xposdif - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > xposdif		Structured version Visualization version GIF version

Theorem xposdif 12701

Description: Extended real version of posdif 11176. (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 12652	. . . 4 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
2		xaddcl 12678	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ -_𝑒𝐵 ∈ ℝ^) → (𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ^)
3	1, 2	sylan2 595	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ^)
4		xlt0neg1 12658	. . 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 12700	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ 𝐵 ≤ 𝐴))
7	6	notbid 321	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (¬ 0 ≤ (𝐴 +_𝑒 -_𝑒𝐵) ↔ ¬ 𝐵 ≤ 𝐴))
8		0xr 10731	. . . 4 ⊢ 0 ∈ ℝ^*
9		xrltnle 10751	. . . 4 ⊢ (((𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ^* ∧ 0 ∈ ℝ^*) → ((𝐴 +_𝑒 -_𝑒𝐵) < 0 ↔ ¬ 0 ≤ (𝐴 +_𝑒 -_𝑒𝐵)))
10	3, 8, 9	sylancl 589	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 -_𝑒𝐵) < 0 ↔ ¬ 0 ≤ (𝐴 +_𝑒 -_𝑒𝐵)))
11		xrltnle 10751	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴))
12	7, 10, 11	3bitr4d 314	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 -_𝑒𝐵) < 0 ↔ 𝐴 < 𝐵))
13		xnegdi 12687	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ -_𝑒𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 -_𝑒𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒-_𝑒𝐵))
14	1, 13	sylan2 595	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 -_𝑒𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒-_𝑒𝐵))
15		xnegneg 12653	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → -_𝑒-_𝑒𝐵 = 𝐵)
16	15	oveq2d 7171	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒𝐴 +_𝑒 -_𝑒-_𝑒𝐵) = (-_𝑒𝐴 +_𝑒 𝐵))
17	16	adantl 485	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 +_𝑒 -_𝑒-_𝑒𝐵) = (-_𝑒𝐴 +_𝑒 𝐵))
18		xnegcl 12652	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → -_𝑒𝐴 ∈ ℝ^*)
19		xaddcom 12679	. . . . 5 ⊢ ((-_𝑒𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 -_𝑒𝐴))
20	18, 19	sylan 583	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 -_𝑒𝐴))
21	14, 17, 20	3eqtrd 2797	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 -_𝑒𝐵) = (𝐵 +_𝑒 -_𝑒𝐴))
22	21	breq2d 5047	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (0 < -_𝑒(𝐴 +_𝑒 -_𝑒𝐵) ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴)))
23	5, 12, 22	3bitr3d 312	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +_𝑒 -_𝑒𝐴)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 class class class wbr 5035 (class class class)co 7155 0cc0 10580 ℝ^*cxr 10717 < clt 10718 ≤ cle 10719 -_𝑒cxne 12550 +_𝑒 cxad 12551

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5172 ax-nul 5179 ax-pow 5237 ax-pr 5301 ax-un 7464 ax-cnex 10636 ax-resscn 10637 ax-1cn 10638 ax-icn 10639 ax-addcl 10640 ax-addrcl 10641 ax-mulcl 10642 ax-mulrcl 10643 ax-mulcom 10644 ax-addass 10645 ax-mulass 10646 ax-distr 10647 ax-i2m1 10648 ax-1ne0 10649 ax-1rid 10650 ax-rnegex 10651 ax-rrecex 10652 ax-cnre 10653 ax-pre-lttri 10654 ax-pre-lttrn 10655 ax-pre-ltadd 10656

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-iun 4888 df-br 5036 df-opab 5098 df-mpt 5116 df-id 5433 df-po 5446 df-so 5447 df-xp 5533 df-rel 5534 df-cnv 5535 df-co 5536 df-dm 5537 df-rn 5538 df-res 5539 df-ima 5540 df-iota 6298 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7698 df-2nd 7699 df-er 8304 df-en 8533 df-dom 8534 df-sdom 8535 df-pnf 10720 df-mnf 10721 df-xr 10722 df-ltxr 10723 df-le 10724 df-sub 10915 df-neg 10916 df-xneg 12553 df-xadd 12554

This theorem is referenced by: blcld 23212 metdstri 23557

W3C validator