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Theorem xltnegi 13268

Description: Forward direction of xltneg 13269. (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 13167	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 13167	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		ltneg 11738	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ -𝐵 < -𝐴))
4		rexneg 13263	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → -_𝑒𝐵 = -𝐵)
5		rexneg 13263	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 = -𝐴)
6	4, 5	breqan12rd 5120	. . . . . . . . 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 13259	. . . . . . . . . . 11 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -_𝑒+∞)
10		xnegpnf 13261	. . . . . . . . . . 11 ⊢ -_𝑒+∞ = -∞
11	9, 10	eqtrdi 2811	. . . . . . . . . 10 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -∞)
12	11	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -_𝑒𝐵 = -∞)
13		renegcl 11545	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
14	5, 13	eqeltrd 2860	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 ∈ ℝ)
15	14	mnfltd 13175	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → -∞ < -_𝑒𝐴)
16	15	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -∞ < -_𝑒𝐴)
17	12, 16	eqbrtrd 5127	. . . . . . . 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 5115	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
21		rexr 11279	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
22		nltmnf 13180	. . . . . . . . . . 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 5113	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
32		pnfnlt 13179	. . . . . . . 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 5106	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
38	37	anbi2d 642	. . . . 5 ⊢ (𝐴 = -∞ → ((𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ↔ (𝐵 ∈ ℝ^* ∧ -∞ < 𝐵)))
39		renegcl 11545	. . . . . . . . . . 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 13172	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ -∞ < 𝐵) → -_𝑒𝐵 < +∞)
43	11	adantr 486	. . . . . . . . 9 ⊢ ((𝐵 = +∞ ∧ -∞ < 𝐵) → -_𝑒𝐵 = -∞)
44		mnfltpnf 13177	. . . . . . . . 9 ⊢ -∞ < +∞
45	43, 44	eqbrtrdi 5144	. . . . . . . 8 ⊢ ((𝐵 = +∞ ∧ -∞ < 𝐵) → -_𝑒𝐵 < +∞)
46		breq2 5107	. . . . . . . . . 10 ⊢ (𝐵 = -∞ → (-∞ < 𝐵 ↔ -∞ < -∞))
47		mnfxr 11290	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
48		nltmnf 13180	. . . . . . . . . . . 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 13259	. . . . . . . 8 ⊢ (𝐴 = -∞ → -_𝑒𝐴 = -_𝑒-∞)
56		xnegmnf 13262	. . . . . . . 8 ⊢ -_𝑒-∞ = +∞
57	55, 56	eqtrdi 2811	. . . . . . 7 ⊢ (𝐴 = -∞ → -_𝑒𝐴 = +∞)
58	57	breq2d 5115	. . . . . 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 1570 ∈ wcel 2145 class class class wbr 5103 ℝcr 11123 +∞cpnf 11264 -∞cmnf 11265 ℝ^*cxr 11266 < clt 11267 -cneg 11466 -_𝑒cxne 13160

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200

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 2100 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 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-po 5563 df-so 5564 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-er 8696 df-en 8953 df-dom 8954 df-sdom 8955 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-xneg 13163

This theorem is used by: xltneg 13269 xrsdsreclblem 21626

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