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Theorem xaddcom 13207

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 13083	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 13083	. . . 4 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		recn 11165	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
4		recn 11165	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ)
5		addcom 11367	. . . . . . 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 13199	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +_𝑒 𝐵) = (𝐴 + 𝐵))
8		rexadd 13199	. . . . . . 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 2775	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
11		oveq2 7398	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 +∞))
12		rexr 11227	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
13		renemnf 11230	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
14		xaddpnf1 13193	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 +∞) = +∞)
15	12, 13, 14	syl2anc 584	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 +∞) = +∞)
16	11, 15	sylan9eqr 2787	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = +∞)
17		oveq1 7397	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐵 +_𝑒 𝐴) = (+∞ +_𝑒 𝐴))
18		xaddpnf2 13194	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (+∞ +_𝑒 𝐴) = +∞)
19	12, 13, 18	syl2anc 584	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (+∞ +_𝑒 𝐴) = +∞)
20	17, 19	sylan9eqr 2787	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐵 +_𝑒 𝐴) = +∞)
21	16, 20	eqtr4d 2768	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
22		oveq2 7398	. . . . . . 7 ⊢ (𝐵 = -∞ → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 -∞))
23		renepnf 11229	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
24		xaddmnf1 13195	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (𝐴 +_𝑒 -∞) = -∞)
25	12, 23, 24	syl2anc 584	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 -∞) = -∞)
26	22, 25	sylan9eqr 2787	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 +_𝑒 𝐵) = -∞)
27		oveq1 7397	. . . . . . 7 ⊢ (𝐵 = -∞ → (𝐵 +_𝑒 𝐴) = (-∞ +_𝑒 𝐴))
28		xaddmnf2 13196	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (-∞ +_𝑒 𝐴) = -∞)
29	12, 23, 28	syl2anc 584	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (-∞ +_𝑒 𝐴) = -∞)
30	27, 29	sylan9eqr 2787	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐵 +_𝑒 𝐴) = -∞)
31	26, 30	eqtr4d 2768	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
32	10, 21, 31	3jaodan 1433	. . . 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 13197	. . . . . . . 8 ⊢ (+∞ +_𝑒 -∞) = 0
35		mnfaddpnf 13198	. . . . . . . 8 ⊢ (-∞ +_𝑒 +∞) = 0
36	34, 35	eqtr4i 2756	. . . . . . 7 ⊢ (+∞ +_𝑒 -∞) = (-∞ +_𝑒 +∞)
37		simpr 484	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → 𝐵 = -∞)
38	37	oveq2d 7406	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (+∞ +_𝑒 𝐵) = (+∞ +_𝑒 -∞))
39	37	oveq1d 7405	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐵 +_𝑒 +∞) = (-∞ +_𝑒 +∞))
40	36, 38, 39	3eqtr4a 2791	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (+∞ +_𝑒 𝐵) = (𝐵 +_𝑒 +∞))
41		xaddpnf2 13194	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = +∞)
42		xaddpnf1 13193	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (𝐵 +_𝑒 +∞) = +∞)
43	41, 42	eqtr4d 2768	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = (𝐵 +_𝑒 +∞))
44	40, 43	pm2.61dane 3013	. . . . 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 7405	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (+∞ +_𝑒 𝐵))
48	46	oveq2d 7406	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐵 +_𝑒 𝐴) = (𝐵 +_𝑒 +∞))
49	45, 47, 48	3eqtr4d 2775	. . 3 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
50	35, 34	eqtr4i 2756	. . . . . . 7 ⊢ (-∞ +_𝑒 +∞) = (+∞ +_𝑒 -∞)
51		simpr 484	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → 𝐵 = +∞)
52	51	oveq2d 7406	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (-∞ +_𝑒 𝐵) = (-∞ +_𝑒 +∞))
53	51	oveq1d 7405	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (𝐵 +_𝑒 -∞) = (+∞ +_𝑒 -∞))
54	50, 52, 53	3eqtr4a 2791	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (-∞ +_𝑒 𝐵) = (𝐵 +_𝑒 -∞))
55		xaddmnf2 13196	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = -∞)
56		xaddmnf1 13195	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (𝐵 +_𝑒 -∞) = -∞)
57	55, 56	eqtr4d 2768	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = (𝐵 +_𝑒 -∞))
58	54, 57	pm2.61dane 3013	. . . . 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 7405	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (-∞ +_𝑒 𝐵))
62	60	oveq2d 7406	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐵 +_𝑒 𝐴) = (𝐵 +_𝑒 -∞))
63	59, 61, 62	3eqtr4d 2775	. . 3 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
64	33, 49, 63	3jaoian 1432	. 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 395 ∨ w3o 1085 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 (class class class)co 7390 ℂcc 11073 ℝcr 11074 0cc0 11075 + caddc 11078 +∞cpnf 11212 -∞cmnf 11213 ℝ^*cxr 11214 +_𝑒 cxad 13077

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-po 5549 df-so 5550 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-er 8674 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-xadd 13080

This theorem is referenced by: xaddlid 13209 xleadd2a 13221 xltadd2 13224 xposdif 13229 xadd4d 13270 hashunx 14358 xrsnsgrp 21326 xrs1cmn 21330 blcld 24400 xrsxmet 24705 metdstri 24747 vtxdginducedm1 29478 xaddeq0 32683 xlt2addrd 32689 xrge0npcan 32968 esumle 34055 esumlef 34059 measun 34208 difelcarsg 34308 xaddcomd 45327

W3C validator