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Theorem xaddrid 13223

Description: Extended real version of addrid 11395. (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 13099	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		0re 11217	. . . . 5 ⊢ 0 ∈ ℝ
3		rexadd 13214	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 +_𝑒 0) = (𝐴 + 0))
4	2, 3	mpan2 688	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 0) = (𝐴 + 0))
5		recn 11199	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
6	5	addridd 11415	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 + 0) = 𝐴)
7	4, 6	eqtrd 2766	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 0) = 𝐴)
8		0xr 11262	. . . . 5 ⊢ 0 ∈ ℝ^*
9		renemnf 11264	. . . . . 6 ⊢ (0 ∈ ℝ → 0 ≠ -∞)
10	2, 9	ax-mp 5	. . . . 5 ⊢ 0 ≠ -∞
11		xaddpnf2 13209	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 0 ≠ -∞) → (+∞ +_𝑒 0) = +∞)
12	8, 10, 11	mp2an 689	. . . 4 ⊢ (+∞ +_𝑒 0) = +∞
13		oveq1 7411	. . . 4 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 0) = (+∞ +_𝑒 0))
14		id 22	. . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞)
15	12, 13, 14	3eqtr4a 2792	. . 3 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 0) = 𝐴)
16		renepnf 11263	. . . . . 6 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
17	2, 16	ax-mp 5	. . . . 5 ⊢ 0 ≠ +∞
18		xaddmnf2 13211	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 0 ≠ +∞) → (-∞ +_𝑒 0) = -∞)
19	8, 17, 18	mp2an 689	. . . 4 ⊢ (-∞ +_𝑒 0) = -∞
20		oveq1 7411	. . . 4 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 0) = (-∞ +_𝑒 0))
21		id 22	. . . 4 ⊢ (𝐴 = -∞ → 𝐴 = -∞)
22	19, 20, 21	3eqtr4a 2792	. . 3 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 0) = 𝐴)
23	7, 15, 22	3jaoi 1424	. 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 1083 = wceq 1533 ∈ wcel 2098 ≠ wne 2934 (class class class)co 7404 ℝcr 11108 0cc0 11109 + caddc 11112 +∞cpnf 11246 -∞cmnf 11247 ℝ^*cxr 11248 +_𝑒 cxad 13093

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-xadd 13096

This theorem is referenced by: xaddlid 13224 xaddridd 13225 xnn0xadd0 13229 xpncan 13233 xadddi 13277 xrsnsgrp 21292 imasdsf1olem 24230 vtxdlfgrval 29247 vtxdginducedm1 29305 xraddge02 32474 xlt2addrd 32476 xrs0 32679 xrge0addgt0 32693 xrge0npcan 32696 metideq 33403 metider 33404 esumpad 33583 esumpr2 33595 esumpfinvallem 33602 esumpmono 33607 ddemeas 33764 aean 33772 baselcarsg 33835 carsgclctunlem2 33848 xadd0ge 44583 sge0tsms 45649 sge0ss 45681

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