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Theorem xltnegi 12608

Description: Forward direction of xltneg 12609. (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 12510	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 12510	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		ltneg 11139	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ -𝐵 < -𝐴))
4		rexneg 12603	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → -_𝑒𝐵 = -𝐵)
5		rexneg 12603	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 = -𝐴)
6	4, 5	breqan12rd 5082	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (-_𝑒𝐵 < -_𝑒𝐴 ↔ -𝐵 < -𝐴))
7	3, 6	bitr4d 284	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ -_𝑒𝐵 < -_𝑒𝐴))
8	7	biimpd 231	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → -_𝑒𝐵 < -_𝑒𝐴))
9		xnegeq 12599	. . . . . . . . . . 11 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -_𝑒+∞)
10		xnegpnf 12601	. . . . . . . . . . 11 ⊢ -_𝑒+∞ = -∞
11	9, 10	syl6eq 2872	. . . . . . . . . 10 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -∞)
12	11	adantl 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -_𝑒𝐵 = -∞)
13		renegcl 10948	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
14	5, 13	eqeltrd 2913	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 ∈ ℝ)
15	14	mnfltd 12518	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → -∞ < -_𝑒𝐴)
16	15	adantr 483	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -∞ < -_𝑒𝐴)
17	12, 16	eqbrtrd 5087	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -_𝑒𝐵 < -_𝑒𝐴)
18	17	a1d 25	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 → -_𝑒𝐵 < -_𝑒𝐴))
19		simpr 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → 𝐵 = -∞)
20	19	breq2d 5077	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
21		rexr 10686	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
22		nltmnf 12523	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ^* → ¬ 𝐴 < -∞)
23	21, 22	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < -∞)
24	23	adantr 483	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → ¬ 𝐴 < -∞)
25	24	pm2.21d 121	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < -∞ → -_𝑒𝐵 < -_𝑒𝐴))
26	20, 25	sylbid 242	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 → -_𝑒𝐵 < -_𝑒𝐴))
27	8, 18, 26	3jaodan 1426	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (𝐴 < 𝐵 → -_𝑒𝐵 < -_𝑒𝐴))
28	2, 27	sylan2b 595	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → -_𝑒𝐵 < -_𝑒𝐴))
29	28	expimpd 456	. . . 4 ⊢ (𝐴 ∈ ℝ → ((𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → -_𝑒𝐵 < -_𝑒𝐴))
30		simpl 485	. . . . . . 7 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → 𝐴 = +∞)
31	30	breq1d 5075	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
32		pnfnlt 12522	. . . . . . . 8 ⊢ (𝐵 ∈ ℝ^* → ¬ +∞ < 𝐵)
33	32	adantl 484	. . . . . . 7 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → ¬ +∞ < 𝐵)
34	33	pm2.21d 121	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (+∞ < 𝐵 → -_𝑒𝐵 < -_𝑒𝐴))
35	31, 34	sylbid 242	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 → -_𝑒𝐵 < -_𝑒𝐴))
36	35	expimpd 456	. . . 4 ⊢ (𝐴 = +∞ → ((𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → -_𝑒𝐵 < -_𝑒𝐴))
37		breq1 5068	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
38	37	anbi2d 630	. . . . 5 ⊢ (𝐴 = -∞ → ((𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ↔ (𝐵 ∈ ℝ^* ∧ -∞ < 𝐵)))
39		renegcl 10948	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ)
40	4, 39	eqeltrd 2913	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → -_𝑒𝐵 ∈ ℝ)
41	40	adantr 483	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ -∞ < 𝐵) → -_𝑒𝐵 ∈ ℝ)
42	41	ltpnfd 12515	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ -∞ < 𝐵) → -_𝑒𝐵 < +∞)
43	11	adantr 483	. . . . . . . . 9 ⊢ ((𝐵 = +∞ ∧ -∞ < 𝐵) → -_𝑒𝐵 = -∞)
44		mnfltpnf 12520	. . . . . . . . 9 ⊢ -∞ < +∞
45	43, 44	eqbrtrdi 5104	. . . . . . . 8 ⊢ ((𝐵 = +∞ ∧ -∞ < 𝐵) → -_𝑒𝐵 < +∞)
46		breq2 5069	. . . . . . . . . 10 ⊢ (𝐵 = -∞ → (-∞ < 𝐵 ↔ -∞ < -∞))
47		mnfxr 10697	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
48		nltmnf 12523	. . . . . . . . . . . 12 ⊢ (-∞ ∈ ℝ^* → ¬ -∞ < -∞)
49	47, 48	ax-mp 5	. . . . . . . . . . 11 ⊢ ¬ -∞ < -∞
50	49	pm2.21i 119	. . . . . . . . . 10 ⊢ (-∞ < -∞ → -_𝑒𝐵 < +∞)
51	46, 50	syl6bi 255	. . . . . . . . 9 ⊢ (𝐵 = -∞ → (-∞ < 𝐵 → -_𝑒𝐵 < +∞))
52	51	imp 409	. . . . . . . 8 ⊢ ((𝐵 = -∞ ∧ -∞ < 𝐵) → -_𝑒𝐵 < +∞)
53	42, 45, 52	3jaoian 1425	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞) ∧ -∞ < 𝐵) → -_𝑒𝐵 < +∞)
54	2, 53	sylanb 583	. . . . . 6 ⊢ ((𝐵 ∈ ℝ^* ∧ -∞ < 𝐵) → -_𝑒𝐵 < +∞)
55		xnegeq 12599	. . . . . . . 8 ⊢ (𝐴 = -∞ → -_𝑒𝐴 = -_𝑒-∞)
56		xnegmnf 12602	. . . . . . . 8 ⊢ -_𝑒-∞ = +∞
57	55, 56	syl6eq 2872	. . . . . . 7 ⊢ (𝐴 = -∞ → -_𝑒𝐴 = +∞)
58	57	breq2d 5077	. . . . . 6 ⊢ (𝐴 = -∞ → (-_𝑒𝐵 < -_𝑒𝐴 ↔ -_𝑒𝐵 < +∞))
59	54, 58	syl5ibr 248	. . . . 5 ⊢ (𝐴 = -∞ → ((𝐵 ∈ ℝ^* ∧ -∞ < 𝐵) → -_𝑒𝐵 < -_𝑒𝐴))
60	38, 59	sylbid 242	. . . 4 ⊢ (𝐴 = -∞ → ((𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → -_𝑒𝐵 < -_𝑒𝐴))
61	29, 36, 60	3jaoi 1423	. . 3 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → ((𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → -_𝑒𝐵 < -_𝑒𝐴))
62	1, 61	sylbi 219	. 2 ⊢ (𝐴 ∈ ℝ^* → ((𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → -_𝑒𝐵 < -_𝑒𝐴))
63	62	3impib 1112	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) → -_𝑒𝐵 < -_𝑒𝐴)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∨ w3o 1082 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 class class class wbr 5065 ℝcr 10535 +∞cpnf 10671 -∞cmnf 10672 ℝ^*cxr 10673 < clt 10674 -cneg 10870 -_𝑒cxne 12503

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-xneg 12506

This theorem is referenced by: xltneg 12609 xrsdsreclblem 20590

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