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Theorem xposdif 13304

Description: Extended real version of posdif 11756. (Contributed by Mario Carneiro, 24-Aug-2015.)

**Assertion**
Ref	Expression
xposdif	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴)))

**Proof of Theorem xposdif**
Step	Hyp	Ref	Expression
1		xnegcl 13255	. . . 4 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
2		xaddcl 13281	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ -_𝑒𝐵 ∈ ℝ^) → (𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ^)
3	1, 2	sylan2 593	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ^)
4		xlt0neg1 13261	. . 3 ⊢ ((𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ^* → ((𝐴 +_𝑒 -_𝑒𝐵) < 0 ↔ 0 < -_𝑒(𝐴 +_𝑒 -_𝑒𝐵)))
5	3, 4	syl 17	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 -_𝑒𝐵) < 0 ↔ 0 < -_𝑒(𝐴 +_𝑒 -_𝑒𝐵)))
6		xsubge0 13303	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐴))
7	6	notbid 318	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (¬ 0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ ¬ 𝐵 ≤ 𝐴))
8		0xr 11308	. . . 4 ⊢ 0 ∈ ℝ^*
9		xrltnle 11328	. . . 4 ⊢ (((𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ^* ∧ 0 ∈ ℝ^*) → ((𝐴 +_𝑒 -_𝑒𝐵) < 0 ↔ ¬ 0 ≤ (𝐴 +_𝑒 -_𝑒𝐵)))
10	3, 8, 9	sylancl 586	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 -_𝑒𝐵) < 0 ↔ ¬ 0 ≤ (𝐴 +_𝑒 -_𝑒𝐵)))
11		xrltnle 11328	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
12	7, 10, 11	3bitr4d 311	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 -_𝑒𝐵) < 0 ↔ 𝐴 < 𝐵))
13		xnegdi 13290	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ -_𝑒𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 -_𝑒𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒-_𝑒𝐵))
14	1, 13	sylan2 593	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 -_𝑒𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒-_𝑒𝐵))
15		xnegneg 13256	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → -_𝑒-_𝑒𝐵 = 𝐵)
16	15	oveq2d 7447	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒𝐴 +_𝑒 -_𝑒-_𝑒𝐵) = (-_𝑒𝐴 +_𝑒 𝐵))
17	16	adantl 481	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 +_𝑒 -_𝑒-_𝑒𝐵) = (-_𝑒𝐴 +_𝑒 𝐵))
18		xnegcl 13255	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → -_𝑒𝐴 ∈ ℝ^*)
19		xaddcom 13282	. . . . 5 ⊢ ((-_𝑒𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 -_𝑒𝐴))
20	18, 19	sylan 580	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 -_𝑒𝐴))
21	14, 17, 20	3eqtrd 2781	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 -_𝑒𝐵) = (𝐵 +_𝑒 -_𝑒𝐴))
22	21	breq2d 5155	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 < -_𝑒(𝐴 +_𝑒 -_𝑒𝐵) ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴)))
23	5, 12, 22	3bitr3d 309	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 (class class class)co 7431 0cc0 11155 ℝ^*cxr 11294 < clt 11295 ≤ cle 11296 -_𝑒cxne 13151 +_𝑒 cxad 13152

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-xneg 13154 df-xadd 13155

This theorem is referenced by: blcld 24518 metdstri 24873

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