ 𝐷 = +∞) ∧ ¬ 𝐸 = +∞) →
(Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸)) |
| 140 | 83, 139 | pm2.61dan 813 |
. 2
⊢ ((𝜑 ∧ ¬ 𝐷 = +∞) →
(Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸)) |
| 141 | 59, 140 | pm2.61dan 813 |
1
⊢ (𝜑 →
(Σ^‘(𝑘 ∈ {𝐴, 𝐵} ↦ 𝐶)) = (𝐷 +𝑒 𝐸)) |