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Theorem xnegdi 10287

Description: Extended real version of negdi 8590. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xnegdi	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -_e(𝐴 +_e 𝐵) = (-_e𝐴 +_e -_e𝐵))

**Proof of Theorem xnegdi**
Step	Hyp	Ref	Expression
1		elxr 10195	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 10195	. . . 4 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		recn 8318	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
4		recn 8318	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ)
5		negdi 8590	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = ( -𝐴 + -𝐵))
6	3, 4, 5	syl2an 289	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -(𝐴 + 𝐵) = ( -𝐴 + -𝐵))
7		readdcl 8311	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
8		rexneg 10249	. . . . . . . 8 ⊢ ((𝐴 + 𝐵) ∈ ℝ → -_e(𝐴 + 𝐵) = -(𝐴 + 𝐵))
9	7, 8	syl 14	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -_e(𝐴 + 𝐵) = -(𝐴 + 𝐵))
10		renegcl 8594	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
11		renegcl 8594	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ)
12		rexadd 10271	. . . . . . . 8 ⊢ (( -𝐴 ∈ ℝ ∧ -𝐵 ∈ ℝ) → ( -𝐴 +_e -𝐵) = ( -𝐴 + -𝐵))
13	10, 11, 12	syl2an 289	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ( -𝐴 +_e -𝐵) = ( -𝐴 + -𝐵))
14	6, 9, 13	3eqtr4d 2281	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -_e(𝐴 + 𝐵) = ( -𝐴 +_e -𝐵))
15		rexadd 10271	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +_e 𝐵) = (𝐴 + 𝐵))
16		xnegeq 10246	. . . . . . 7 ⊢ ((𝐴 +_e 𝐵) = (𝐴 + 𝐵) → -_e(𝐴 +_e 𝐵) = -_e(𝐴 + 𝐵))
17	15, 16	syl 14	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -_e(𝐴 +_e 𝐵) = -_e(𝐴 + 𝐵))
18		rexneg 10249	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → -_e𝐴 = -𝐴)
19		rexneg 10249	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → -_e𝐵 = -𝐵)
20	18, 19	oveqan12d 6107	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-_e𝐴 +_e -_e𝐵) = ( -𝐴 +_e -𝐵))
21	14, 17, 20	3eqtr4d 2281	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -_e(𝐴 +_e 𝐵) = (-_e𝐴 +_e -_e𝐵))
22		xnegpnf 10247	. . . . . 6 ⊢ -_e+∞ = -∞
23		oveq2 6096	. . . . . . . 8 ⊢ (𝐵 = +∞ → (𝐴 +_e 𝐵) = (𝐴 +_e +∞))
24		rexr 8377	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
25		renemnf 8380	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
26		xaddpnf1 10265	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 +_e +∞) = +∞)
27	24, 25, 26	syl2anc 415	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 +_e +∞) = +∞)
28	23, 27	sylan9eqr 2293	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 +_e 𝐵) = +∞)
29		xnegeq 10246	. . . . . . 7 ⊢ ((𝐴 +_e 𝐵) = +∞ → -_e(𝐴 +_e 𝐵) = -_e+∞)
30	28, 29	syl 14	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -_e(𝐴 +_e 𝐵) = -_e+∞)
31		xnegeq 10246	. . . . . . . . 9 ⊢ (𝐵 = +∞ → -_e𝐵 = -_e+∞)
32	31, 22	eqtrdi 2287	. . . . . . . 8 ⊢ (𝐵 = +∞ → -_e𝐵 = -∞)
33	32	oveq2d 6104	. . . . . . 7 ⊢ (𝐵 = +∞ → (-_e𝐴 +_e -_e𝐵) = (-_e𝐴 +_e -∞))
34	18, 10	eqeltrd 2315	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → -_e𝐴 ∈ ℝ)
35		rexr 8377	. . . . . . . . 9 ⊢ (-_e𝐴 ∈ ℝ → -_e𝐴 ∈ ℝ^*)
36		renepnf 8379	. . . . . . . . 9 ⊢ (-_e𝐴 ∈ ℝ → -_e𝐴 ≠ +∞)
37		xaddmnf1 10267	. . . . . . . . 9 ⊢ ((-_e𝐴 ∈ ℝ^* ∧ -_e𝐴 ≠ +∞) → (-_e𝐴 +_e -∞) = -∞)
38	35, 36, 37	syl2anc 415	. . . . . . . 8 ⊢ (-_e𝐴 ∈ ℝ → (-_e𝐴 +_e -∞) = -∞)
39	34, 38	syl 14	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (-_e𝐴 +_e -∞) = -∞)
40	33, 39	sylan9eqr 2293	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (-_e𝐴 +_e -_e𝐵) = -∞)
41	22, 30, 40	3eqtr4a 2297	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -_e(𝐴 +_e 𝐵) = (-_e𝐴 +_e -_e𝐵))
42		xnegmnf 10248	. . . . . 6 ⊢ -_e-∞ = +∞
43		oveq2 6096	. . . . . . . 8 ⊢ (𝐵 = -∞ → (𝐴 +_e 𝐵) = (𝐴 +_e -∞))
44		renepnf 8379	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
45		xaddmnf1 10267	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (𝐴 +_e -∞) = -∞)
46	24, 44, 45	syl2anc 415	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 +_e -∞) = -∞)
47	43, 46	sylan9eqr 2293	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 +_e 𝐵) = -∞)
48		xnegeq 10246	. . . . . . 7 ⊢ ((𝐴 +_e 𝐵) = -∞ → -_e(𝐴 +_e 𝐵) = -_e-∞)
49	47, 48	syl 14	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → -_e(𝐴 +_e 𝐵) = -_e-∞)
50		xnegeq 10246	. . . . . . . . 9 ⊢ (𝐵 = -∞ → -_e𝐵 = -_e-∞)
51	50, 42	eqtrdi 2287	. . . . . . . 8 ⊢ (𝐵 = -∞ → -_e𝐵 = +∞)
52	51	oveq2d 6104	. . . . . . 7 ⊢ (𝐵 = -∞ → (-_e𝐴 +_e -_e𝐵) = (-_e𝐴 +_e +∞))
53		renemnf 8380	. . . . . . . . 9 ⊢ (-_e𝐴 ∈ ℝ → -_e𝐴 ≠ -∞)
54		xaddpnf1 10265	. . . . . . . . 9 ⊢ ((-_e𝐴 ∈ ℝ^* ∧ -_e𝐴 ≠ -∞) → (-_e𝐴 +_e +∞) = +∞)
55	35, 53, 54	syl2anc 415	. . . . . . . 8 ⊢ (-_e𝐴 ∈ ℝ → (-_e𝐴 +_e +∞) = +∞)
56	34, 55	syl 14	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (-_e𝐴 +_e +∞) = +∞)
57	52, 56	sylan9eqr 2293	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (-_e𝐴 +_e -_e𝐵) = +∞)
58	42, 49, 57	3eqtr4a 2297	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → -_e(𝐴 +_e 𝐵) = (-_e𝐴 +_e -_e𝐵))
59	21, 41, 58	3jaodan 1347	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → -_e(𝐴 +_e 𝐵) = (-_e𝐴 +_e -_e𝐵))
60	2, 59	sylan2b 287	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → -_e(𝐴 +_e 𝐵) = (-_e𝐴 +_e -_e𝐵))
61		xneg0 10250	. . . . . . 7 ⊢ -_e0 = 0
62		simpr 110	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → 𝐵 = -∞)
63	62	oveq2d 6104	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (+∞ +_e 𝐵) = (+∞ +_e -∞))
64		pnfaddmnf 10269	. . . . . . . . 9 ⊢ (+∞ +_e -∞) = 0
65	63, 64	eqtrdi 2287	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (+∞ +_e 𝐵) = 0)
66		xnegeq 10246	. . . . . . . 8 ⊢ ((+∞ +_e 𝐵) = 0 → -_e(+∞ +_e 𝐵) = -_e0)
67	65, 66	syl 14	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → -_e(+∞ +_e 𝐵) = -_e0)
68	51	adantl 277	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → -_e𝐵 = +∞)
69	68	oveq2d 6104	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (-∞ +_e -_e𝐵) = (-∞ +_e +∞))
70		mnfaddpnf 10270	. . . . . . . 8 ⊢ (-∞ +_e +∞) = 0
71	69, 70	eqtrdi 2287	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (-∞ +_e -_e𝐵) = 0)
72	61, 67, 71	3eqtr4a 2297	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → -_e(+∞ +_e 𝐵) = (-∞ +_e -_e𝐵))
73		xaddpnf2 10266	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_e 𝐵) = +∞)
74		xnegeq 10246	. . . . . . . 8 ⊢ ((+∞ +_e 𝐵) = +∞ → -_e(+∞ +_e 𝐵) = -_e+∞)
75	73, 74	syl 14	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → -_e(+∞ +_e 𝐵) = -_e+∞)
76		xnegcl 10251	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → -_e𝐵 ∈ ℝ^*)
77		xnegeq 10246	. . . . . . . . . . . 12 ⊢ (-_e𝐵 = +∞ → -_e-_e𝐵 = -_e+∞)
78	77, 22	eqtrdi 2287	. . . . . . . . . . 11 ⊢ (-_e𝐵 = +∞ → -_e-_e𝐵 = -∞)
79		xnegneg 10252	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℝ^* → -_e-_e𝐵 = 𝐵)
80	79	eqeq1d 2247	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ^* → (-_e-_e𝐵 = -∞ ↔ 𝐵 = -∞))
81	78, 80	imbitrid 154	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ^* → (-_e𝐵 = +∞ → 𝐵 = -∞))
82	81	necon3d 2464	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → (𝐵 ≠ -∞ → -_e𝐵 ≠ +∞))
83	82	imp 124	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → -_e𝐵 ≠ +∞)
84		xaddmnf2 10268	. . . . . . . 8 ⊢ ((-_e𝐵 ∈ ℝ^* ∧ -_e𝐵 ≠ +∞) → (-∞ +_e -_e𝐵) = -∞)
85	76, 83, 84	syl2an2r 603	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (-∞ +_e -_e𝐵) = -∞)
86	22, 75, 85	3eqtr4a 2297	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → -_e(+∞ +_e 𝐵) = (-∞ +_e -_e𝐵))
87		xrmnfdc 10262	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → DECID 𝐵 = -∞)
88		exmiddc 848	. . . . . . . 8 ⊢ (DECID 𝐵 = -∞ → (𝐵 = -∞ ∨ ¬ 𝐵 = -∞))
89	87, 88	syl 14	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → (𝐵 = -∞ ∨ ¬ 𝐵 = -∞))
90		df-ne 2421	. . . . . . . 8 ⊢ (𝐵 ≠ -∞ ↔ ¬ 𝐵 = -∞)
91	90	orbi2i 774	. . . . . . 7 ⊢ ((𝐵 = -∞ ∨ 𝐵 ≠ -∞) ↔ (𝐵 = -∞ ∨ ¬ 𝐵 = -∞))
92	89, 91	sylibr 134	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → (𝐵 = -∞ ∨ 𝐵 ≠ -∞))
93	72, 86, 92	mpjaodan 810	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → -_e(+∞ +_e 𝐵) = (-∞ +_e -_e𝐵))
94	93	adantl 277	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_e(+∞ +_e 𝐵) = (-∞ +_e -_e𝐵))
95		simpl 109	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → 𝐴 = +∞)
96	95	oveq1d 6103	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_e 𝐵) = (+∞ +_e 𝐵))
97		xnegeq 10246	. . . . 5 ⊢ ((𝐴 +_e 𝐵) = (+∞ +_e 𝐵) → -_e(𝐴 +_e 𝐵) = -_e(+∞ +_e 𝐵))
98	96, 97	syl 14	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_e(𝐴 +_e 𝐵) = -_e(+∞ +_e 𝐵))
99		xnegeq 10246	. . . . . . 7 ⊢ (𝐴 = +∞ → -_e𝐴 = -_e+∞)
100	99	adantr 276	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_e𝐴 = -_e+∞)
101	100, 22	eqtrdi 2287	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_e𝐴 = -∞)
102	101	oveq1d 6103	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (-_e𝐴 +_e -_e𝐵) = (-∞ +_e -_e𝐵))
103	94, 98, 102	3eqtr4d 2281	. . 3 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_e(𝐴 +_e 𝐵) = (-_e𝐴 +_e -_e𝐵))
104		simpr 110	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → 𝐵 = +∞)
105	104	oveq2d 6104	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (-∞ +_e 𝐵) = (-∞ +_e +∞))
106	105, 70	eqtrdi 2287	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (-∞ +_e 𝐵) = 0)
107		xnegeq 10246	. . . . . . . 8 ⊢ ((-∞ +_e 𝐵) = 0 → -_e(-∞ +_e 𝐵) = -_e0)
108	106, 107	syl 14	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → -_e(-∞ +_e 𝐵) = -_e0)
109	32	adantl 277	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → -_e𝐵 = -∞)
110	109	oveq2d 6104	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (+∞ +_e -_e𝐵) = (+∞ +_e -∞))
111	110, 64	eqtrdi 2287	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (+∞ +_e -_e𝐵) = 0)
112	61, 108, 111	3eqtr4a 2297	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → -_e(-∞ +_e 𝐵) = (+∞ +_e -_e𝐵))
113		xaddmnf2 10268	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_e 𝐵) = -∞)
114		xnegeq 10246	. . . . . . . 8 ⊢ ((-∞ +_e 𝐵) = -∞ → -_e(-∞ +_e 𝐵) = -_e-∞)
115	113, 114	syl 14	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → -_e(-∞ +_e 𝐵) = -_e-∞)
116		xnegeq 10246	. . . . . . . . . . . 12 ⊢ (-_e𝐵 = -∞ → -_e-_e𝐵 = -_e-∞)
117	116, 42	eqtrdi 2287	. . . . . . . . . . 11 ⊢ (-_e𝐵 = -∞ → -_e-_e𝐵 = +∞)
118	79	eqeq1d 2247	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ^* → (-_e-_e𝐵 = +∞ ↔ 𝐵 = +∞))
119	117, 118	imbitrid 154	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ^* → (-_e𝐵 = -∞ → 𝐵 = +∞))
120	119	necon3d 2464	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → (𝐵 ≠ +∞ → -_e𝐵 ≠ -∞))
121	120	imp 124	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → -_e𝐵 ≠ -∞)
122		xaddpnf2 10266	. . . . . . . 8 ⊢ ((-_e𝐵 ∈ ℝ^* ∧ -_e𝐵 ≠ -∞) → (+∞ +_e -_e𝐵) = +∞)
123	76, 121, 122	syl2an2r 603	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (+∞ +_e -_e𝐵) = +∞)
124	42, 115, 123	3eqtr4a 2297	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → -_e(-∞ +_e 𝐵) = (+∞ +_e -_e𝐵))
125		xrpnfdc 10261	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → DECID 𝐵 = +∞)
126		exmiddc 848	. . . . . . . 8 ⊢ (DECID 𝐵 = +∞ → (𝐵 = +∞ ∨ ¬ 𝐵 = +∞))
127	125, 126	syl 14	. . . . . . 7 ⊢ (𝐵 ∈ ℝ^* → (𝐵 = +∞ ∨ ¬ 𝐵 = +∞))
128		df-ne 2421	. . . . . . . 8 ⊢ (𝐵 ≠ +∞ ↔ ¬ 𝐵 = +∞)
129	128	orbi2i 774	. . . . . . 7 ⊢ ((𝐵 = +∞ ∨ 𝐵 ≠ +∞) ↔ (𝐵 = +∞ ∨ ¬ 𝐵 = +∞))
130	127, 129	sylibr 134	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → (𝐵 = +∞ ∨ 𝐵 ≠ +∞))
131	112, 124, 130	mpjaodan 810	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → -_e(-∞ +_e 𝐵) = (+∞ +_e -_e𝐵))
132	131	adantl 277	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_e(-∞ +_e 𝐵) = (+∞ +_e -_e𝐵))
133		simpl 109	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → 𝐴 = -∞)
134	133	oveq1d 6103	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_e 𝐵) = (-∞ +_e 𝐵))
135		xnegeq 10246	. . . . 5 ⊢ ((𝐴 +_e 𝐵) = (-∞ +_e 𝐵) → -_e(𝐴 +_e 𝐵) = -_e(-∞ +_e 𝐵))
136	134, 135	syl 14	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_e(𝐴 +_e 𝐵) = -_e(-∞ +_e 𝐵))
137		xnegeq 10246	. . . . . . 7 ⊢ (𝐴 = -∞ → -_e𝐴 = -_e-∞)
138	137	adantr 276	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_e𝐴 = -_e-∞)
139	138, 42	eqtrdi 2287	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_e𝐴 = +∞)
140	139	oveq1d 6103	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (-_e𝐴 +_e -_e𝐵) = (+∞ +_e -_e𝐵))
141	132, 136, 140	3eqtr4d 2281	. . 3 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_e(𝐴 +_e 𝐵) = (-_e𝐴 +_e -_e𝐵))
142	60, 103, 141	3jaoian 1346	. 2 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^*) → -_e(𝐴 +_e 𝐵) = (-_e𝐴 +_e -_e𝐵))
143	1, 142	sylanb 284	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -_e(𝐴 +_e 𝐵) = (-_e𝐴 +_e -_e𝐵))

Colors of variables: wff set class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 104 ∨ wo 720 DECID wdc 846 ∨ w3o 1008 = wceq 1402 ∈ wcel 2209 ≠ wne 2420 (class class class)co 6088 ℂcc 8183 ℝcr 8184 0cc0 8185 + caddc 8188 +∞cpnf 8363 -∞cmnf 8364 ℝ^*cxr 8365 -cneg 8505 -_ecxne 10188 +_e cxad 10189

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4251 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-addcom 8285 ax-addass 8287 ax-distr 8289 ax-i2m1 8290 ax-0id 8293 ax-rnegex 8294 ax-cnre 8296

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-br 4132 df-opab 4194 df-id 4440 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-iota 5340 df-fun 5382 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-pnf 8368 df-mnf 8369 df-xr 8370 df-sub 8506 df-neg 8507 df-xneg 10191 df-xadd 10192

This theorem is used by: xaddass2 10289 xrminadd 12066

W3C validator