xposdif - Intuitionistic Logic Explorer

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Theorem xposdif 10267

Description: Extended real version of posdif 8777. (Contributed by Mario Carneiro, 24-Aug-2015.) (Revised by Jim Kingdon, 17-Apr-2023.)

**Assertion**
Ref	Expression
xposdif	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴)))

**Proof of Theorem xposdif**
Step	Hyp	Ref	Expression
1		elxr 10161	. . 3 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
2		elxr 10161	. . . . 5 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
3		posdif 8777	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴)))
4		rexsub 10238	. . . . . . . . . 10 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 +_𝑒 -_𝑒𝐴) = (𝐵 − 𝐴))
5	4	ancoms 268	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 +_𝑒 -_𝑒𝐴) = (𝐵 − 𝐴))
6	5	breq2d 4140	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < (𝐵 +_𝑒 -_𝑒𝐴) ↔ 0 < (𝐵 − 𝐴)))
7	3, 6	bitr4d 191	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴)))
8	7	ex 115	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐵 ∈ ℝ → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴))))
9		rexr 8365	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
10		pnfnlt 10172	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ^* → ¬ +∞ < 𝐵)
11	10	adantl 277	. . . . . . . . . 10 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ¬ +∞ < 𝐵)
12	9, 11	sylan2 286	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → ¬ +∞ < 𝐵)
13		simpl 109	. . . . . . . . . 10 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → 𝐴 = +∞)
14	13	breq1d 4138	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
15	12, 14	mtbird 684	. . . . . . . 8 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → ¬ 𝐴 < 𝐵)
16		0xr 8366	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
17		nltmnf 10173	. . . . . . . . . 10 ⊢ (0 ∈ ℝ^* → ¬ 0 < -∞)
18	16, 17	ax-mp 5	. . . . . . . . 9 ⊢ ¬ 0 < -∞
19		xnegeq 10212	. . . . . . . . . . . . . 14 ⊢ (𝐴 = +∞ → -_𝑒𝐴 = -_𝑒+∞)
20	19	adantr 276	. . . . . . . . . . . . 13 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → -_𝑒𝐴 = -_𝑒+∞)
21		xnegpnf 10213	. . . . . . . . . . . . 13 ⊢ -_𝑒+∞ = -∞
22	20, 21	eqtrdi 2287	. . . . . . . . . . . 12 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → -_𝑒𝐴 = -∞)
23	22	oveq2d 6095	. . . . . . . . . . 11 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐵 +_𝑒 -_𝑒𝐴) = (𝐵 +_𝑒 -∞))
24		renepnf 8367	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ +∞)
25	24	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → 𝐵 ≠ +∞)
26		xaddmnf1 10233	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ +∞) → (𝐵 +_𝑒 -∞) = -∞)
27	9, 25, 26	syl2an2 602	. . . . . . . . . . 11 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐵 +_𝑒 -∞) = -∞)
28	23, 27	eqtrd 2271	. . . . . . . . . 10 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐵 +_𝑒 -_𝑒𝐴) = -∞)
29	28	breq2d 4140	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (0 < (𝐵 +_𝑒 -_𝑒𝐴) ↔ 0 < -∞))
30	18, 29	mtbiri 686	. . . . . . . 8 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → ¬ 0 < (𝐵 +_𝑒 -_𝑒𝐴))
31	15, 30	2falsed 714	. . . . . . 7 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴)))
32	31	ex 115	. . . . . 6 ⊢ (𝐴 = +∞ → (𝐵 ∈ ℝ → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴))))
33		simpl 109	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → 𝐴 = -∞)
34		mnflt 10168	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → -∞ < 𝐵)
35	34	adantl 277	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → -∞ < 𝐵)
36	33, 35	eqbrtrd 4150	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → 𝐴 < 𝐵)
37		0ltpnf 10167	. . . . . . . . 9 ⊢ 0 < +∞
38		xnegeq 10212	. . . . . . . . . . . . 13 ⊢ (𝐴 = -∞ → -_𝑒𝐴 = -_𝑒-∞)
39		xnegmnf 10214	. . . . . . . . . . . . 13 ⊢ -_𝑒-∞ = +∞
40	38, 39	eqtrdi 2287	. . . . . . . . . . . 12 ⊢ (𝐴 = -∞ → -_𝑒𝐴 = +∞)
41	40	oveq2d 6095	. . . . . . . . . . 11 ⊢ (𝐴 = -∞ → (𝐵 +_𝑒 -_𝑒𝐴) = (𝐵 +_𝑒 +∞))
42	41	adantr 276	. . . . . . . . . 10 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐵 +_𝑒 -_𝑒𝐴) = (𝐵 +_𝑒 +∞))
43		renemnf 8368	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ ℝ → 𝐵 ≠ -∞)
44	43	adantl 277	. . . . . . . . . . 11 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → 𝐵 ≠ -∞)
45		xaddpnf1 10231	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐵 ≠ -∞) → (𝐵 +_𝑒 +∞) = +∞)
46	9, 44, 45	syl2an2 602	. . . . . . . . . 10 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐵 +_𝑒 +∞) = +∞)
47	42, 46	eqtrd 2271	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐵 +_𝑒 -_𝑒𝐴) = +∞)
48	37, 47	breqtrrid 4166	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → 0 < (𝐵 +_𝑒 -_𝑒𝐴))
49	36, 48	2thd 175	. . . . . . 7 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴)))
50	49	ex 115	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐵 ∈ ℝ → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴))))
51	8, 32, 50	3jaoi 1344	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐵 ∈ ℝ → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴))))
52	2, 51	sylbi 121	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (𝐵 ∈ ℝ → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴))))
53		ltpnf 10165	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
54	53	adantr 276	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < +∞)
55		simpr 110	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐵 = +∞)
56	54, 55	breqtrrd 4156	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
57	55	oveq1d 6094	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐵 +_𝑒 -_𝑒𝐴) = (+∞ +_𝑒 -_𝑒𝐴))
58		rexneg 10215	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 = -𝐴)
59		renegcl 8581	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
60	58, 59	eqeltrd 2315	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 ∈ ℝ)
61	60	rexrd 8369	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 ∈ ℝ^*)
62	61	adantr 276	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -_𝑒𝐴 ∈ ℝ^*)
63	60	renemnfd 8371	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 ≠ -∞)
64	63	adantr 276	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -_𝑒𝐴 ≠ -∞)
65		xaddpnf2 10232	. . . . . . . . . . 11 ⊢ ((-_𝑒𝐴 ∈ ℝ^* ∧ -_𝑒𝐴 ≠ -∞) → (+∞ +_𝑒 -_𝑒𝐴) = +∞)
66	62, 64, 65	syl2anc 415	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (+∞ +_𝑒 -_𝑒𝐴) = +∞)
67	57, 66	eqtrd 2271	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐵 +_𝑒 -_𝑒𝐴) = +∞)
68	37, 67	breqtrrid 4166	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 0 < (𝐵 +_𝑒 -_𝑒𝐴))
69	56, 68	2thd 175	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴)))
70	69	ex 115	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐵 = +∞ → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴))))
71		pnfxr 8372	. . . . . . . . . 10 ⊢ +∞ ∈ ℝ^*
72		xrltnr 10164	. . . . . . . . . 10 ⊢ (+∞ ∈ ℝ^* → ¬ +∞ < +∞)
73	71, 72	ax-mp 5	. . . . . . . . 9 ⊢ ¬ +∞ < +∞
74		breq12 4133	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ +∞ < +∞))
75	73, 74	mtbiri 686	. . . . . . . 8 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → ¬ 𝐴 < 𝐵)
76		0re 8320	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
77	76	ltnri 8412	. . . . . . . . 9 ⊢ ¬ 0 < 0
78		simpr 110	. . . . . . . . . . . 12 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → 𝐵 = +∞)
79	19, 21	eqtrdi 2287	. . . . . . . . . . . . 13 ⊢ (𝐴 = +∞ → -_𝑒𝐴 = -∞)
80	79	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → -_𝑒𝐴 = -∞)
81	78, 80	oveq12d 6097	. . . . . . . . . . 11 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → (𝐵 +_𝑒 -_𝑒𝐴) = (+∞ +_𝑒 -∞))
82		pnfaddmnf 10235	. . . . . . . . . . 11 ⊢ (+∞ +_𝑒 -∞) = 0
83	81, 82	eqtrdi 2287	. . . . . . . . . 10 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → (𝐵 +_𝑒 -_𝑒𝐴) = 0)
84	83	breq2d 4140	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → (0 < (𝐵 +_𝑒 -_𝑒𝐴) ↔ 0 < 0))
85	77, 84	mtbiri 686	. . . . . . . 8 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → ¬ 0 < (𝐵 +_𝑒 -_𝑒𝐴))
86	75, 85	2falsed 714	. . . . . . 7 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴)))
87	86	ex 115	. . . . . 6 ⊢ (𝐴 = +∞ → (𝐵 = +∞ → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴))))
88		mnfltpnf 10170	. . . . . . . . 9 ⊢ -∞ < +∞
89		breq12 4133	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ -∞ < +∞))
90	88, 89	mpbiri 168	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
91		oveq1 6086	. . . . . . . . . . 11 ⊢ (𝐵 = +∞ → (𝐵 +_𝑒 +∞) = (+∞ +_𝑒 +∞))
92	41, 91	sylan9eq 2291	. . . . . . . . . 10 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐵 +_𝑒 -_𝑒𝐴) = (+∞ +_𝑒 +∞))
93		pnfnemnf 8374	. . . . . . . . . . 11 ⊢ +∞ ≠ -∞
94		xaddpnf1 10231	. . . . . . . . . . 11 ⊢ ((+∞ ∈ ℝ^* ∧ +∞ ≠ -∞) → (+∞ +_𝑒 +∞) = +∞)
95	71, 93, 94	mp2an 430	. . . . . . . . . 10 ⊢ (+∞ +_𝑒 +∞) = +∞
96	92, 95	eqtrdi 2287	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐵 +_𝑒 -_𝑒𝐴) = +∞)
97	37, 96	breqtrrid 4166	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → 0 < (𝐵 +_𝑒 -_𝑒𝐴))
98	90, 97	2thd 175	. . . . . . 7 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴)))
99	98	ex 115	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐵 = +∞ → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴))))
100	70, 87, 99	3jaoi 1344	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐵 = +∞ → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴))))
101	2, 100	sylbi 121	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (𝐵 = +∞ → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴))))
102		rexr 8365	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
103	102	adantr 276	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → 𝐴 ∈ ℝ^*)
104		nltmnf 10173	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < -∞)
105	103, 104	syl 14	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
106		simpr 110	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → 𝐵 = -∞)
107	106	breq2d 4140	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
108	105, 107	mtbird 684	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
109	106	oveq1d 6094	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐵 +_𝑒 -_𝑒𝐴) = (-∞ +_𝑒 -_𝑒𝐴))
110		rexr 8365	. . . . . . . . . . . . . 14 ⊢ (-_𝑒𝐴 ∈ ℝ → -_𝑒𝐴 ∈ ℝ^*)
111		renepnf 8367	. . . . . . . . . . . . . 14 ⊢ (-_𝑒𝐴 ∈ ℝ → -_𝑒𝐴 ≠ +∞)
112		xaddmnf2 10234	. . . . . . . . . . . . . 14 ⊢ ((-_𝑒𝐴 ∈ ℝ^* ∧ -_𝑒𝐴 ≠ +∞) → (-∞ +_𝑒 -_𝑒𝐴) = -∞)
113	110, 111, 112	syl2anc 415	. . . . . . . . . . . . 13 ⊢ (-_𝑒𝐴 ∈ ℝ → (-∞ +_𝑒 -_𝑒𝐴) = -∞)
114	60, 113	syl 14	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → (-∞ +_𝑒 -_𝑒𝐴) = -∞)
115	114	adantr 276	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (-∞ +_𝑒 -_𝑒𝐴) = -∞)
116	109, 115	eqtrd 2271	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐵 +_𝑒 -_𝑒𝐴) = -∞)
117	116	breq2d 4140	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (0 < (𝐵 +_𝑒 -_𝑒𝐴) ↔ 0 < -∞))
118	18, 117	mtbiri 686	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 0 < (𝐵 +_𝑒 -_𝑒𝐴))
119	108, 118	2falsed 714	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴)))
120	119	ex 115	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (𝐵 = -∞ → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴))))
121		eleq1 2301	. . . . . . . . . . . 12 ⊢ (𝐴 = +∞ → (𝐴 ∈ ℝ^* ↔ +∞ ∈ ℝ^*))
122	71, 121	mpbiri 168	. . . . . . . . . . 11 ⊢ (𝐴 = +∞ → 𝐴 ∈ ℝ^*)
123	122	adantr 276	. . . . . . . . . 10 ⊢ ((𝐴 = +∞ ∧ 𝐵 = -∞) → 𝐴 ∈ ℝ^*)
124	123, 104	syl 14	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
125		simpr 110	. . . . . . . . . 10 ⊢ ((𝐴 = +∞ ∧ 𝐵 = -∞) → 𝐵 = -∞)
126	125	breq2d 4140	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
127	124, 126	mtbird 684	. . . . . . . 8 ⊢ ((𝐴 = +∞ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
128	79	oveq2d 6095	. . . . . . . . . . . 12 ⊢ (𝐴 = +∞ → (𝐵 +_𝑒 -_𝑒𝐴) = (𝐵 +_𝑒 -∞))
129	128	adantr 276	. . . . . . . . . . 11 ⊢ ((𝐴 = +∞ ∧ 𝐵 = -∞) → (𝐵 +_𝑒 -_𝑒𝐴) = (𝐵 +_𝑒 -∞))
130		mnfxr 8376	. . . . . . . . . . . . 13 ⊢ -∞ ∈ ℝ^*
131		eleq1 2301	. . . . . . . . . . . . 13 ⊢ (𝐵 = -∞ → (𝐵 ∈ ℝ^* ↔ -∞ ∈ ℝ^*))
132	130, 131	mpbiri 168	. . . . . . . . . . . 12 ⊢ (𝐵 = -∞ → 𝐵 ∈ ℝ^*)
133		mnfnepnf 8375	. . . . . . . . . . . . . 14 ⊢ -∞ ≠ +∞
134		neeq1 2433	. . . . . . . . . . . . . 14 ⊢ (𝐵 = -∞ → (𝐵 ≠ +∞ ↔ -∞ ≠ +∞))
135	133, 134	mpbiri 168	. . . . . . . . . . . . 13 ⊢ (𝐵 = -∞ → 𝐵 ≠ +∞)
136	135	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝐴 = +∞ ∧ 𝐵 = -∞) → 𝐵 ≠ +∞)
137	132, 136, 26	syl2an2 602	. . . . . . . . . . 11 ⊢ ((𝐴 = +∞ ∧ 𝐵 = -∞) → (𝐵 +_𝑒 -∞) = -∞)
138	129, 137	eqtrd 2271	. . . . . . . . . 10 ⊢ ((𝐴 = +∞ ∧ 𝐵 = -∞) → (𝐵 +_𝑒 -_𝑒𝐴) = -∞)
139	138	breq2d 4140	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 = -∞) → (0 < (𝐵 +_𝑒 -_𝑒𝐴) ↔ 0 < -∞))
140	18, 139	mtbiri 686	. . . . . . . 8 ⊢ ((𝐴 = +∞ ∧ 𝐵 = -∞) → ¬ 0 < (𝐵 +_𝑒 -_𝑒𝐴))
141	127, 140	2falsed 714	. . . . . . 7 ⊢ ((𝐴 = +∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴)))
142	141	ex 115	. . . . . 6 ⊢ (𝐴 = +∞ → (𝐵 = -∞ → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴))))
143		xrltnr 10164	. . . . . . . . . 10 ⊢ (-∞ ∈ ℝ^* → ¬ -∞ < -∞)
144	130, 143	ax-mp 5	. . . . . . . . 9 ⊢ ¬ -∞ < -∞
145		breq12 4133	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ -∞ < -∞))
146	144, 145	mtbiri 686	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → ¬ 𝐴 < 𝐵)
147		oveq1 6086	. . . . . . . . . . . 12 ⊢ (𝐵 = -∞ → (𝐵 +_𝑒 +∞) = (-∞ +_𝑒 +∞))
148	41, 147	sylan9eq 2291	. . . . . . . . . . 11 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐵 +_𝑒 -_𝑒𝐴) = (-∞ +_𝑒 +∞))
149		mnfaddpnf 10236	. . . . . . . . . . 11 ⊢ (-∞ +_𝑒 +∞) = 0
150	148, 149	eqtrdi 2287	. . . . . . . . . 10 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐵 +_𝑒 -_𝑒𝐴) = 0)
151	150	breq2d 4140	. . . . . . . . 9 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → (0 < (𝐵 +_𝑒 -_𝑒𝐴) ↔ 0 < 0))
152	77, 151	mtbiri 686	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → ¬ 0 < (𝐵 +_𝑒 -_𝑒𝐴))
153	146, 152	2falsed 714	. . . . . . 7 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴)))
154	153	ex 115	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐵 = -∞ → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴))))
155	120, 142, 154	3jaoi 1344	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐵 = -∞ → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴))))
156	2, 155	sylbi 121	. . . 4 ⊢ (𝐴 ∈ ℝ^* → (𝐵 = -∞ → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴))))
157	52, 101, 156	3jaod 1345	. . 3 ⊢ (𝐴 ∈ ℝ^* → ((𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴))))
158	1, 157	biimtrid 152	. 2 ⊢ (𝐴 ∈ ℝ^* → (𝐵 ∈ ℝ^* → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴))))
159	158	imp 124	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ w3o 1008 = wceq 1402 ∈ wcel 2209 ≠ wne 2420 class class class wbr 4128 (class class class)co 6079 ℝcr 8172 0cc0 8173 +∞cpnf 8351 -∞cmnf 8352 ℝ^*cxr 8353 < clt 8354 − cmin 8491 -cneg 8492 -_𝑒cxne 10154 +_𝑒 cxad 10155

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 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 4247 ax-pow 4309 ax-pr 4344 ax-un 4576 ax-setind 4682 ax-cnex 8264 ax-resscn 8265 ax-1cn 8266 ax-1re 8267 ax-icn 8268 ax-addcl 8269 ax-addrcl 8270 ax-mulcl 8271 ax-addcom 8273 ax-addass 8275 ax-distr 8277 ax-i2m1 8278 ax-0id 8281 ax-rnegex 8282 ax-cnre 8284 ax-pre-ltirr 8285 ax-pre-ltadd 8289

This theorem 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 3714 df-pr 3715 df-op 3717 df-uni 3934 df-br 4129 df-opab 4191 df-id 4436 df-xp 4778 df-rel 4779 df-cnv 4780 df-co 4781 df-dm 4782 df-iota 5335 df-fun 5377 df-fv 5383 df-riota 6032 df-ov 6082 df-oprab 6083 df-mpo 6084 df-pnf 8356 df-mnf 8357 df-xr 8358 df-ltxr 8359 df-sub 8493 df-neg 8494 df-xneg 10157 df-xadd 10158

This theorem is referenced by: (None)

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