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Theorem xltnegi 13386

Description: Forward direction of xltneg 13387. (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 13285	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 13285	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		ltneg 11856	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ -𝐵 < -𝐴))
4		rexneg 13381	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → -_e𝐵 = -𝐵)
5		rexneg 13381	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → -_e𝐴 = -𝐴)
6	4, 5	breqan12rd 5125	. . . . . . . . 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 13377	. . . . . . . . . . 11 ⊢ (𝐵 = +∞ → -_e𝐵 = -_e+∞)
10		xnegpnf 13379	. . . . . . . . . . 11 ⊢ -_e+∞ = -∞
11	9, 10	eqtrdi 2814	. . . . . . . . . 10 ⊢ (𝐵 = +∞ → -_e𝐵 = -∞)
12	11	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -_e𝐵 = -∞)
13		renegcl 11660	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
14	5, 13	eqeltrd 2864	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → -_e𝐴 ∈ ℝ)
15	14	mnfltd 13293	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → -∞ < -_e𝐴)
16	15	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -∞ < -_e𝐴)
17	12, 16	eqbrtrd 5133	. . . . . . . 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 5120	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
21		rexr 11394	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
22		nltmnf 13298	. . . . . . . . . . 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 1459	. . . . . 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 5118	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
32		pnfnlt 13297	. . . . . . . 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 5111	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
38	37	anbi2d 642	. . . . 5 ⊢ (𝐴 = -∞ → ((𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ↔ (𝐵 ∈ ℝ^* ∧ -∞ < 𝐵)))
39		renegcl 11660	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ)
40	4, 39	eqeltrd 2864	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → -_e𝐵 ∈ ℝ)
41	40	adantr 486	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ -∞ < 𝐵) → -_e𝐵 ∈ ℝ)
42	41	ltpnfd 13290	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ -∞ < 𝐵) → -_e𝐵 < +∞)
43	11	adantr 486	. . . . . . . . 9 ⊢ ((𝐵 = +∞ ∧ -∞ < 𝐵) → -_e𝐵 = -∞)
44		mnfltpnf 13295	. . . . . . . . 9 ⊢ -∞ < +∞
45	43, 44	eqbrtrdi 5150	. . . . . . . 8 ⊢ ((𝐵 = +∞ ∧ -∞ < 𝐵) → -_e𝐵 < +∞)
46		breq2 5112	. . . . . . . . . 10 ⊢ (𝐵 = -∞ → (-∞ < 𝐵 ↔ -∞ < -∞))
47		mnfxr 11405	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
48		nltmnf 13298	. . . . . . . . . . . 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 1458	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞) ∧ -∞ < 𝐵) → -_e𝐵 < +∞)
54	2, 53	sylanb 593	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ -∞ < 𝐵) → -_e𝐵 < +∞)
55		xnegeq 13377	. . . . . . . 8 ⊢ (𝐴 = -∞ → -_e𝐴 = -_e-∞)
56		xnegmnf 13380	. . . . . . . 8 ⊢ -_e-∞ = +∞
57	55, 56	eqtrdi 2814	. . . . . . 7 ⊢ (𝐴 = -∞ → -_e𝐴 = +∞)
58	57	breq2d 5120	. . . . . 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 1455	. . 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 1570 ∈ wcel 2147 class class class wbr 5108 ℝcr 11238 +∞cpnf 11379 -∞cmnf 11380 ℝ^*cxr 11381 < clt 11382 -cneg 11581 -_ecxne 13278

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4841 df-uni 4873 df-br 5109 df-opab 5174 df-mpt 5193 df-id 5553 df-po 5566 df-so 5567 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-er 8755 df-en 9012 df-dom 9013 df-sdom 9014 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-xneg 13281

This theorem is used by: xltneg 13387 xrsdsreclblem 21771

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