xaddcom - Metamath Proof Explorer

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Theorem xaddcom 13215

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 13092	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 13092	. . . 4 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		recn 11196	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
4		recn 11196	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ)
5		addcom 11396	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
6	3, 4, 5	syl2an 596	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
7		rexadd 13207	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +_𝑒 𝐵) = (𝐴 + 𝐵))
8		rexadd 13207	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 +_𝑒 𝐴) = (𝐵 + 𝐴))
9	8	ancoms 459	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 +_𝑒 𝐴) = (𝐵 + 𝐴))
10	6, 7, 9	3eqtr4d 2782	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
11		oveq2 7413	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 +∞))
12		rexr 11256	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
13		renemnf 11259	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
14		xaddpnf1 13201	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 +∞) = +∞)
15	12, 13, 14	syl2anc 584	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 +∞) = +∞)
16	11, 15	sylan9eqr 2794	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = +∞)
17		oveq1 7412	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐵 +_𝑒 𝐴) = (+∞ +_𝑒 𝐴))
18		xaddpnf2 13202	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (+∞ +_𝑒 𝐴) = +∞)
19	12, 13, 18	syl2anc 584	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (+∞ +_𝑒 𝐴) = +∞)
20	17, 19	sylan9eqr 2794	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐵 +_𝑒 𝐴) = +∞)
21	16, 20	eqtr4d 2775	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
22		oveq2 7413	. . . . . . 7 ⊢ (𝐵 = -∞ → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 -∞))
23		renepnf 11258	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
24		xaddmnf1 13203	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (𝐴 +_𝑒 -∞) = -∞)
25	12, 23, 24	syl2anc 584	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 -∞) = -∞)
26	22, 25	sylan9eqr 2794	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 +_𝑒 𝐵) = -∞)
27		oveq1 7412	. . . . . . 7 ⊢ (𝐵 = -∞ → (𝐵 +_𝑒 𝐴) = (-∞ +_𝑒 𝐴))
28		xaddmnf2 13204	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (-∞ +_𝑒 𝐴) = -∞)
29	12, 23, 28	syl2anc 584	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (-∞ +_𝑒 𝐴) = -∞)
30	27, 29	sylan9eqr 2794	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐵 +_𝑒 𝐴) = -∞)
31	26, 30	eqtr4d 2775	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
32	10, 21, 31	3jaodan 1430	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
33	2, 32	sylan2b 594	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
34		pnfaddmnf 13205	. . . . . . . 8 ⊢ (+∞ +_𝑒 -∞) = 0
35		mnfaddpnf 13206	. . . . . . . 8 ⊢ (-∞ +_𝑒 +∞) = 0
36	34, 35	eqtr4i 2763	. . . . . . 7 ⊢ (+∞ +_𝑒 -∞) = (-∞ +_𝑒 +∞)
37		simpr 485	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → 𝐵 = -∞)
38	37	oveq2d 7421	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (+∞ +_𝑒 𝐵) = (+∞ +_𝑒 -∞))
39	37	oveq1d 7420	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐵 +_𝑒 +∞) = (-∞ +_𝑒 +∞))
40	36, 38, 39	3eqtr4a 2798	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (+∞ +_𝑒 𝐵) = (𝐵 +_𝑒 +∞))
41		xaddpnf2 13202	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = +∞)
42		xaddpnf1 13201	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (𝐵 +_𝑒 +∞) = +∞)
43	41, 42	eqtr4d 2775	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = (𝐵 +_𝑒 +∞))
44	40, 43	pm2.61dane 3029	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (+∞ +_𝑒 𝐵) = (𝐵 +_𝑒 +∞))
45	44	adantl 482	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (+∞ +_𝑒 𝐵) = (𝐵 +_𝑒 +∞))
46		simpl 483	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → 𝐴 = +∞)
47	46	oveq1d 7420	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (+∞ +_𝑒 𝐵))
48	46	oveq2d 7421	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐵 +_𝑒 𝐴) = (𝐵 +_𝑒 +∞))
49	45, 47, 48	3eqtr4d 2782	. . 3 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
50	35, 34	eqtr4i 2763	. . . . . . 7 ⊢ (-∞ +_𝑒 +∞) = (+∞ +_𝑒 -∞)
51		simpr 485	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → 𝐵 = +∞)
52	51	oveq2d 7421	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (-∞ +_𝑒 𝐵) = (-∞ +_𝑒 +∞))
53	51	oveq1d 7420	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (𝐵 +_𝑒 -∞) = (+∞ +_𝑒 -∞))
54	50, 52, 53	3eqtr4a 2798	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (-∞ +_𝑒 𝐵) = (𝐵 +_𝑒 -∞))
55		xaddmnf2 13204	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = -∞)
56		xaddmnf1 13203	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (𝐵 +_𝑒 -∞) = -∞)
57	55, 56	eqtr4d 2775	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = (𝐵 +_𝑒 -∞))
58	54, 57	pm2.61dane 3029	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (-∞ +_𝑒 𝐵) = (𝐵 +_𝑒 -∞))
59	58	adantl 482	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (-∞ +_𝑒 𝐵) = (𝐵 +_𝑒 -∞))
60		simpl 483	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → 𝐴 = -∞)
61	60	oveq1d 7420	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (-∞ +_𝑒 𝐵))
62	60	oveq2d 7421	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐵 +_𝑒 𝐴) = (𝐵 +_𝑒 -∞))
63	59, 61, 62	3eqtr4d 2782	. . 3 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
64	33, 49, 63	3jaoian 1429	. 2 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
65	1, 64	sylanb 581	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∨ w3o 1086 = wceq 1541 ∈ wcel 2106 ≠ wne 2940 (class class class)co 7405 ℂcc 11104 ℝcr 11105 0cc0 11106 + caddc 11109 +∞cpnf 11241 -∞cmnf 11242 ℝ^*cxr 11243 +_𝑒 cxad 13086

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7408 df-oprab 7409 df-mpo 7410 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-xadd 13089

This theorem is referenced by: xaddlid 13217 xleadd2a 13229 xltadd2 13232 xposdif 13237 xadd4d 13278 hashunx 14342 xrsnsgrp 20973 xrs1cmn 20977 blcld 24005 xrsxmet 24316 metdstri 24358 vtxdginducedm1 28789 xaddeq0 31953 xlt2addrd 31958 xrge0npcan 32182 esumle 33044 esumlef 33048 measun 33197 difelcarsg 33297 xaddcomd 44020

W3C validator