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Theorem xposdif 13283

Description: Extended real version of posdif 11735. (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 13234	. . . 4 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
2		xaddcl 13260	. . . 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 13240	. . 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 13282	. . . 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 11287	. . . 4 ⊢ 0 ∈ ℝ^*
9		xrltnle 11307	. . . 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 11307	. . 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 13269	. . . . 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 13235	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → -_𝑒-_𝑒𝐵 = 𝐵)
16	15	oveq2d 7426	. . . . 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 13234	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → -_𝑒𝐴 ∈ ℝ^*)
19		xaddcom 13261	. . . . 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 2775	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 -_𝑒𝐵) = (𝐵 +_𝑒 -_𝑒𝐴))
22	21	breq2d 5136	. 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 2109 class class class wbr 5124 (class class class)co 7410 0cc0 11134 ℝ^*cxr 11273 < clt 11274 ≤ cle 11275 -_𝑒cxne 13130 +_𝑒 cxad 13131

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2708 ax-sep 5271 ax-nul 5281 ax-pow 5340 ax-pr 5407 ax-un 7734 ax-cnex 11190 ax-resscn 11191 ax-1cn 11192 ax-icn 11193 ax-addcl 11194 ax-addrcl 11195 ax-mulcl 11196 ax-mulrcl 11197 ax-mulcom 11198 ax-addass 11199 ax-mulass 11200 ax-distr 11201 ax-i2m1 11202 ax-1ne0 11203 ax-1rid 11204 ax-rnegex 11205 ax-rrecex 11206 ax-cnre 11207 ax-pre-lttri 11208 ax-pre-lttrn 11209 ax-pre-ltadd 11210

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2810 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4889 df-iun 4974 df-br 5125 df-opab 5187 df-mpt 5207 df-id 5553 df-po 5566 df-so 5567 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-iota 6489 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-1st 7993 df-2nd 7994 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11276 df-mnf 11277 df-xr 11278 df-ltxr 11279 df-le 11280 df-sub 11473 df-neg 11474 df-xneg 13133 df-xadd 13134

This theorem is referenced by: blcld 24449 metdstri 24796

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