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Mirrors > Home > MPE Home > Th. List > xposdif | Structured version Visualization version GIF version |
Description: Extended real version of posdif 11649. (Contributed by Mario Carneiro, 24-Aug-2015.) |
Ref | Expression |
---|---|
xposdif | ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +𝑒 -𝑒𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xnegcl 13133 | . . . 4 ⊢ (𝐵 ∈ ℝ* → -𝑒𝐵 ∈ ℝ*) | |
2 | xaddcl 13159 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ -𝑒𝐵 ∈ ℝ*) → (𝐴 +𝑒 -𝑒𝐵) ∈ ℝ*) | |
3 | 1, 2 | sylan2 594 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 -𝑒𝐵) ∈ ℝ*) |
4 | xlt0neg1 13139 | . . 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 13181 | . . . 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 11203 | . . . 4 ⊢ 0 ∈ ℝ* | |
9 | xrltnle 11223 | . . . 4 ⊢ (((𝐴 +𝑒 -𝑒𝐵) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((𝐴 +𝑒 -𝑒𝐵) < 0 ↔ ¬ 0 ≤ (𝐴 +𝑒 -𝑒𝐵))) | |
10 | 3, 8, 9 | sylancl 587 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 +𝑒 -𝑒𝐵) < 0 ↔ ¬ 0 ≤ (𝐴 +𝑒 -𝑒𝐵))) |
11 | xrltnle 11223 | . . 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 13168 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ -𝑒𝐵 ∈ ℝ*) → -𝑒(𝐴 +𝑒 -𝑒𝐵) = (-𝑒𝐴 +𝑒 -𝑒-𝑒𝐵)) | |
14 | 1, 13 | sylan2 594 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → -𝑒(𝐴 +𝑒 -𝑒𝐵) = (-𝑒𝐴 +𝑒 -𝑒-𝑒𝐵)) |
15 | xnegneg 13134 | . . . . . 6 ⊢ (𝐵 ∈ ℝ* → -𝑒-𝑒𝐵 = 𝐵) | |
16 | 15 | oveq2d 7374 | . . . . 5 ⊢ (𝐵 ∈ ℝ* → (-𝑒𝐴 +𝑒 -𝑒-𝑒𝐵) = (-𝑒𝐴 +𝑒 𝐵)) |
17 | 16 | adantl 483 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝐴 +𝑒 -𝑒-𝑒𝐵) = (-𝑒𝐴 +𝑒 𝐵)) |
18 | xnegcl 13133 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*) | |
19 | xaddcom 13160 | . . . . 5 ⊢ ((-𝑒𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝐴 +𝑒 𝐵) = (𝐵 +𝑒 -𝑒𝐴)) | |
20 | 18, 19 | sylan 581 | . . . 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 5118 | . 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 class class class wbr 5106 (class class class)co 7358 0cc0 11052 ℝ*cxr 11189 < clt 11190 ≤ cle 11191 -𝑒cxne 13031 +𝑒 cxad 13032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-po 5546 df-so 5547 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-xneg 13034 df-xadd 13035 |
This theorem is referenced by: blcld 23864 metdstri 24217 |
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