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Theorem xaddcom 12636

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 12514	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 12514	. . . 4 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		recn 10630	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
4		recn 10630	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ)
5		addcom 10829	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
6	3, 4, 5	syl2an 597	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
7		rexadd 12628	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +_𝑒 𝐵) = (𝐴 + 𝐵))
8		rexadd 12628	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 +_𝑒 𝐴) = (𝐵 + 𝐴))
9	8	ancoms 461	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 +_𝑒 𝐴) = (𝐵 + 𝐴))
10	6, 7, 9	3eqtr4d 2869	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
11		oveq2 7167	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 +∞))
12		rexr 10690	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
13		renemnf 10693	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
14		xaddpnf1 12622	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 +∞) = +∞)
15	12, 13, 14	syl2anc 586	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 +∞) = +∞)
16	11, 15	sylan9eqr 2881	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = +∞)
17		oveq1 7166	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐵 +_𝑒 𝐴) = (+∞ +_𝑒 𝐴))
18		xaddpnf2 12623	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (+∞ +_𝑒 𝐴) = +∞)
19	12, 13, 18	syl2anc 586	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (+∞ +_𝑒 𝐴) = +∞)
20	17, 19	sylan9eqr 2881	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐵 +_𝑒 𝐴) = +∞)
21	16, 20	eqtr4d 2862	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
22		oveq2 7167	. . . . . . 7 ⊢ (𝐵 = -∞ → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 -∞))
23		renepnf 10692	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
24		xaddmnf1 12624	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (𝐴 +_𝑒 -∞) = -∞)
25	12, 23, 24	syl2anc 586	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 -∞) = -∞)
26	22, 25	sylan9eqr 2881	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 +_𝑒 𝐵) = -∞)
27		oveq1 7166	. . . . . . 7 ⊢ (𝐵 = -∞ → (𝐵 +_𝑒 𝐴) = (-∞ +_𝑒 𝐴))
28		xaddmnf2 12625	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (-∞ +_𝑒 𝐴) = -∞)
29	12, 23, 28	syl2anc 586	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (-∞ +_𝑒 𝐴) = -∞)
30	27, 29	sylan9eqr 2881	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐵 +_𝑒 𝐴) = -∞)
31	26, 30	eqtr4d 2862	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
32	10, 21, 31	3jaodan 1426	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
33	2, 32	sylan2b 595	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
34		pnfaddmnf 12626	. . . . . . . 8 ⊢ (+∞ +_𝑒 -∞) = 0
35		mnfaddpnf 12627	. . . . . . . 8 ⊢ (-∞ +_𝑒 +∞) = 0
36	34, 35	eqtr4i 2850	. . . . . . 7 ⊢ (+∞ +_𝑒 -∞) = (-∞ +_𝑒 +∞)
37		simpr 487	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → 𝐵 = -∞)
38	37	oveq2d 7175	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (+∞ +_𝑒 𝐵) = (+∞ +_𝑒 -∞))
39	37	oveq1d 7174	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐵 +_𝑒 +∞) = (-∞ +_𝑒 +∞))
40	36, 38, 39	3eqtr4a 2885	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (+∞ +_𝑒 𝐵) = (𝐵 +_𝑒 +∞))
41		xaddpnf2 12623	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = +∞)
42		xaddpnf1 12622	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (𝐵 +_𝑒 +∞) = +∞)
43	41, 42	eqtr4d 2862	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = (𝐵 +_𝑒 +∞))
44	40, 43	pm2.61dane 3107	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (+∞ +_𝑒 𝐵) = (𝐵 +_𝑒 +∞))
45	44	adantl 484	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (+∞ +_𝑒 𝐵) = (𝐵 +_𝑒 +∞))
46		simpl 485	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → 𝐴 = +∞)
47	46	oveq1d 7174	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (+∞ +_𝑒 𝐵))
48	46	oveq2d 7175	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐵 +_𝑒 𝐴) = (𝐵 +_𝑒 +∞))
49	45, 47, 48	3eqtr4d 2869	. . 3 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
50	35, 34	eqtr4i 2850	. . . . . . 7 ⊢ (-∞ +_𝑒 +∞) = (+∞ +_𝑒 -∞)
51		simpr 487	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → 𝐵 = +∞)
52	51	oveq2d 7175	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (-∞ +_𝑒 𝐵) = (-∞ +_𝑒 +∞))
53	51	oveq1d 7174	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (𝐵 +_𝑒 -∞) = (+∞ +_𝑒 -∞))
54	50, 52, 53	3eqtr4a 2885	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (-∞ +_𝑒 𝐵) = (𝐵 +_𝑒 -∞))
55		xaddmnf2 12625	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = -∞)
56		xaddmnf1 12624	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (𝐵 +_𝑒 -∞) = -∞)
57	55, 56	eqtr4d 2862	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = (𝐵 +_𝑒 -∞))
58	54, 57	pm2.61dane 3107	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (-∞ +_𝑒 𝐵) = (𝐵 +_𝑒 -∞))
59	58	adantl 484	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (-∞ +_𝑒 𝐵) = (𝐵 +_𝑒 -∞))
60		simpl 485	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → 𝐴 = -∞)
61	60	oveq1d 7174	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (-∞ +_𝑒 𝐵))
62	60	oveq2d 7175	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐵 +_𝑒 𝐴) = (𝐵 +_𝑒 -∞))
63	59, 61, 62	3eqtr4d 2869	. . 3 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
64	33, 49, 63	3jaoian 1425	. 2 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
65	1, 64	sylanb 583	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∨ w3o 1082 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 (class class class)co 7159 ℂcc 10538 ℝcr 10539 0cc0 10540 + caddc 10543 +∞cpnf 10675 -∞cmnf 10676 ℝ^*cxr 10677 +_𝑒 cxad 12508

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-po 5477 df-so 5478 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-xadd 12511

This theorem is referenced by: xaddid2 12638 xleadd2a 12650 xltadd2 12653 xposdif 12658 xadd4d 12699 hashunx 13750 xrsnsgrp 20584 xrs1cmn 20588 blcld 23118 xrsxmet 23420 metdstri 23462 vtxdginducedm1 27328 xaddeq0 30480 xlt2addrd 30485 xrge0npcan 30685 esumle 31321 esumlef 31325 measun 31474 difelcarsg 31572 xaddcomd 41598

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