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Mirrors > Home > MPE Home > Th. List > xaddnepnf | Structured version Visualization version GIF version |
Description: Closure of extended real addition in the subset ℝ* / {+∞}. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddnepnf | ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ∧ (𝐵 ∈ ℝ* ∧ 𝐵 ≠ +∞)) → (𝐴 +𝑒 𝐵) ≠ +∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrnepnf 12516 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = -∞)) | |
2 | xrnepnf 12516 | . . . 4 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐵 ≠ +∞) ↔ (𝐵 ∈ ℝ ∨ 𝐵 = -∞)) | |
3 | rexadd 12628 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) = (𝐴 + 𝐵)) | |
4 | readdcl 10622 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ) | |
5 | 3, 4 | eqeltrd 2915 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) ∈ ℝ) |
6 | 5 | renepnfd 10694 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +𝑒 𝐵) ≠ +∞) |
7 | oveq2 7166 | . . . . . . 7 ⊢ (𝐵 = -∞ → (𝐴 +𝑒 𝐵) = (𝐴 +𝑒 -∞)) | |
8 | rexr 10689 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
9 | renepnf 10691 | . . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞) | |
10 | xaddmnf1 12624 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) → (𝐴 +𝑒 -∞) = -∞) | |
11 | 8, 9, 10 | syl2anc 586 | . . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 +𝑒 -∞) = -∞) |
12 | 7, 11 | sylan9eqr 2880 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 +𝑒 𝐵) = -∞) |
13 | mnfnepnf 10699 | . . . . . . 7 ⊢ -∞ ≠ +∞ | |
14 | 13 | a1i 11 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → -∞ ≠ +∞) |
15 | 12, 14 | eqnetrd 3085 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 +𝑒 𝐵) ≠ +∞) |
16 | 6, 15 | jaodan 954 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = -∞)) → (𝐴 +𝑒 𝐵) ≠ +∞) |
17 | 2, 16 | sylan2b 595 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ* ∧ 𝐵 ≠ +∞)) → (𝐴 +𝑒 𝐵) ≠ +∞) |
18 | oveq1 7165 | . . . . 5 ⊢ (𝐴 = -∞ → (𝐴 +𝑒 𝐵) = (-∞ +𝑒 𝐵)) | |
19 | xaddmnf2 12625 | . . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐵 ≠ +∞) → (-∞ +𝑒 𝐵) = -∞) | |
20 | 18, 19 | sylan9eq 2878 | . . . 4 ⊢ ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ* ∧ 𝐵 ≠ +∞)) → (𝐴 +𝑒 𝐵) = -∞) |
21 | 13 | a1i 11 | . . . 4 ⊢ ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ* ∧ 𝐵 ≠ +∞)) → -∞ ≠ +∞) |
22 | 20, 21 | eqnetrd 3085 | . . 3 ⊢ ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ* ∧ 𝐵 ≠ +∞)) → (𝐴 +𝑒 𝐵) ≠ +∞) |
23 | 17, 22 | jaoian 953 | . 2 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ* ∧ 𝐵 ≠ +∞)) → (𝐴 +𝑒 𝐵) ≠ +∞) |
24 | 1, 23 | sylanb 583 | 1 ⊢ (((𝐴 ∈ ℝ* ∧ 𝐴 ≠ +∞) ∧ (𝐵 ∈ ℝ* ∧ 𝐵 ≠ +∞)) → (𝐴 +𝑒 𝐵) ≠ +∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 (class class class)co 7158 ℝcr 10538 + caddc 10542 +∞cpnf 10674 -∞cmnf 10675 ℝ*cxr 10676 +𝑒 cxad 12508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-i2m1 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-xadd 12511 |
This theorem is referenced by: xlt2add 12656 |
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