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Theorem xposdif 13318

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 13269	. . . 4 ⊢ (𝐵 ∈ ℝ^* → -_𝑒𝐵 ∈ ℝ^*)
2		xaddcl 13295	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ -_𝑒𝐵 ∈ ℝ^) → (𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ^)
3	1, 2	sylan2 605	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^) → (𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ^)
4		xlt0neg1 13275	. . 3 ⊢ ((𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ^* → ((𝐴 +_𝑒 -_𝑒𝐵) < 0 ↔ 0 < -_𝑒(𝐴 +_𝑒 -_𝑒𝐵)))
5	3, 4	syl 18	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 -_𝑒𝐵) < 0 ↔ 0 < -_𝑒(𝐴 +_𝑒 -_𝑒𝐵)))
6		xsubge0 13317	. . . 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 11284	. . . 4 ⊢ 0 ∈ ℝ^*
9		xrltnle 11304	. . . 4 ⊢ (((𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ^* ∧ 0 ∈ ℝ^*) → ((𝐴 +_𝑒 -_𝑒𝐵) < 0 ↔ ¬ 0 ≤ (𝐴 +_𝑒 -_𝑒𝐵)))
10	3, 8, 9	sylancl 598	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 +_𝑒 -_𝑒𝐵) < 0 ↔ ¬ 0 ≤ (𝐴 +_𝑒 -_𝑒𝐵)))
11		xrltnle 11304	. . 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 13304	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ -_𝑒𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 -_𝑒𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒-_𝑒𝐵))
14	1, 13	sylan2 605	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 -_𝑒𝐵) = (-_𝑒𝐴 +_𝑒 -_𝑒-_𝑒𝐵))
15		xnegneg 13270	. . . . . 6 ⊢ (𝐵 ∈ ℝ^* → -_𝑒-_𝑒𝐵 = 𝐵)
16	15	oveq2d 7433	. . . . 5 ⊢ (𝐵 ∈ ℝ^* → (-_𝑒𝐴 +_𝑒 -_𝑒-_𝑒𝐵) = (-_𝑒𝐴 +_𝑒 𝐵))
17	16	adantl 487	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 +_𝑒 -_𝑒-_𝑒𝐵) = (-_𝑒𝐴 +_𝑒 𝐵))
18		xnegcl 13269	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → -_𝑒𝐴 ∈ ℝ^*)
19		xaddcom 13296	. . . . 5 ⊢ ((-_𝑒𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 -_𝑒𝐴))
20	18, 19	sylan 592	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (-_𝑒𝐴 +_𝑒 𝐵) = (𝐵 +_𝑒 -_𝑒𝐴))
21	14, 17, 20	3eqtrd 2801	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → -_𝑒(𝐴 +_𝑒 -_𝑒𝐵) = (𝐵 +_𝑒 -_𝑒𝐴))
22	21	breq2d 5119	. 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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 class class class wbr 5107 (class class class)co 7417 0cc0 11128 ℝ^*cxr 11270 < clt 11271 ≤ cle 11272 -_𝑒cxne 13164 +_𝑒 cxad 13165

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-id 5554 df-po 5567 df-so 5568 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 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-1st 7990 df-2nd 7991 df-er 8700 df-en 8957 df-dom 8958 df-sdom 8959 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-xneg 13167 df-xadd 13168

This theorem is used by: blcld 24737 metdstri 25084

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