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Mirrors > Home > MPE Home > Th. List > xposdif | Structured version Visualization version GIF version |
Description: Extended real version of posdif 10400. (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 11918 | . . . 4 ⊢ (𝐵 ∈ ℝ* → -𝑒𝐵 ∈ ℝ*) | |
2 | xaddcl 11944 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ -𝑒𝐵 ∈ ℝ*) → (𝐴 +𝑒 -𝑒𝐵) ∈ ℝ*) | |
3 | 1, 2 | sylan2 490 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 +𝑒 -𝑒𝐵) ∈ ℝ*) |
4 | xlt0neg1 11924 | . . 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 11963 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (0 ≤ (𝐴 +𝑒 -𝑒𝐵) ↔ 𝐵 ≤ 𝐴)) | |
7 | 6 | notbid 307 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (¬ 0 ≤ (𝐴 +𝑒 -𝑒𝐵) ↔ ¬ 𝐵 ≤ 𝐴)) |
8 | 0xr 9965 | . . . 4 ⊢ 0 ∈ ℝ* | |
9 | xrltnle 9984 | . . . 4 ⊢ (((𝐴 +𝑒 -𝑒𝐵) ∈ ℝ* ∧ 0 ∈ ℝ*) → ((𝐴 +𝑒 -𝑒𝐵) < 0 ↔ ¬ 0 ≤ (𝐴 +𝑒 -𝑒𝐵))) | |
10 | 3, 8, 9 | sylancl 693 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 +𝑒 -𝑒𝐵) < 0 ↔ ¬ 0 ≤ (𝐴 +𝑒 -𝑒𝐵))) |
11 | xrltnle 9984 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ 𝐵 ≤ 𝐴)) | |
12 | 7, 10, 11 | 3bitr4d 299 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 +𝑒 -𝑒𝐵) < 0 ↔ 𝐴 < 𝐵)) |
13 | xnegdi 11950 | . . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ -𝑒𝐵 ∈ ℝ*) → -𝑒(𝐴 +𝑒 -𝑒𝐵) = (-𝑒𝐴 +𝑒 -𝑒-𝑒𝐵)) | |
14 | 1, 13 | sylan2 490 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → -𝑒(𝐴 +𝑒 -𝑒𝐵) = (-𝑒𝐴 +𝑒 -𝑒-𝑒𝐵)) |
15 | xnegneg 11919 | . . . . . 6 ⊢ (𝐵 ∈ ℝ* → -𝑒-𝑒𝐵 = 𝐵) | |
16 | 15 | oveq2d 6565 | . . . . 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 11918 | . . . . 5 ⊢ (𝐴 ∈ ℝ* → -𝑒𝐴 ∈ ℝ*) | |
19 | xaddcom 11945 | . . . . 5 ⊢ ((-𝑒𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝐴 +𝑒 𝐵) = (𝐵 +𝑒 -𝑒𝐴)) | |
20 | 18, 19 | sylan 487 | . . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (-𝑒𝐴 +𝑒 𝐵) = (𝐵 +𝑒 -𝑒𝐴)) |
21 | 14, 17, 20 | 3eqtrd 2648 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → -𝑒(𝐴 +𝑒 -𝑒𝐵) = (𝐵 +𝑒 -𝑒𝐴)) |
22 | 21 | breq2d 4595 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (0 < -𝑒(𝐴 +𝑒 -𝑒𝐵) ↔ 0 < (𝐵 +𝑒 -𝑒𝐴))) |
23 | 5, 12, 22 | 3bitr3d 297 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ 0 < (𝐵 +𝑒 -𝑒𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 (class class class)co 6549 0cc0 9815 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 -𝑒cxne 11819 +𝑒 cxad 11820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-xneg 11822 df-xadd 11823 |
This theorem is referenced by: blcld 22120 metdstri 22462 |
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