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Mirrors > Home > ILE Home > Th. List > genpelvl | Unicode version |
Description: Membership in lower cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.) |
Ref | Expression |
---|---|
genpelvl.1 |
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genpelvl.2 |
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Ref | Expression |
---|---|
genpelvl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | genpelvl.1 |
. . . . . . 7
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2 | genpelvl.2 |
. . . . . . 7
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3 | 1, 2 | genipv 7537 |
. . . . . 6
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4 | 3 | fveq2d 5538 |
. . . . 5
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5 | nqex 7391 |
. . . . . . 7
![]() ![]() ![]() ![]() | |
6 | 5 | rabex 4162 |
. . . . . 6
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7 | 5 | rabex 4162 |
. . . . . 6
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8 | 6, 7 | op1st 6170 |
. . . . 5
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9 | 4, 8 | eqtrdi 2238 |
. . . 4
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10 | 9 | eleq2d 2259 |
. . 3
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11 | elrabi 2905 |
. . 3
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12 | 10, 11 | biimtrdi 163 |
. 2
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13 | prop 7503 |
. . . . . . 7
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14 | elprnql 7509 |
. . . . . . 7
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15 | 13, 14 | sylan 283 |
. . . . . 6
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16 | prop 7503 |
. . . . . . 7
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17 | elprnql 7509 |
. . . . . . 7
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18 | 16, 17 | sylan 283 |
. . . . . 6
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19 | 2 | caovcl 6050 |
. . . . . 6
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20 | 15, 18, 19 | syl2an 289 |
. . . . 5
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21 | 20 | an4s 588 |
. . . 4
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22 | eleq1 2252 |
. . . 4
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23 | 21, 22 | syl5ibrcom 157 |
. . 3
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24 | 23 | rexlimdvva 2615 |
. 2
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25 | eqeq1 2196 |
. . . . . 6
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26 | 25 | 2rexbidv 2515 |
. . . . 5
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27 | 26 | elrab3 2909 |
. . . 4
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28 | 10, 27 | sylan9bb 462 |
. . 3
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29 | 28 | ex 115 |
. 2
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30 | 12, 24, 29 | pm5.21ndd 706 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-ov 5898 df-oprab 5899 df-mpo 5900 df-1st 6164 df-2nd 6165 df-qs 6564 df-ni 7332 df-nqqs 7376 df-inp 7494 |
This theorem is referenced by: genpprecll 7542 genpcdl 7547 genprndl 7549 genpdisj 7551 genpassl 7552 addnqprlemrl 7585 mulnqprlemrl 7601 distrlem1prl 7610 distrlem5prl 7614 1idprl 7618 ltexprlemfl 7637 recexprlem1ssl 7661 recexprlemss1l 7663 cauappcvgprlemladdfl 7683 |
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