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Theorem xposdif 13153

Description: Extended real version of posdif 11602. (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 13104	. . . 4 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
2		xaddcl 13130	. . . 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 13110	. . 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 13152	. . . 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 11151	. . . 4 ⊢ 0 ∈ ℝ^*
9		xrltnle 11171	. . . 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 11171	. . 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 13139	. . . . 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 13105	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → -_𝑒-_𝑒𝐵 = 𝐵)
16	15	oveq2d 7357	. . . . 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 13104	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → -_𝑒𝐴 ∈ ℝ^*)
19		xaddcom 13131	. . . . 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 2769	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 -_𝑒𝐵) = (𝐵 +_𝑒 -_𝑒𝐴))
22	21	breq2d 5101	. 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 1541 ∈ wcel 2110 class class class wbr 5089 (class class class)co 7341 0cc0 10998 ℝ^*cxr 11137 < clt 11138 ≤ cle 11139 -_𝑒cxne 13000 +_𝑒 cxad 13001

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663 ax-cnex 11054 ax-resscn 11055 ax-1cn 11056 ax-icn 11057 ax-addcl 11058 ax-addrcl 11059 ax-mulcl 11060 ax-mulrcl 11061 ax-mulcom 11062 ax-addass 11063 ax-mulass 11064 ax-distr 11065 ax-i2m1 11066 ax-1ne0 11067 ax-1rid 11068 ax-rnegex 11069 ax-rrecex 11070 ax-cnre 11071 ax-pre-lttri 11072 ax-pre-lttrn 11073 ax-pre-ltadd 11074

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-po 5522 df-so 5523 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-riota 7298 df-ov 7344 df-oprab 7345 df-mpo 7346 df-1st 7916 df-2nd 7917 df-er 8617 df-en 8865 df-dom 8866 df-sdom 8867 df-pnf 11140 df-mnf 11141 df-xr 11142 df-ltxr 11143 df-le 11144 df-sub 11338 df-neg 11339 df-xneg 13003 df-xadd 13004

This theorem is referenced by: blcld 24413 metdstri 24760

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