xnegdi - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > xnegdi		Structured version Visualization version GIF version

Theorem xnegdi 12367

Description: Extended real version of xnegdi 12367. (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 12237	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 12237	. . . 4 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		recn 10343	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
4		recn 10343	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ)
5		negdi 10660	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → -(𝐴 + 𝐵) = (-𝐴 + -𝐵))
6	3, 4, 5	syl2an 591	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -(𝐴 + 𝐵) = (-𝐴 + -𝐵))
7		readdcl 10336	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + 𝐵) ∈ ℝ)
8		rexneg 12331	. . . . . . . 8 ⊢ ((𝐴 + 𝐵) ∈ ℝ → -_𝑒(𝐴 + 𝐵) = -(𝐴 + 𝐵))
9	7, 8	syl 17	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -_𝑒(𝐴 + 𝐵) = -(𝐴 + 𝐵))
10		renegcl 10666	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
11		renegcl 10666	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ)
12		rexadd 12352	. . . . . . . 8 ⊢ ((-𝐴 ∈ ℝ ∧ -𝐵 ∈ ℝ) → (-𝐴 +_𝑒 -𝐵) = (-𝐴 + -𝐵))
13	10, 11, 12	syl2an 591	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-𝐴 +_𝑒 -𝐵) = (-𝐴 + -𝐵))
14	6, 9, 13	3eqtr4d 2872	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -_𝑒(𝐴 + 𝐵) = (-𝐴 +_𝑒 -𝐵))
15		rexadd 12352	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 +_𝑒 𝐵) = (𝐴 + 𝐵))
16		xnegeq 12327	. . . . . . 7 ⊢ ((𝐴 +_𝑒 𝐵) = (𝐴 + 𝐵) → -_𝑒(𝐴 +_𝑒 𝐵) = -_𝑒(𝐴 + 𝐵))
17	15, 16	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -_𝑒(𝐴 +_𝑒 𝐵) = -_𝑒(𝐴 + 𝐵))
18		rexneg 12331	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 = -𝐴)
19		rexneg 12331	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → -_𝑒𝐵 = -𝐵)
20	18, 19	oveqan12d 6925	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = (-𝐴 +_𝑒 -𝐵))
21	14, 17, 20	3eqtr4d 2872	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
22		xnegpnf 12329	. . . . . 6 ⊢ -_𝑒+∞ = -∞
23		oveq2 6914	. . . . . . . 8 ⊢ (𝐵 = +∞ → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 +∞))
24		rexr 10403	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
25		renemnf 10406	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
26		xaddpnf1 12346	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ -∞) → (𝐴 +_𝑒 +∞) = +∞)
27	24, 25, 26	syl2anc 581	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 +∞) = +∞)
28	23, 27	sylan9eqr 2884	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 +_𝑒 𝐵) = +∞)
29		xnegeq 12327	. . . . . . 7 ⊢ ((𝐴 +_𝑒 𝐵) = +∞ → -_𝑒(𝐴 +_𝑒 𝐵) = -_𝑒+∞)
30	28, 29	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -_𝑒(𝐴 +_𝑒 𝐵) = -_𝑒+∞)
31		xnegeq 12327	. . . . . . . . 9 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -_𝑒+∞)
32	31, 22	syl6eq 2878	. . . . . . . 8 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -∞)
33	32	oveq2d 6922	. . . . . . 7 ⊢ (𝐵 = +∞ → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = (-_𝑒𝐴 +_𝑒 -∞))
34	18, 10	eqeltrd 2907	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 ∈ ℝ)
35		rexr 10403	. . . . . . . . 9 ⊢ (-_𝑒𝐴 ∈ ℝ → -_𝑒𝐴 ∈ ℝ^*)
36		renepnf 10405	. . . . . . . . 9 ⊢ (-_𝑒𝐴 ∈ ℝ → -_𝑒𝐴 ≠ +∞)
37		xaddmnf1 12348	. . . . . . . . 9 ⊢ ((-_𝑒𝐴 ∈ ℝ^* ∧ -_𝑒𝐴 ≠ +∞) → (-_𝑒𝐴 +_𝑒 -∞) = -∞)
38	35, 36, 37	syl2anc 581	. . . . . . . 8 ⊢ (-_𝑒𝐴 ∈ ℝ → (-_𝑒𝐴 +_𝑒 -∞) = -∞)
39	34, 38	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (-_𝑒𝐴 +_𝑒 -∞) = -∞)
40	33, 39	sylan9eqr 2884	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = -∞)
41	22, 30, 40	3eqtr4a 2888	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
42		xnegmnf 12330	. . . . . 6 ⊢ -_𝑒-∞ = +∞
43		oveq2 6914	. . . . . . . 8 ⊢ (𝐵 = -∞ → (𝐴 +_𝑒 𝐵) = (𝐴 +_𝑒 -∞))
44		renepnf 10405	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
45		xaddmnf1 12348	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≠ +∞) → (𝐴 +_𝑒 -∞) = -∞)
46	24, 44, 45	syl2anc 581	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ → (𝐴 +_𝑒 -∞) = -∞)
47	43, 46	sylan9eqr 2884	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 +_𝑒 𝐵) = -∞)
48		xnegeq 12327	. . . . . . 7 ⊢ ((𝐴 +_𝑒 𝐵) = -∞ → -_𝑒(𝐴 +_𝑒 𝐵) = -_𝑒-∞)
49	47, 48	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → -_𝑒(𝐴 +_𝑒 𝐵) = -_𝑒-∞)
50		xnegeq 12327	. . . . . . . . 9 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = -_𝑒-∞)
51	50, 42	syl6eq 2878	. . . . . . . 8 ⊢ (𝐵 = -∞ → -_𝑒𝐵 = +∞)
52	51	oveq2d 6922	. . . . . . 7 ⊢ (𝐵 = -∞ → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = (-_𝑒𝐴 +_𝑒 +∞))
53		renemnf 10406	. . . . . . . . 9 ⊢ (-_𝑒𝐴 ∈ ℝ → -_𝑒𝐴 ≠ -∞)
54		xaddpnf1 12346	. . . . . . . . 9 ⊢ ((-_𝑒𝐴 ∈ ℝ^* ∧ -_𝑒𝐴 ≠ -∞) → (-_𝑒𝐴 +_𝑒 +∞) = +∞)
55	35, 53, 54	syl2anc 581	. . . . . . . 8 ⊢ (-_𝑒𝐴 ∈ ℝ → (-_𝑒𝐴 +_𝑒 +∞) = +∞)
56	34, 55	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℝ → (-_𝑒𝐴 +_𝑒 +∞) = +∞)
57	52, 56	sylan9eqr 2884	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = +∞)
58	42, 49, 57	3eqtr4a 2888	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
59	21, 41, 58	3jaodan 1561	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
60	2, 59	sylan2b 589	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
61		xneg0 12332	. . . . . . 7 ⊢ -_𝑒0 = 0
62		simpr 479	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → 𝐵 = -∞)
63	62	oveq2d 6922	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (+∞ +_𝑒 𝐵) = (+∞ +_𝑒 -∞))
64		pnfaddmnf 12350	. . . . . . . . 9 ⊢ (+∞ +_𝑒 -∞) = 0
65	63, 64	syl6eq 2878	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (+∞ +_𝑒 𝐵) = 0)
66		xnegeq 12327	. . . . . . . 8 ⊢ ((+∞ +_𝑒 𝐵) = 0 → -_𝑒(+∞ +_𝑒 𝐵) = -_𝑒0)
67	65, 66	syl 17	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → -_𝑒(+∞ +_𝑒 𝐵) = -_𝑒0)
68	51	adantl 475	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → -_𝑒𝐵 = +∞)
69	68	oveq2d 6922	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (-∞ +_𝑒 -_𝑒𝐵) = (-∞ +_𝑒 +∞))
70		mnfaddpnf 12351	. . . . . . . 8 ⊢ (-∞ +_𝑒 +∞) = 0
71	69, 70	syl6eq 2878	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → (-∞ +_𝑒 -_𝑒𝐵) = 0)
72	61, 67, 71	3eqtr4a 2888	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = -∞) → -_𝑒(+∞ +_𝑒 𝐵) = (-∞ +_𝑒 -_𝑒𝐵))
73		xaddpnf2 12347	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (+∞ +_𝑒 𝐵) = +∞)
74		xnegeq 12327	. . . . . . . 8 ⊢ ((+∞ +_𝑒 𝐵) = +∞ → -_𝑒(+∞ +_𝑒 𝐵) = -_𝑒+∞)
75	73, 74	syl 17	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → -_𝑒(+∞ +_𝑒 𝐵) = -_𝑒+∞)
76		xnegcl 12333	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
77	76	adantr 474	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → -_𝑒𝐵 ∈ ℝ^*)
78		xnegeq 12327	. . . . . . . . . . . 12 ⊢ (-_𝑒𝐵 = +∞ → -_𝑒-_𝑒𝐵 = -_𝑒+∞)
79	78, 22	syl6eq 2878	. . . . . . . . . . 11 ⊢ (-_𝑒𝐵 = +∞ → -_𝑒-_𝑒𝐵 = -∞)
80		xnegneg 12334	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℝ^* → -_𝑒-_𝑒𝐵 = 𝐵)
81	80	eqeq1d 2828	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒-_𝑒𝐵 = -∞ ↔ 𝐵 = -∞))
82	79, 81	syl5ib 236	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒𝐵 = +∞ → 𝐵 = -∞))
83	82	necon3d 3021	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → (𝐵 ≠ -∞ → -_𝑒𝐵 ≠ +∞))
84	83	imp 397	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → -_𝑒𝐵 ≠ +∞)
85		xaddmnf2 12349	. . . . . . . 8 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐵 ≠ +∞) → (-∞ +_𝑒 -_𝑒𝐵) = -∞)
86	77, 84, 85	syl2anc 581	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (-∞ +_𝑒 -_𝑒𝐵) = -∞)
87	22, 75, 86	3eqtr4a 2888	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → -_𝑒(+∞ +_𝑒 𝐵) = (-∞ +_𝑒 -_𝑒𝐵))
88	72, 87	pm2.61dane 3087	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → -_𝑒(+∞ +_𝑒 𝐵) = (-∞ +_𝑒 -_𝑒𝐵))
89	88	adantl 475	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(+∞ +_𝑒 𝐵) = (-∞ +_𝑒 -_𝑒𝐵))
90		simpl 476	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → 𝐴 = +∞)
91	90	oveq1d 6921	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (+∞ +_𝑒 𝐵))
92		xnegeq 12327	. . . . 5 ⊢ ((𝐴 +_𝑒 𝐵) = (+∞ +_𝑒 𝐵) → -_𝑒(𝐴 +_𝑒 𝐵) = -_𝑒(+∞ +_𝑒 𝐵))
93	91, 92	syl 17	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = -_𝑒(+∞ +_𝑒 𝐵))
94		xnegeq 12327	. . . . . . 7 ⊢ (𝐴 = +∞ → -_𝑒𝐴 = -_𝑒+∞)
95	94	adantr 474	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒𝐴 = -_𝑒+∞)
96	95, 22	syl6eq 2878	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒𝐴 = -∞)
97	96	oveq1d 6921	. . . 4 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = (-∞ +_𝑒 -_𝑒𝐵))
98	89, 93, 97	3eqtr4d 2872	. . 3 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
99		simpr 479	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → 𝐵 = +∞)
100	99	oveq2d 6922	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (-∞ +_𝑒 𝐵) = (-∞ +_𝑒 +∞))
101	100, 70	syl6eq 2878	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (-∞ +_𝑒 𝐵) = 0)
102		xnegeq 12327	. . . . . . . 8 ⊢ ((-∞ +_𝑒 𝐵) = 0 → -_𝑒(-∞ +_𝑒 𝐵) = -_𝑒0)
103	101, 102	syl 17	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → -_𝑒(-∞ +_𝑒 𝐵) = -_𝑒0)
104	32	adantl 475	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → -_𝑒𝐵 = -∞)
105	104	oveq2d 6922	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (+∞ +_𝑒 -_𝑒𝐵) = (+∞ +_𝑒 -∞))
106	105, 64	syl6eq 2878	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → (+∞ +_𝑒 -_𝑒𝐵) = 0)
107	61, 103, 106	3eqtr4a 2888	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 = +∞) → -_𝑒(-∞ +_𝑒 𝐵) = (+∞ +_𝑒 -_𝑒𝐵))
108		xaddmnf2 12349	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (-∞ +_𝑒 𝐵) = -∞)
109		xnegeq 12327	. . . . . . . 8 ⊢ ((-∞ +_𝑒 𝐵) = -∞ → -_𝑒(-∞ +_𝑒 𝐵) = -_𝑒-∞)
110	108, 109	syl 17	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → -_𝑒(-∞ +_𝑒 𝐵) = -_𝑒-∞)
111	76	adantr 474	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → -_𝑒𝐵 ∈ ℝ^*)
112		xnegeq 12327	. . . . . . . . . . . 12 ⊢ (-_𝑒𝐵 = -∞ → -_𝑒-_𝑒𝐵 = -_𝑒-∞)
113	112, 42	syl6eq 2878	. . . . . . . . . . 11 ⊢ (-_𝑒𝐵 = -∞ → -_𝑒-_𝑒𝐵 = +∞)
114	80	eqeq1d 2828	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒-_𝑒𝐵 = +∞ ↔ 𝐵 = +∞))
115	113, 114	syl5ib 236	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒𝐵 = -∞ → 𝐵 = +∞))
116	115	necon3d 3021	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ^* → (𝐵 ≠ +∞ → -_𝑒𝐵 ≠ -∞))
117	116	imp 397	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → -_𝑒𝐵 ≠ -∞)
118		xaddpnf2 12347	. . . . . . . 8 ⊢ ((-_𝑒𝐵 ∈ ℝ^* ∧ -_𝑒𝐵 ≠ -∞) → (+∞ +_𝑒 -_𝑒𝐵) = +∞)
119	111, 117, 118	syl2anc 581	. . . . . . 7 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (+∞ +_𝑒 -_𝑒𝐵) = +∞)
120	42, 110, 119	3eqtr4a 2888	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → -_𝑒(-∞ +_𝑒 𝐵) = (+∞ +_𝑒 -_𝑒𝐵))
121	107, 120	pm2.61dane 3087	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → -_𝑒(-∞ +_𝑒 𝐵) = (+∞ +_𝑒 -_𝑒𝐵))
122	121	adantl 475	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(-∞ +_𝑒 𝐵) = (+∞ +_𝑒 -_𝑒𝐵))
123		simpl 476	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → 𝐴 = -∞)
124	123	oveq1d 6921	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 +_𝑒 𝐵) = (-∞ +_𝑒 𝐵))
125		xnegeq 12327	. . . . 5 ⊢ ((𝐴 +_𝑒 𝐵) = (-∞ +_𝑒 𝐵) → -_𝑒(𝐴 +_𝑒 𝐵) = -_𝑒(-∞ +_𝑒 𝐵))
126	124, 125	syl 17	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = -_𝑒(-∞ +_𝑒 𝐵))
127		xnegeq 12327	. . . . . . 7 ⊢ (𝐴 = -∞ → -_𝑒𝐴 = -_𝑒-∞)
128	127	adantr 474	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒𝐴 = -_𝑒-∞)
129	128, 42	syl6eq 2878	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒𝐴 = +∞)
130	129	oveq1d 6921	. . . 4 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 +_𝑒 -_𝑒𝐵) = (+∞ +_𝑒 -_𝑒𝐵))
131	122, 126, 130	3eqtr4d 2872	. . 3 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
132	60, 98, 131	3jaoian 1560	. 2 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))
133	1, 132	sylanb 578	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 ∨ w3o 1112 = wceq 1658 ∈ wcel 2166 ≠ wne 3000 (class class class)co 6906 ℂcc 10251 ℝcr 10252 0cc0 10253 + caddc 10256 +∞cpnf 10389 -∞cmnf 10390 ℝ^*cxr 10391 -cneg 10587 -_𝑒cxne 12230 +_𝑒 cxad 12231

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2804 ax-sep 5006 ax-nul 5014 ax-pow 5066 ax-pr 5128 ax-un 7210 ax-cnex 10309 ax-resscn 10310 ax-1cn 10311 ax-icn 10312 ax-addcl 10313 ax-addrcl 10314 ax-mulcl 10315 ax-mulrcl 10316 ax-mulcom 10317 ax-addass 10318 ax-mulass 10319 ax-distr 10320 ax-i2m1 10321 ax-1ne0 10322 ax-1rid 10323 ax-rnegex 10324 ax-rrecex 10325 ax-cnre 10326 ax-pre-lttri 10327 ax-pre-lttrn 10328 ax-pre-ltadd 10329

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2606 df-eu 2641 df-clab 2813 df-cleq 2819 df-clel 2822 df-nfc 2959 df-ne 3001 df-nel 3104 df-ral 3123 df-rex 3124 df-reu 3125 df-rab 3127 df-v 3417 df-sbc 3664 df-csb 3759 df-dif 3802 df-un 3804 df-in 3806 df-ss 3813 df-nul 4146 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4660 df-br 4875 df-opab 4937 df-mpt 4954 df-id 5251 df-po 5264 df-so 5265 df-xp 5349 df-rel 5350 df-cnv 5351 df-co 5352 df-dm 5353 df-rn 5354 df-res 5355 df-ima 5356 df-iota 6087 df-fun 6126 df-fn 6127 df-f 6128 df-f1 6129 df-fo 6130 df-f1o 6131 df-fv 6132 df-riota 6867 df-ov 6909 df-oprab 6910 df-mpt2 6911 df-er 8010 df-en 8224 df-dom 8225 df-sdom 8226 df-pnf 10394 df-mnf 10395 df-xr 10396 df-ltxr 10397 df-sub 10588 df-neg 10589 df-xneg 12233 df-xadd 12234

This theorem is referenced by: xaddass2 12369 xposdif 12381 xadddi 12414 xrsxmet 22983

W3C validator