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Theorem xaddrid 13184

Description: Extended real version of addrid 11317. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xaddrid	⊢ (𝐴 ∈ ℝ^* → (𝐴 +_𝑒 0) = 𝐴)

**Proof of Theorem xaddrid**
Step	Hyp	Ref	Expression
1		elxr 13058	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		0re 11137	. . . . 5 ⊢ 0 ∈ ℝ
3		rexadd 13175	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 +_𝑒 0) = (𝐴 + 0))
4	2, 3	mpan2 697	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 0) = (𝐴 + 0))
5		recn 11119	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
6	5	addridd 11337	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 + 0) = 𝐴)
7	4, 6	eqtrd 2774	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 0) = 𝐴)
8		0xr 11183	. . . . 5 ⊢ 0 ∈ ℝ^*
9		renemnf 11185	. . . . . 6 ⊢ (0 ∈ ℝ → 0 ≠ -∞)
10	2, 9	ax-mp 5	. . . . 5 ⊢ 0 ≠ -∞
11		xaddpnf2 13170	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 0 ≠ -∞) → (+∞ +_𝑒 0) = +∞)
12	8, 10, 11	mp2an 698	. . . 4 ⊢ (+∞ +_𝑒 0) = +∞
13		oveq1 7363	. . . 4 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 0) = (+∞ +_𝑒 0))
14		id 22	. . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞)
15	12, 13, 14	3eqtr4a 2800	. . 3 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 0) = 𝐴)
16		renepnf 11184	. . . . . 6 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
17	2, 16	ax-mp 5	. . . . 5 ⊢ 0 ≠ +∞
18		xaddmnf2 13172	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 0 ≠ +∞) → (-∞ +_𝑒 0) = -∞)
19	8, 17, 18	mp2an 698	. . . 4 ⊢ (-∞ +_𝑒 0) = -∞
20		oveq1 7363	. . . 4 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 0) = (-∞ +_𝑒 0))
21		id 22	. . . 4 ⊢ (𝐴 = -∞ → 𝐴 = -∞)
22	19, 20, 21	3eqtr4a 2800	. . 3 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 0) = 𝐴)
23	7, 15, 22	3jaoi 1436	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 +_𝑒 0) = 𝐴)
24	1, 23	sylbi 218	1 ⊢ (𝐴 ∈ ℝ^* → (𝐴 +_𝑒 0) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ w3o 1091 = wceq 1547 ∈ wcel 2119 ≠ wne 2934 (class class class)co 7356 ℝcr 11028 0cc0 11029 + caddc 11032 +∞cpnf 11167 -∞cmnf 11168 ℝ^*cxr 11169 +_𝑒 cxad 13052

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-br 5073 df-opab 5135 df-mpt 5154 df-id 5513 df-po 5526 df-so 5527 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-ov 7359 df-oprab 7360 df-mpo 7361 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-xadd 13055

This theorem is referenced by: xaddlid 13185 xaddridd 13186 xnn0xadd0 13190 xpncan 13194 xadddi 13238 xrsnsgrp 21383 imasdsf1olem 24356 vtxdlfgrval 29572 vtxdginducedm1 29630 xraddge02 32849 xlt2addrd 32851 xrs0 33085 xrge0addgt0 33096 xrge0npcan 33099 metideq 34077 metider 34078 esumpad 34239 esumpr2 34251 esumpfinvallem 34258 esumpmono 34263 ddemeas 34420 aean 34428 baselcarsg 34490 carsgclctunlem2 34503 xadd0ge 45767 sge0tsms 46823 sge0ss 46855

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