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Theorem xaddrid 13268

Description: Extended real version of addrid 11391. (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 13142	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		0re 11211	. . . . 5 ⊢ 0 ∈ ℝ
3		rexadd 13259	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 +_𝑒 0) = (𝐴 + 0))
4	2, 3	mpan2 703	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 0) = (𝐴 + 0))
5		recn 11191	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
6	5	addridd 11411	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 + 0) = 𝐴)
7	4, 6	eqtrd 2798	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 0) = 𝐴)
8		0xr 11257	. . . . 5 ⊢ 0 ∈ ℝ^*
9		renemnf 11259	. . . . . 6 ⊢ (0 ∈ ℝ → 0 ≠ -∞)
10	2, 9	ax-mp 5	. . . . 5 ⊢ 0 ≠ -∞
11		xaddpnf2 13254	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 0 ≠ -∞) → (+∞ +_𝑒 0) = +∞)
12	8, 10, 11	mp2an 704	. . . 4 ⊢ (+∞ +_𝑒 0) = +∞
13		oveq1 7419	. . . 4 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 0) = (+∞ +_𝑒 0))
14		id 23	. . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞)
15	12, 13, 14	3eqtr4a 2824	. . 3 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 0) = 𝐴)
16		renepnf 11258	. . . . . 6 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
17	2, 16	ax-mp 5	. . . . 5 ⊢ 0 ≠ +∞
18		xaddmnf2 13256	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 0 ≠ +∞) → (-∞ +_𝑒 0) = -∞)
19	8, 17, 18	mp2an 704	. . . 4 ⊢ (-∞ +_𝑒 0) = -∞
20		oveq1 7419	. . . 4 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 0) = (-∞ +_𝑒 0))
21		id 23	. . . 4 ⊢ (𝐴 = -∞ → 𝐴 = -∞)
22	19, 20, 21	3eqtr4a 2824	. . 3 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 0) = 𝐴)
23	7, 15, 22	3jaoi 1454	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 +_𝑒 0) = 𝐴)
24	1, 23	sylbi 220	1 ⊢ (𝐴 ∈ ℝ^* → (𝐴 +_𝑒 0) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ w3o 1102 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 (class class class)co 7412 ℝcr 11100 0cc0 11101 + caddc 11104 +∞cpnf 11241 -∞cmnf 11242 ℝ^*cxr 11243 +_𝑒 cxad 13136

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-br 5111 df-opab 5175 df-mpt 5194 df-id 5558 df-po 5571 df-so 5572 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8695 df-en 8945 df-dom 8946 df-sdom 8947 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-xadd 13139

This theorem is referenced by: xaddlid 13269 xaddridd 13270 xnn0xadd0 13274 xpncan 13278 xadddi 13322 xrsnsgrp 21539 imasdsf1olem 24511 vtxdlfgrval 29816 vtxdginducedm1 29874 xraddge02 33083 xlt2addrd 33085 xrs0 33307 xrge0addgt0 33318 xrge0npcan 33321 metideq 34264 metider 34265 esumpad 34426 esumpr2 34438 esumpfinvallem 34445 esumpmono 34450 ddemeas 34607 aean 34615 baselcarsg 34677 carsgclctunlem2 34690 xadd0ge 46021 sge0tsms 47077 sge0ss 47109

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