xaddcom - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > xaddcom		GIF version

Theorem xaddcom 10213

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 10128	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 10128	. . . 4 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		recn 8276	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
4		recn 8276	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ)
5		addcom 8426	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
6	3, 4, 5	syl2an 289	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) = (𝐵 + 𝐴))
7		rexadd 10204	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +_𝑒 𝐵) = (𝐴 + 𝐵))
8		rexadd 10204	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 +_𝑒 𝐴) = (𝐵 + 𝐴))
9	8	ancoms 268	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 +_𝑒 𝐴) = (𝐵 + 𝐴))
10	6, 7, 9	3eqtr4d 2277	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
11		oveq2 6066	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 +∞))
12		rexr 8335	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
13		renemnf 8338	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
14		xaddpnf1 10198	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 +∞) = +∞)
15	12, 13, 14	syl2anc 411	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 +∞) = +∞)
16	11, 15	sylan9eqr 2289	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = +∞)
17		oveq1 6065	. . . . . . 7 ⊢ (𝐵 = +∞ → (𝐵 +_𝑒 𝐴) = (+∞ +_𝑒 𝐴))
18		xaddpnf2 10199	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (+∞ +_𝑒 𝐴) = +∞)
19	12, 13, 18	syl2anc 411	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (+∞ +_𝑒 𝐴) = +∞)
20	17, 19	sylan9eqr 2289	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐵 +_𝑒 𝐴) = +∞)
21	16, 20	eqtr4d 2270	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
22		oveq2 6066	. . . . . . 7 ⊢ (𝐵 = -∞ → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 -∞))
23		renepnf 8337	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
24		xaddmnf1 10200	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (𝐴 +_𝑒 -∞) = -∞)
25	12, 23, 24	syl2anc 411	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 -∞) = -∞)
26	22, 25	sylan9eqr 2289	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 +_𝑒 𝐵) = -∞)
27		oveq1 6065	. . . . . . 7 ⊢ (𝐵 = -∞ → (𝐵 +_𝑒 𝐴) = (-∞ +_𝑒 𝐴))
28		xaddmnf2 10201	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (-∞ +_𝑒 𝐴) = -∞)
29	12, 23, 28	syl2anc 411	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (-∞ +_𝑒 𝐴) = -∞)
30	27, 29	sylan9eqr 2289	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐵 +_𝑒 𝐴) = -∞)
31	26, 30	eqtr4d 2270	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
32	10, 21, 31	3jaodan 1343	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
33	2, 32	sylan2b 287	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
34		pnfaddmnf 10202	. . . . . . . 8 ⊢ (+∞ +_𝑒 -∞) = 0
35		mnfaddpnf 10203	. . . . . . . 8 ⊢ (-∞ +_𝑒 +∞) = 0
36	34, 35	eqtr4i 2258	. . . . . . 7 ⊢ (+∞ +_𝑒 -∞) = (-∞ +_𝑒 +∞)
37		simpr 110	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → 𝐵 = -∞)
38	37	oveq2d 6074	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (+∞ +_𝑒 𝐵) = (+∞ +_𝑒 -∞))
39	37	oveq1d 6073	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (𝐵 +_𝑒 +∞) = (-∞ +_𝑒 +∞))
40	36, 38, 39	3eqtr4a 2293	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (+∞ +_𝑒 𝐵) = (𝐵 +_𝑒 +∞))
41		xaddpnf2 10199	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = +∞)
42		xaddpnf1 10198	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (𝐵 +_𝑒 +∞) = +∞)
43	41, 42	eqtr4d 2270	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = (𝐵 +_𝑒 +∞))
44		xrmnfdc 10195	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → DECID 𝐵 = -∞)
45		exmiddc 844	. . . . . . . 8 ⊢ (DECID 𝐵 = -∞ → (𝐵 = -∞ ∨ ¬ 𝐵 = -∞))
46	44, 45	syl 14	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → (𝐵 = -∞ ∨ ¬ 𝐵 = -∞))
47		df-ne 2415	. . . . . . . 8 ⊢ (𝐵 ≠ -∞ ↔ ¬ 𝐵 = -∞)
48	47	orbi2i 770	. . . . . . 7 ⊢ ((𝐵 = -∞ ∨ 𝐵 ≠ -∞) ↔ (𝐵 = -∞ ∨ ¬ 𝐵 = -∞))
49	46, 48	sylibr 134	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → (𝐵 = -∞ ∨ 𝐵 ≠ -∞))
50	40, 43, 49	mpjaodan 806	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (+∞ +_𝑒 𝐵) = (𝐵 +_𝑒 +∞))
51	50	adantl 277	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (+∞ +_𝑒 𝐵) = (𝐵 +_𝑒 +∞))
52		simpl 109	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → 𝐴 = +∞)
53	52	oveq1d 6073	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (+∞ +_𝑒 𝐵))
54	52	oveq2d 6074	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐵 +_𝑒 𝐴) = (𝐵 +_𝑒 +∞))
55	51, 53, 54	3eqtr4d 2277	. . 3 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
56	35, 34	eqtr4i 2258	. . . . . . 7 ⊢ (-∞ +_𝑒 +∞) = (+∞ +_𝑒 -∞)
57		simpr 110	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → 𝐵 = +∞)
58	57	oveq2d 6074	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (-∞ +_𝑒 𝐵) = (-∞ +_𝑒 +∞))
59	57	oveq1d 6073	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (𝐵 +_𝑒 -∞) = (+∞ +_𝑒 -∞))
60	56, 58, 59	3eqtr4a 2293	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (-∞ +_𝑒 𝐵) = (𝐵 +_𝑒 -∞))
61		xaddmnf2 10201	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = -∞)
62		xaddmnf1 10200	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (𝐵 +_𝑒 -∞) = -∞)
63	61, 62	eqtr4d 2270	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = (𝐵 +_𝑒 -∞))
64		xrpnfdc 10194	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → DECID 𝐵 = +∞)
65		exmiddc 844	. . . . . . . 8 ⊢ (DECID 𝐵 = +∞ → (𝐵 = +∞ ∨ ¬ 𝐵 = +∞))
66	64, 65	syl 14	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → (𝐵 = +∞ ∨ ¬ 𝐵 = +∞))
67		df-ne 2415	. . . . . . . 8 ⊢ (𝐵 ≠ +∞ ↔ ¬ 𝐵 = +∞)
68	67	orbi2i 770	. . . . . . 7 ⊢ ((𝐵 = +∞ ∨ 𝐵 ≠ +∞) ↔ (𝐵 = +∞ ∨ ¬ 𝐵 = +∞))
69	66, 68	sylibr 134	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → (𝐵 = +∞ ∨ 𝐵 ≠ +∞))
70	60, 63, 69	mpjaodan 806	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (-∞ +_𝑒 𝐵) = (𝐵 +_𝑒 -∞))
71	70	adantl 277	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (-∞ +_𝑒 𝐵) = (𝐵 +_𝑒 -∞))
72		simpl 109	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → 𝐴 = -∞)
73	72	oveq1d 6073	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (-∞ +_𝑒 𝐵))
74	72	oveq2d 6074	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐵 +_𝑒 𝐴) = (𝐵 +_𝑒 -∞))
75	71, 73, 74	3eqtr4d 2277	. . 3 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
76	33, 55, 75	3jaoian 1342	. 2 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))
77	1, 76	sylanb 284	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 𝐴))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 716 DECID wdc 842 ∨ w3o 1004 = wceq 1398 ∈ wcel 2205 ≠ wne 2414 (class class class)co 6058 ℂcc 8141 ℝcr 8142 0cc0 8143 + caddc 8146 +∞cpnf 8321 -∞cmnf 8322 ℝ^*cxr 8323 +_𝑒 cxad 10122

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-cnex 8234 ax-resscn 8235 ax-1re 8237 ax-addrcl 8240 ax-addcom 8243 ax-rnegex 8252

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-rab 2531 df-v 2817 df-sbc 3046 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-if 3625 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-br 4115 df-opab 4177 df-id 4419 df-xp 4760 df-rel 4761 df-cnv 4762 df-co 4763 df-dm 4764 df-iota 5317 df-fun 5359 df-fv 5365 df-ov 6061 df-oprab 6062 df-mpo 6063 df-pnf 8326 df-mnf 8327 df-xr 8328 df-xadd 10125

This theorem is referenced by: xaddid2 10215 xleadd2a 10226 xltadd2 10229 xadd4d 10237 xrmaxaddlem 11970

W3C validator