xaddrid - Metamath Proof Explorer

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Theorem xaddrid 13293

Description: Extended real version of addrid 11414. (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 13167	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		0re 11234	. . . . 5 ⊢ 0 ∈ ℝ
3		rexadd 13284	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 +_𝑒 0) = (𝐴 + 0))
4	2, 3	mpan2 704	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 0) = (𝐴 + 0))
5		recn 11214	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
6	5	addridd 11434	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 + 0) = 𝐴)
7	4, 6	eqtrd 2795	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 0) = 𝐴)
8		0xr 11280	. . . . 5 ⊢ 0 ∈ ℝ^*
9		renemnf 11282	. . . . . 6 ⊢ (0 ∈ ℝ → 0 ≠ -∞)
10	2, 9	ax-mp 5	. . . . 5 ⊢ 0 ≠ -∞
11		xaddpnf2 13279	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 0 ≠ -∞) → (+∞ +_𝑒 0) = +∞)
12	8, 10, 11	mp2an 705	. . . 4 ⊢ (+∞ +_𝑒 0) = +∞
13		oveq1 7420	. . . 4 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 0) = (+∞ +_𝑒 0))
14		id 23	. . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞)
15	12, 13, 14	3eqtr4a 2821	. . 3 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 0) = 𝐴)
16		renepnf 11281	. . . . . 6 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
17	2, 16	ax-mp 5	. . . . 5 ⊢ 0 ≠ +∞
18		xaddmnf2 13281	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 0 ≠ +∞) → (-∞ +_𝑒 0) = -∞)
19	8, 17, 18	mp2an 705	. . . 4 ⊢ (-∞ +_𝑒 0) = -∞
20		oveq1 7420	. . . 4 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 0) = (-∞ +_𝑒 0))
21		id 23	. . . 4 ⊢ (𝐴 = -∞ → 𝐴 = -∞)
22	19, 20, 21	3eqtr4a 2821	. . 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

This proof depends on syntax axioms: → wi 4 ∨ w3o 1102 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 (class class class)co 7413 ℝcr 11123 0cc0 11124 + caddc 11127 +∞cpnf 11264 -∞cmnf 11265 ℝ^*cxr 11266 +_𝑒 cxad 13161

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-po 5563 df-so 5564 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7416 df-oprab 7417 df-mpo 7418 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-xadd 13164

This theorem is used by: xaddlid 13294 xaddridd 13295 xnn0xadd0 13299 xpncan 13303 xadddi 13347 xrsnsgrp 21621 imasdsf1olem 24599 vtxdlfgrval 29945 vtxdginducedm1 30003 xraddge02 33228 xlt2addrd 33230 xrs0 33446 xrge0addgt0 33457 xrge0npcan 33460 metideq 34403 metider 34404 esumpad 34565 esumpr2 34577 esumpfinvallem 34584 esumpmono 34589 ddemeas 34747 aean 34755 baselcarsg 34817 carsgclctunlem2 34830 xadd0ge 46152 sge0tsms 47208 sge0ss 47240

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