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Mirrors > Home > MPE Home > Th. List > posdif | Structured version Visualization version GIF version |
Description: Comparison of two numbers whose difference is positive. (Contributed by NM, 17-Nov-2004.) |
Ref | Expression |
---|---|
posdif | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resubcl 11555 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 − 𝐴) ∈ ℝ) | |
2 | 1 | ancoms 458 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 − 𝐴) ∈ ℝ) |
3 | simpl 482 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ) | |
4 | ltaddpos 11735 | . . 3 ⊢ (((𝐵 − 𝐴) ∈ ℝ ∧ 𝐴 ∈ ℝ) → (0 < (𝐵 − 𝐴) ↔ 𝐴 < (𝐴 + (𝐵 − 𝐴)))) | |
5 | 2, 3, 4 | syl2anc 583 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (0 < (𝐵 − 𝐴) ↔ 𝐴 < (𝐴 + (𝐵 − 𝐴)))) |
6 | recn 11229 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
7 | recn 11229 | . . . 4 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
8 | pncan3 11499 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + (𝐵 − 𝐴)) = 𝐵) | |
9 | 6, 7, 8 | syl2an 595 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + (𝐵 − 𝐴)) = 𝐵) |
10 | 9 | breq2d 5160 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < (𝐴 + (𝐵 − 𝐴)) ↔ 𝐴 < 𝐵)) |
11 | 5, 10 | bitr2d 280 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 0 < (𝐵 − 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 class class class wbr 5148 (class class class)co 7420 ℂcc 11137 ℝcr 11138 0cc0 11139 + caddc 11142 < clt 11279 − cmin 11475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-ltxr 11284 df-sub 11477 df-neg 11478 |
This theorem is referenced by: posdifi 11795 posdifd 11832 nnsub 12287 nn0sub 12553 znnsub 12639 rpnnen1lem5 12996 difrp 13045 qbtwnre 13211 eluzgtdifelfzo 13727 subfzo0 13787 expnbnd 14227 expmulnbnd 14230 pfxccatin12lem3 14715 eflt 16094 cos01gt0 16168 ndvdsadd 16387 nn0seqcvgd 16541 prmgaplem7 17026 cshwshashlem2 17066 dvcvx 25966 abelthlem7 26388 sinq12gt0 26455 cosq14gt0 26458 cosne0 26476 tanregt0 26486 logdivlti 26567 logcnlem4 26592 scvxcvx 26931 perfectlem2 27176 rplogsumlem2 27431 dchrisum0flblem1 27454 crctcshwlkn0lem3 29636 crctcshwlkn0lem7 29640 mblfinlem3 37132 mblfinlem4 37133 dvasin 37177 geomcau 37232 bfp 37297 perfectALTVlem2 47062 |
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