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Theorem xltnegi 13258

Description: Forward direction of xltneg 13259. (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 13157	. . 3 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		elxr 13157	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
3		ltneg 11729	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ -𝐵 < -𝐴))
4		rexneg 13253	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → -_𝑒𝐵 = -𝐵)
5		rexneg 13253	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 = -𝐴)
6	4, 5	breqan12rd 5128	. . . . . . . . 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 13249	. . . . . . . . . . 11 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -_𝑒+∞)
10		xnegpnf 13251	. . . . . . . . . . 11 ⊢ -_𝑒+∞ = -∞
11	9, 10	eqtrdi 2816	. . . . . . . . . 10 ⊢ (𝐵 = +∞ → -_𝑒𝐵 = -∞)
12	11	adantl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -_𝑒𝐵 = -∞)
13		renegcl 11536	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
14	5, 13	eqeltrd 2865	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → -_𝑒𝐴 ∈ ℝ)
15	14	mnfltd 13165	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → -∞ < -_𝑒𝐴)
16	15	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → -∞ < -_𝑒𝐴)
17	12, 16	eqbrtrd 5135	. . . . . . . 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 5123	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 < 𝐵 ↔ 𝐴 < -∞))
21		rexr 11270	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
22		nltmnf 13170	. . . . . . . . . . 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 5121	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ +∞ < 𝐵))
32		pnfnlt 13169	. . . . . . . 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 5114	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
38	37	anbi2d 642	. . . . 5 ⊢ (𝐴 = -∞ → ((𝐵 ∈ ℝ^* ∧ 𝐴 < 𝐵) ↔ (𝐵 ∈ ℝ^* ∧ -∞ < 𝐵)))
39		renegcl 11536	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ)
40	4, 39	eqeltrd 2865	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → -_𝑒𝐵 ∈ ℝ)
41	40	adantr 486	. . . . . . . . 9 ⊢ ((𝐵 ∈ ℝ ∧ -∞ < 𝐵) → -_𝑒𝐵 ∈ ℝ)
42	41	ltpnfd 13162	. . . . . . . 8 ⊢ ((𝐵 ∈ ℝ ∧ -∞ < 𝐵) → -_𝑒𝐵 < +∞)
43	11	adantr 486	. . . . . . . . 9 ⊢ ((𝐵 = +∞ ∧ -∞ < 𝐵) → -_𝑒𝐵 = -∞)
44		mnfltpnf 13167	. . . . . . . . 9 ⊢ -∞ < +∞
45	43, 44	eqbrtrdi 5152	. . . . . . . 8 ⊢ ((𝐵 = +∞ ∧ -∞ < 𝐵) → -_𝑒𝐵 < +∞)
46		breq2 5115	. . . . . . . . . 10 ⊢ (𝐵 = -∞ → (-∞ < 𝐵 ↔ -∞ < -∞))
47		mnfxr 11281	. . . . . . . . . . . 12 ⊢ -∞ ∈ ℝ^*
48		nltmnf 13170	. . . . . . . . . . . 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 13249	. . . . . . . 8 ⊢ (𝐴 = -∞ → -_𝑒𝐴 = -_𝑒-∞)
56		xnegmnf 13252	. . . . . . . 8 ⊢ -_𝑒-∞ = +∞
57	55, 56	eqtrdi 2816	. . . . . . 7 ⊢ (𝐴 = -∞ → -_𝑒𝐴 = +∞)
58	57	breq2d 5123	. . . . . 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 2146 class class class wbr 5111 ℝcr 11114 +∞cpnf 11255 -∞cmnf 11256 ℝ^*cxr 11257 < clt 11258 -cneg 11457 -_𝑒cxne 13150

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-br 5112 df-opab 5176 df-mpt 5195 df-id 5558 df-po 5571 df-so 5572 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-xneg 13153

This theorem is used by: xltneg 13259 xrsdsreclblem 21613

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