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Mirrors > Home > MPE Home > Th. List > xaddid1 | Structured version Visualization version GIF version |
Description: Extended real version of addid1 11340. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xaddid1 | ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 13042 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | 0re 11162 | . . . . 5 ⊢ 0 ∈ ℝ | |
3 | rexadd 13157 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 +𝑒 0) = (𝐴 + 0)) | |
4 | 2, 3 | mpan2 690 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 +𝑒 0) = (𝐴 + 0)) |
5 | recn 11146 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
6 | 5 | addid1d 11360 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 + 0) = 𝐴) |
7 | 4, 6 | eqtrd 2773 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 +𝑒 0) = 𝐴) |
8 | 0xr 11207 | . . . . 5 ⊢ 0 ∈ ℝ* | |
9 | renemnf 11209 | . . . . . 6 ⊢ (0 ∈ ℝ → 0 ≠ -∞) | |
10 | 2, 9 | ax-mp 5 | . . . . 5 ⊢ 0 ≠ -∞ |
11 | xaddpnf2 13152 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ 0 ≠ -∞) → (+∞ +𝑒 0) = +∞) | |
12 | 8, 10, 11 | mp2an 691 | . . . 4 ⊢ (+∞ +𝑒 0) = +∞ |
13 | oveq1 7365 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 +𝑒 0) = (+∞ +𝑒 0)) | |
14 | id 22 | . . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞) | |
15 | 12, 13, 14 | 3eqtr4a 2799 | . . 3 ⊢ (𝐴 = +∞ → (𝐴 +𝑒 0) = 𝐴) |
16 | renepnf 11208 | . . . . . 6 ⊢ (0 ∈ ℝ → 0 ≠ +∞) | |
17 | 2, 16 | ax-mp 5 | . . . . 5 ⊢ 0 ≠ +∞ |
18 | xaddmnf2 13154 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ 0 ≠ +∞) → (-∞ +𝑒 0) = -∞) | |
19 | 8, 17, 18 | mp2an 691 | . . . 4 ⊢ (-∞ +𝑒 0) = -∞ |
20 | oveq1 7365 | . . . 4 ⊢ (𝐴 = -∞ → (𝐴 +𝑒 0) = (-∞ +𝑒 0)) | |
21 | id 22 | . . . 4 ⊢ (𝐴 = -∞ → 𝐴 = -∞) | |
22 | 19, 20, 21 | 3eqtr4a 2799 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 +𝑒 0) = 𝐴) |
23 | 7, 15, 22 | 3jaoi 1428 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 +𝑒 0) = 𝐴) |
24 | 1, 23 | sylbi 216 | 1 ⊢ (𝐴 ∈ ℝ* → (𝐴 +𝑒 0) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∨ w3o 1087 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 (class class class)co 7358 ℝcr 11055 0cc0 11056 + caddc 11059 +∞cpnf 11191 -∞cmnf 11192 ℝ*cxr 11193 +𝑒 cxad 13036 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-xadd 13039 |
This theorem is referenced by: xaddid2 13167 xaddid1d 13168 xnn0xadd0 13172 xpncan 13176 xadddi 13220 xrsnsgrp 20849 imasdsf1olem 23742 vtxdlfgrval 28475 vtxdginducedm1 28533 xraddge02 31708 xlt2addrd 31710 xrs0 31915 xrge0addgt0 31931 xrge0npcan 31934 metideq 32531 metider 32532 esumpad 32711 esumpr2 32723 esumpfinvallem 32730 esumpmono 32735 ddemeas 32892 aean 32900 baselcarsg 32963 carsgclctunlem2 32976 xadd0ge 43641 sge0tsms 44707 sge0ss 44739 |
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