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Theorem xltnegi 13353

Description: Forward direction of xltneg 13354. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Assertion**
Ref	Expression
xltnegi	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → -_e𝐵 < -_e𝐴)

**Proof of Theorem xltnegi**
Step	Hyp	Ref	Expression
1		elxr 13252	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 13252	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		ltneg 11823	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ -𝐵 < -𝐴))
4		rexneg 13348	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → -_e𝐵 = -𝐵)
5		rexneg 13348	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → -_e𝐴 = -𝐴)
6	4, 5	breqan12rd 5119	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ( -_e𝐵 < -_e𝐴 ↔ -𝐵 < -𝐴))
7	3, 6	bitr4d 285	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ -_e𝐵 < -_e𝐴))
8	7	biimpd 232	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → -_e𝐵 < -_e𝐴))
9		xnegeq 13344	. . . . . . . . . . 11 ⊢ (𝐵 = +∞ → -_e𝐵 = -_e+∞)
10		xnegpnf 13346	. . . . . . . . . . 11 ⊢ -_e+∞ = -∞
11	9, 10	eqtrdi 2811	. . . . . . . . . 10 ⊢ (𝐵 = +∞ → -_e𝐵 = -∞)
12	11	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -_e𝐵 = -∞)
13		renegcl 11628	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
14	5, 13	eqeltrd 2860	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → -_e𝐴 ∈ ℝ)
15	14	mnfltd 13260	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → -∞ < -_e𝐴)
16	15	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -∞ < -_e𝐴)
17	12, 16	eqbrtrd 5126	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -_e𝐵 < -_e𝐴)
18	17	a1d 26	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → -_e𝐵 < -_e𝐴))
19		simpr 490	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → 𝐵 = -∞)
20	19	breq2d 5114	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
21		rexr 11362	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
22		nltmnf 13265	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < -∞)
23	21, 22	syl 18	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < -∞)
24	23	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
25	24	pm2.21d 122	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < -∞ → -_e𝐵 < -_e𝐴))
26	20, 25	sylbid 243	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → -_e𝐵 < -_e𝐴))
27	8, 18, 26	3jaodan 1458	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → -_e𝐵 < -_e𝐴))
28	2, 27	sylan2b 606	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → -_e𝐵 < -_e𝐴))
29	28	expimpd 459	. . . 4 ⊢ (𝐴 ∈ ℝ → ((𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → -_e𝐵 < -_e𝐴))
30		simpl 488	. . . . . . 7 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → 𝐴 = +∞)
31	30	breq1d 5112	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
32		pnfnlt 13264	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → ¬ +∞ < 𝐵)
33	32	adantl 487	. . . . . . 7 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ¬ +∞ < 𝐵)
34	33	pm2.21d 122	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (+∞ < 𝐵 → -_e𝐵 < -_e𝐴))
35	31, 34	sylbid 243	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → -_e𝐵 < -_e𝐴))
36	35	expimpd 459	. . . 4 ⊢ (𝐴 = +∞ → ((𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → -_e𝐵 < -_e𝐴))
37		breq1 5105	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
38	37	anbi2d 642	. . . . 5 ⊢ (𝐴 = -∞ → ((𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ↔ (𝐵 ∈ ℝ^* ∧ -∞ < 𝐵)))
39		renegcl 11628	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ)
40	4, 39	eqeltrd 2860	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → -_e𝐵 ∈ ℝ)
41	40	adantr 486	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ -∞ < 𝐵) → -_e𝐵 ∈ ℝ)
42	41	ltpnfd 13257	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ -∞ < 𝐵) → -_e𝐵 < +∞)
43	11	adantr 486	. . . . . . . . 9 ⊢ ((𝐵 = +∞ ∧ -∞ < 𝐵) → -_e𝐵 = -∞)
44		mnfltpnf 13262	. . . . . . . . 9 ⊢ -∞ < +∞
45	43, 44	eqbrtrdi 5143	. . . . . . . 8 ⊢ ((𝐵 = +∞ ∧ -∞ < 𝐵) → -_e𝐵 < +∞)
46		breq2 5106	. . . . . . . . . 10 ⊢ (𝐵 = -∞ → (-∞ < 𝐵 ↔ -∞ < -∞))
47		mnfxr 11373	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
48		nltmnf 13265	. . . . . . . . . . . 12 ⊢ (-∞ ∈ ℝ^* → ¬ -∞ < -∞)
49	47, 48	ax-mp 5	. . . . . . . . . . 11 ⊢ ¬ -∞ < -∞
50	49	pm2.21i 120	. . . . . . . . . 10 ⊢ (-∞ < -∞ → -_e𝐵 < +∞)
51	46, 50	biimtrdi 256	. . . . . . . . 9 ⊢ (𝐵 = -∞ → (-∞ < 𝐵 → -_e𝐵 < +∞))
52	51	imp 412	. . . . . . . 8 ⊢ ((𝐵 = -∞ ∧ -∞ < 𝐵) → -_e𝐵 < +∞)
53	42, 45, 52	3jaoian 1457	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞) ∧ -∞ < 𝐵) → -_e𝐵 < +∞)
54	2, 53	sylanb 593	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ -∞ < 𝐵) → -_e𝐵 < +∞)
55		xnegeq 13344	. . . . . . . 8 ⊢ (𝐴 = -∞ → -_e𝐴 = -_e-∞)
56		xnegmnf 13347	. . . . . . . 8 ⊢ -_e-∞ = +∞
57	55, 56	eqtrdi 2811	. . . . . . 7 ⊢ (𝐴 = -∞ → -_e𝐴 = +∞)
58	57	breq2d 5114	. . . . . 6 ⊢ (𝐴 = -∞ → ( -_e𝐵 < -_e𝐴 ↔ -_e𝐵 < +∞))
59	54, 58	imbitrrid 249	. . . . 5 ⊢ (𝐴 = -∞ → ((𝐵 ∈ ℝ^* ∧ -∞ < 𝐵) → -_e𝐵 < -_e𝐴))
60	38, 59	sylbid 243	. . . 4 ⊢ (𝐴 = -∞ → ((𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → -_e𝐵 < -_e𝐴))
61	29, 36, 60	3jaoi 1454	. . 3 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → ((𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → -_e𝐵 < -_e𝐴))
62	1, 61	sylbi 220	. 2 ⊢ (𝐴 ∈ ℝ^* → ((𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → -_e𝐵 < -_e𝐴))
63	62	3impib 1134	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → -_e𝐵 < -_e𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∨ w3o 1102 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 class class class wbr 5102 ℝcr 11206 +∞cpnf 11347 -∞cmnf 11348 ℝ^*cxr 11349 < clt 11350 -cneg 11549 -_ecxne 13245

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5546 df-po 5559 df-so 5560 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-er 8724 df-en 8981 df-dom 8982 df-sdom 8983 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-xneg 13248

This theorem is used by: xltneg 13354 xrsdsreclblem 21728

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