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Mirrors > Home > ILE Home > Th. List > xaddid1 | GIF version |
Description: Extended real version of addid1 7868. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddid1 | ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9531 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | 0re 7734 | . . . . 5 ⊢ 0 ∈ ℝ | |
3 | rexadd 9603 | . . . . 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 7721 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
6 | 5 | addid1d 7879 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 + 0) = 𝐴) |
7 | 4, 6 | eqtrd 2150 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 +𝑒 0) = 𝐴) |
8 | 0xr 7780 | . . . . 5 ⊢ 0 ∈ ℝ* | |
9 | renemnf 7782 | . . . . . 6 ⊢ (0 ∈ ℝ → 0 ≠ -∞) | |
10 | 2, 9 | ax-mp 5 | . . . . 5 ⊢ 0 ≠ -∞ |
11 | xaddpnf2 9598 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ 0 ≠ -∞) → (+∞ +𝑒 0) = +∞) | |
12 | 8, 10, 11 | mp2an 422 | . . . 4 ⊢ (+∞ +𝑒 0) = +∞ |
13 | oveq1 5749 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 +𝑒 0) = (+∞ +𝑒 0)) | |
14 | id 19 | . . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞) | |
15 | 12, 13, 14 | 3eqtr4a 2176 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 +𝑒 0) = 𝐴) |
16 | renepnf 7781 | . . . . . 6 ⊢ (0 ∈ ℝ → 0 ≠ +∞) | |
17 | 2, 16 | ax-mp 5 | . . . . 5 ⊢ 0 ≠ +∞ |
18 | xaddmnf2 9600 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ 0 ≠ +∞) → (-∞ +𝑒 0) = -∞) | |
19 | 8, 17, 18 | mp2an 422 | . . . 4 ⊢ (-∞ +𝑒 0) = -∞ |
20 | oveq1 5749 | . . . 4 ⊢ (𝐴 = -∞ → (𝐴 +𝑒 0) = (-∞ +𝑒 0)) | |
21 | id 19 | . . . 4 ⊢ (𝐴 = -∞ → 𝐴 = -∞) | |
22 | 19, 20, 21 | 3eqtr4a 2176 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 +𝑒 0) = 𝐴) |
23 | 7, 15, 22 | 3jaoi 1266 | . 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 946 = wceq 1316 ∈ wcel 1465 ≠ wne 2285 (class class class)co 5742 ℝcr 7587 0cc0 7588 + caddc 7591 +∞cpnf 7765 -∞cmnf 7766 ℝ*cxr 7767 +𝑒 cxad 9525 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-1re 7682 ax-addrcl 7685 ax-0id 7696 ax-rnegex 7697 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-xadd 9528 |
This theorem is referenced by: xaddid2 9614 xaddid1d 9615 xnn0xadd0 9618 xpncan 9622 psmetsym 12425 psmetge0 12427 xmetge0 12461 xmetsym 12464 |
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