xaddcom - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > xaddcom		Structured version Visualization version GIF version

Theorem xaddcom 13302

Description: The extended real addition operation is commutative. (Contributed by NM, 26-Dec-2011.)

**Assertion**
Ref	Expression
xaddcom	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))

**Proof of Theorem xaddcom**
Step	Hyp	Ref	Expression
1		elxr 13179	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 13179	. . . 4 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		recn 11274	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
4		recn 11274	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ)
5		addcom 11476	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
6	3, 4, 5	syl2an 595	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
7		rexadd 13294	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +_𝑒 𝐵) = (𝐴 + 𝐵))
8		rexadd 13294	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 +_𝑒 𝐴) = (𝐵 + 𝐴))
9	8	ancoms 458	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 +_𝑒 𝐴) = (𝐵 + 𝐴))
10	6, 7, 9	3eqtr4d 2790	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
11		oveq2 7456	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 +∞))
12		rexr 11336	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
13		renemnf 11339	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
14		xaddpnf1 13288	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 +∞) = +∞)
15	12, 13, 14	syl2anc 583	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 +∞) = +∞)
16	11, 15	sylan9eqr 2802	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = +∞)
17		oveq1 7455	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐵 +_𝑒 𝐴) = (+∞ +_𝑒 𝐴))
18		xaddpnf2 13289	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (+∞ +_𝑒 𝐴) = +∞)
19	12, 13, 18	syl2anc 583	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (+∞ +_𝑒 𝐴) = +∞)
20	17, 19	sylan9eqr 2802	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐵 +_𝑒 𝐴) = +∞)
21	16, 20	eqtr4d 2783	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
22		oveq2 7456	. . . . . . 7 ⊢ (𝐵 = -∞ → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 -∞))
23		renepnf 11338	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
24		xaddmnf1 13290	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (𝐴 +_𝑒 -∞) = -∞)
25	12, 23, 24	syl2anc 583	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 -∞) = -∞)
26	22, 25	sylan9eqr 2802	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 +_𝑒 𝐵) = -∞)
27		oveq1 7455	. . . . . . 7 ⊢ (𝐵 = -∞ → (𝐵 +_𝑒 𝐴) = (-∞ +_𝑒 𝐴))
28		xaddmnf2 13291	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (-∞ +_𝑒 𝐴) = -∞)
29	12, 23, 28	syl2anc 583	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (-∞ +_𝑒 𝐴) = -∞)
30	27, 29	sylan9eqr 2802	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐵 +_𝑒 𝐴) = -∞)
31	26, 30	eqtr4d 2783	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
32	10, 21, 31	3jaodan 1431	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
33	2, 32	sylan2b 593	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
34		pnfaddmnf 13292	. . . . . . . 8 ⊢ (+∞ +_𝑒 -∞) = 0
35		mnfaddpnf 13293	. . . . . . . 8 ⊢ (-∞ +_𝑒 +∞) = 0
36	34, 35	eqtr4i 2771	. . . . . . 7 ⊢ (+∞ +_𝑒 -∞) = (-∞ +_𝑒 +∞)
37		simpr 484	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → 𝐵 = -∞)
38	37	oveq2d 7464	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (+∞ +_𝑒 𝐵) = (+∞ +_𝑒 -∞))
39	37	oveq1d 7463	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐵 +_𝑒 +∞) = (-∞ +_𝑒 +∞))
40	36, 38, 39	3eqtr4a 2806	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (+∞ +_𝑒 𝐵) = (𝐵 +_𝑒 +∞))
41		xaddpnf2 13289	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = +∞)
42		xaddpnf1 13288	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (𝐵 +_𝑒 +∞) = +∞)
43	41, 42	eqtr4d 2783	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = (𝐵 +_𝑒 +∞))
44	40, 43	pm2.61dane 3035	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (+∞ +_𝑒 𝐵) = (𝐵 +_𝑒 +∞))
45	44	adantl 481	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (+∞ +_𝑒 𝐵) = (𝐵 +_𝑒 +∞))
46		simpl 482	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → 𝐴 = +∞)
47	46	oveq1d 7463	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (+∞ +_𝑒 𝐵))
48	46	oveq2d 7464	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐵 +_𝑒 𝐴) = (𝐵 +_𝑒 +∞))
49	45, 47, 48	3eqtr4d 2790	. . 3 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
50	35, 34	eqtr4i 2771	. . . . . . 7 ⊢ (-∞ +_𝑒 +∞) = (+∞ +_𝑒 -∞)
51		simpr 484	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → 𝐵 = +∞)
52	51	oveq2d 7464	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (-∞ +_𝑒 𝐵) = (-∞ +_𝑒 +∞))
53	51	oveq1d 7463	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (𝐵 +_𝑒 -∞) = (+∞ +_𝑒 -∞))
54	50, 52, 53	3eqtr4a 2806	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (-∞ +_𝑒 𝐵) = (𝐵 +_𝑒 -∞))
55		xaddmnf2 13291	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = -∞)
56		xaddmnf1 13290	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (𝐵 +_𝑒 -∞) = -∞)
57	55, 56	eqtr4d 2783	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = (𝐵 +_𝑒 -∞))
58	54, 57	pm2.61dane 3035	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (-∞ +_𝑒 𝐵) = (𝐵 +_𝑒 -∞))
59	58	adantl 481	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (-∞ +_𝑒 𝐵) = (𝐵 +_𝑒 -∞))
60		simpl 482	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → 𝐴 = -∞)
61	60	oveq1d 7463	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (-∞ +_𝑒 𝐵))
62	60	oveq2d 7464	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐵 +_𝑒 𝐴) = (𝐵 +_𝑒 -∞))
63	59, 61, 62	3eqtr4d 2790	. . 3 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
64	33, 49, 63	3jaoian 1430	. 2 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
65	1, 64	sylanb 580	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∨ w3o 1086 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 + caddc 11187 +∞cpnf 11321 -∞cmnf 11322 ℝ^*cxr 11323 +_𝑒 cxad 13173

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-po 5607 df-so 5608 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-ov 7451 df-oprab 7452 df-mpo 7453 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-xadd 13176

This theorem is referenced by: xaddlid 13304 xleadd2a 13316 xltadd2 13319 xposdif 13324 xadd4d 13365 hashunx 14435 xrsnsgrp 21443 xrs1cmn 21447 blcld 24539 xrsxmet 24850 metdstri 24892 vtxdginducedm1 29579 xaddeq0 32760 xlt2addrd 32765 xrge0npcan 33006 esumle 34022 esumlef 34026 measun 34175 difelcarsg 34275 xaddcomd 45239

W3C validator