Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > xaddid1 | GIF version |
Description: Extended real version of addid1 7900. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddid1 | ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9563 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | 0re 7766 | . . . . 5 ⊢ 0 ∈ ℝ | |
3 | rexadd 9635 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 +𝑒 0) = (𝐴 + 0)) | |
4 | 2, 3 | mpan2 421 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 +𝑒 0) = (𝐴 + 0)) |
5 | recn 7753 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
6 | 5 | addid1d 7911 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 + 0) = 𝐴) |
7 | 4, 6 | eqtrd 2172 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 +𝑒 0) = 𝐴) |
8 | 0xr 7812 | . . . . 5 ⊢ 0 ∈ ℝ* | |
9 | renemnf 7814 | . . . . . 6 ⊢ (0 ∈ ℝ → 0 ≠ -∞) | |
10 | 2, 9 | ax-mp 5 | . . . . 5 ⊢ 0 ≠ -∞ |
11 | xaddpnf2 9630 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ 0 ≠ -∞) → (+∞ +𝑒 0) = +∞) | |
12 | 8, 10, 11 | mp2an 422 | . . . 4 ⊢ (+∞ +𝑒 0) = +∞ |
13 | oveq1 5781 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 +𝑒 0) = (+∞ +𝑒 0)) | |
14 | id 19 | . . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞) | |
15 | 12, 13, 14 | 3eqtr4a 2198 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 +𝑒 0) = 𝐴) |
16 | renepnf 7813 | . . . . . 6 ⊢ (0 ∈ ℝ → 0 ≠ +∞) | |
17 | 2, 16 | ax-mp 5 | . . . . 5 ⊢ 0 ≠ +∞ |
18 | xaddmnf2 9632 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ 0 ≠ +∞) → (-∞ +𝑒 0) = -∞) | |
19 | 8, 17, 18 | mp2an 422 | . . . 4 ⊢ (-∞ +𝑒 0) = -∞ |
20 | oveq1 5781 | . . . 4 ⊢ (𝐴 = -∞ → (𝐴 +𝑒 0) = (-∞ +𝑒 0)) | |
21 | id 19 | . . . 4 ⊢ (𝐴 = -∞ → 𝐴 = -∞) | |
22 | 19, 20, 21 | 3eqtr4a 2198 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 +𝑒 0) = 𝐴) |
23 | 7, 15, 22 | 3jaoi 1281 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 +𝑒 0) = 𝐴) |
24 | 1, 23 | sylbi 120 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ w3o 961 = wceq 1331 ∈ wcel 1480 ≠ wne 2308 (class class class)co 5774 ℝcr 7619 0cc0 7620 + caddc 7623 +∞cpnf 7797 -∞cmnf 7798 ℝ*cxr 7799 +𝑒 cxad 9557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 ax-0id 7728 ax-rnegex 7729 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-xadd 9560 |
This theorem is referenced by: xaddid2 9646 xaddid1d 9647 xnn0xadd0 9650 xpncan 9654 psmetsym 12498 psmetge0 12500 xmetge0 12534 xmetsym 12537 |
Copyright terms: Public domain | W3C validator |