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Mirrors > Home > MPE Home > Th. List > divelunit | Structured version Unicode version |
Description: A condition for a ratio to be a member of the closed unit. (Contributed by Scott Fenton, 11-Jun-2013.) |
Ref | Expression |
---|---|
divelunit |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0re 9496 |
. . . 4
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2 | 1re 9495 |
. . . 4
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3 | 1, 2 | elicc2i 11471 |
. . 3
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4 | df-3an 967 |
. . 3
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5 | 3, 4 | bitri 249 |
. 2
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6 | ledivmul 10315 |
. . . . 5
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7 | 2, 6 | mp3an2 1303 |
. . . 4
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8 | 7 | adantlr 714 |
. . 3
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9 | simpll 753 |
. . . . . 6
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10 | simprl 755 |
. . . . . 6
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11 | gt0ne0 9914 |
. . . . . . 7
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12 | 11 | adantl 466 |
. . . . . 6
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13 | 9, 10, 12 | redivcld 10269 |
. . . . 5
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14 | divge0 10308 |
. . . . 5
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15 | 13, 14 | jca 532 |
. . . 4
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16 | 15 | biantrurd 508 |
. . 3
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17 | recn 9482 |
. . . . . 6
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18 | 17 | ad2antrl 727 |
. . . . 5
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19 | 18 | mulid1d 9513 |
. . . 4
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20 | 19 | breq2d 4411 |
. . 3
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21 | 8, 16, 20 | 3bitr3d 283 |
. 2
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22 | 5, 21 | syl5bb 257 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4520 ax-nul 4528 ax-pow 4577 ax-pr 4638 ax-un 6481 ax-cnex 9448 ax-resscn 9449 ax-1cn 9450 ax-icn 9451 ax-addcl 9452 ax-addrcl 9453 ax-mulcl 9454 ax-mulrcl 9455 ax-mulcom 9456 ax-addass 9457 ax-mulass 9458 ax-distr 9459 ax-i2m1 9460 ax-1ne0 9461 ax-1rid 9462 ax-rnegex 9463 ax-rrecex 9464 ax-cnre 9465 ax-pre-lttri 9466 ax-pre-lttrn 9467 ax-pre-ltadd 9468 ax-pre-mulgt0 9469 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2649 df-nel 2650 df-ral 2803 df-rex 2804 df-reu 2805 df-rmo 2806 df-rab 2807 df-v 3078 df-sbc 3293 df-csb 3395 df-dif 3438 df-un 3440 df-in 3442 df-ss 3449 df-nul 3745 df-if 3899 df-pw 3969 df-sn 3985 df-pr 3987 df-op 3991 df-uni 4199 df-br 4400 df-opab 4458 df-mpt 4459 df-id 4743 df-po 4748 df-so 4749 df-xp 4953 df-rel 4954 df-cnv 4955 df-co 4956 df-dm 4957 df-rn 4958 df-res 4959 df-ima 4960 df-iota 5488 df-fun 5527 df-fn 5528 df-f 5529 df-f1 5530 df-fo 5531 df-f1o 5532 df-fv 5533 df-riota 6160 df-ov 6202 df-oprab 6203 df-mpt2 6204 df-er 7210 df-en 7420 df-dom 7421 df-sdom 7422 df-pnf 9530 df-mnf 9531 df-xr 9532 df-ltxr 9533 df-le 9534 df-sub 9707 df-neg 9708 df-div 10104 df-icc 11417 |
This theorem is referenced by: brbtwn2 23302 axsegconlem7 23320 axcontlem2 23362 axcontlem4 23364 axcontlem7 23367 axcontlem8 23368 |
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