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Mirrors > Home > ILE Home > Th. List > genpelvu | Unicode version |
Description: Membership in upper cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 15-Oct-2019.) |
Ref | Expression |
---|---|
genpelvl.1 |
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genpelvl.2 |
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Ref | Expression |
---|---|
genpelvu |
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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 7341 |
. . . . . 6
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4 | 3 | fveq2d 5433 |
. . . . 5
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5 | nqex 7195 |
. . . . . . 7
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6 | 5 | rabex 4080 |
. . . . . 6
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7 | 5 | rabex 4080 |
. . . . . 6
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8 | 6, 7 | op2nd 6053 |
. . . . 5
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9 | 4, 8 | eqtrdi 2189 |
. . . 4
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10 | 9 | eleq2d 2210 |
. . 3
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11 | elrabi 2841 |
. . 3
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12 | 10, 11 | syl6bi 162 |
. 2
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13 | prop 7307 |
. . . . . . 7
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14 | elprnqu 7314 |
. . . . . . 7
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15 | 13, 14 | sylan 281 |
. . . . . 6
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16 | prop 7307 |
. . . . . . 7
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17 | elprnqu 7314 |
. . . . . . 7
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18 | 16, 17 | sylan 281 |
. . . . . 6
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19 | 2 | caovcl 5933 |
. . . . . 6
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20 | 15, 18, 19 | syl2an 287 |
. . . . 5
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21 | 20 | an4s 578 |
. . . 4
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22 | eleq1 2203 |
. . . 4
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23 | 21, 22 | syl5ibrcom 156 |
. . 3
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24 | 23 | rexlimdvva 2560 |
. 2
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25 | eqeq1 2147 |
. . . . . 6
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26 | 25 | 2rexbidv 2463 |
. . . . 5
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27 | 26 | elrab3 2845 |
. . . 4
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28 | 10, 27 | sylan9bb 458 |
. . 3
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29 | 28 | ex 114 |
. 2
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30 | 12, 24, 29 | pm5.21ndd 695 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-qs 6443 df-ni 7136 df-nqqs 7180 df-inp 7298 |
This theorem is referenced by: genppreclu 7347 genpcuu 7352 genprndu 7354 genpdisj 7355 genpassu 7357 addnqprlemru 7390 mulnqprlemru 7406 distrlem1pru 7415 distrlem5pru 7419 1idpru 7423 ltexprlemfu 7443 recexprlem1ssu 7466 recexprlemss1u 7468 cauappcvgprlemladdfu 7486 caucvgprlemladdfu 7509 |
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