xaddid1 - Metamath Proof Explorer

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Theorem xaddid1 12015

Description: Extended real version of addid1 10160. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xaddid1	⊢ (𝐴 ∈ ℝ^* → (𝐴 +_𝑒 0) = 𝐴)

**Proof of Theorem xaddid1**
Step	Hyp	Ref	Expression
1		elxr 11894	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		0re 9984	. . . . 5 ⊢ 0 ∈ ℝ
3		rexadd 12006	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ∈ ℝ) → (𝐴 +_𝑒 0) = (𝐴 + 0))
4	2, 3	mpan2 706	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 0) = (𝐴 + 0))
5		recn 9970	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
6	5	addid1d 10180	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 + 0) = 𝐴)
7	4, 6	eqtrd 2655	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 0) = 𝐴)
8		0xr 10030	. . . . 5 ⊢ 0 ∈ ℝ^*
9		renemnf 10032	. . . . . 6 ⊢ (0 ∈ ℝ → 0 ≠ -∞)
10	2, 9	ax-mp 5	. . . . 5 ⊢ 0 ≠ -∞
11		xaddpnf2 12001	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 0 ≠ -∞) → (+∞ +_𝑒 0) = +∞)
12	8, 10, 11	mp2an 707	. . . 4 ⊢ (+∞ +_𝑒 0) = +∞
13		oveq1 6611	. . . 4 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 0) = (+∞ +_𝑒 0))
14		id 22	. . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞)
15	12, 13, 14	3eqtr4a 2681	. . 3 ⊢ (𝐴 = +∞ → (𝐴 +_𝑒 0) = 𝐴)
16		renepnf 10031	. . . . . 6 ⊢ (0 ∈ ℝ → 0 ≠ +∞)
17	2, 16	ax-mp 5	. . . . 5 ⊢ 0 ≠ +∞
18		xaddmnf2 12003	. . . . 5 ⊢ ((0 ∈ ℝ^* ∧ 0 ≠ +∞) → (-∞ +_𝑒 0) = -∞)
19	8, 17, 18	mp2an 707	. . . 4 ⊢ (-∞ +_𝑒 0) = -∞
20		oveq1 6611	. . . 4 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 0) = (-∞ +_𝑒 0))
21		id 22	. . . 4 ⊢ (𝐴 = -∞ → 𝐴 = -∞)
22	19, 20, 21	3eqtr4a 2681	. . 3 ⊢ (𝐴 = -∞ → (𝐴 +_𝑒 0) = 𝐴)
23	7, 15, 22	3jaoi 1388	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 +_𝑒 0) = 𝐴)
24	1, 23	sylbi 207	1 ⊢ (𝐴 ∈ ℝ^* → (𝐴 +_𝑒 0) = 𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∨ w3o 1035 = wceq 1480 ∈ wcel 1987 ≠ wne 2790 (class class class)co 6604 ℝcr 9879 0cc0 9880 + caddc 9883 +∞cpnf 10015 -∞cmnf 10016 ℝ^*cxr 10017 +_𝑒 cxad 11888

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1719 ax-4 1734 ax-5 1836 ax-6 1885 ax-7 1932 ax-8 1989 ax-9 1996 ax-10 2016 ax-11 2031 ax-12 2044 ax-13 2245 ax-ext 2601 ax-sep 4741 ax-nul 4749 ax-pow 4803 ax-pr 4867 ax-un 6902 ax-cnex 9936 ax-resscn 9937 ax-1cn 9938 ax-icn 9939 ax-addcl 9940 ax-addrcl 9941 ax-mulcl 9942 ax-mulrcl 9943 ax-mulcom 9944 ax-addass 9945 ax-mulass 9946 ax-distr 9947 ax-i2m1 9948 ax-1ne0 9949 ax-1rid 9950 ax-rnegex 9951 ax-rrecex 9952 ax-cnre 9953 ax-pre-lttri 9954 ax-pre-lttrn 9955 ax-pre-ltadd 9956

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1037 df-3an 1038 df-tru 1483 df-ex 1702 df-nf 1707 df-sb 1878 df-eu 2473 df-mo 2474 df-clab 2608 df-cleq 2614 df-clel 2617 df-nfc 2750 df-ne 2791 df-nel 2894 df-ral 2912 df-rex 2913 df-rab 2916 df-v 3188 df-sbc 3418 df-csb 3515 df-dif 3558 df-un 3560 df-in 3562 df-ss 3569 df-nul 3892 df-if 4059 df-pw 4132 df-sn 4149 df-pr 4151 df-op 4155 df-uni 4403 df-br 4614 df-opab 4674 df-mpt 4675 df-id 4989 df-po 4995 df-so 4996 df-xp 5080 df-rel 5081 df-cnv 5082 df-co 5083 df-dm 5084 df-rn 5085 df-res 5086 df-ima 5087 df-iota 5810 df-fun 5849 df-fn 5850 df-f 5851 df-f1 5852 df-fo 5853 df-f1o 5854 df-fv 5855 df-ov 6607 df-oprab 6608 df-mpt2 6609 df-er 7687 df-en 7900 df-dom 7901 df-sdom 7902 df-pnf 10020 df-mnf 10021 df-xr 10022 df-ltxr 10023 df-xadd 11891

This theorem is referenced by: xaddid2 12016 xaddid1d 12017 xnn0xadd0 12020 xpncan 12024 xadddi 12068 xadddi2 12070 xrsnsgrp 19701 xrs1mnd 19703 xrs10 19704 psmetsym 22025 xmetsym 22062 imasdsf1olem 22088 stdbdxmet 22230 xrge0gsumle 22544 metdsle 22563 metnrmlem1 22570 vtxdlfgrval 26267 xraddge02 29362 xlt2addrd 29364 xrs0 29457 xrge0addgt0 29473 xrge0npcan 29476 metideq 29715 metider 29716 esumpad 29895 esumpr2 29907 esumpfinvallem 29914 esumpmono 29919 ddemeas 30077 aean 30085 baselcarsg 30146 carsgclctunlem2 30159 xadd0ge 38997 sge0tsms 39901 sge0ss 39933

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