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Mirrors > Home > ILE Home > Th. List > genpmu | Unicode version |
Description: The upper cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Dec-2019.) |
Ref | Expression |
---|---|
genpelvl.1 |
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genpelvl.2 |
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Ref | Expression |
---|---|
genpmu |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prop 7499 |
. . . 4
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2 | prmu 7502 |
. . . 4
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3 | rexex 2536 |
. . . 4
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4 | 1, 2, 3 | 3syl 17 |
. . 3
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5 | 4 | adantr 276 |
. 2
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6 | prop 7499 |
. . . . 5
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7 | prmu 7502 |
. . . . 5
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8 | rexex 2536 |
. . . . 5
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9 | 6, 7, 8 | 3syl 17 |
. . . 4
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10 | 9 | ad2antlr 489 |
. . 3
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11 | genpelvl.1 |
. . . . . . 7
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12 | genpelvl.2 |
. . . . . . 7
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13 | 11, 12 | genppreclu 7539 |
. . . . . 6
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14 | 13 | imp 124 |
. . . . 5
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15 | elprnqu 7506 |
. . . . . . . . . 10
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16 | 1, 15 | sylan 283 |
. . . . . . . . 9
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17 | elprnqu 7506 |
. . . . . . . . . 10
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18 | 6, 17 | sylan 283 |
. . . . . . . . 9
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19 | 16, 18 | anim12i 338 |
. . . . . . . 8
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20 | 19 | an4s 588 |
. . . . . . 7
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21 | 12 | caovcl 6047 |
. . . . . . 7
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22 | 20, 21 | syl 14 |
. . . . . 6
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23 | simpr 110 |
. . . . . . 7
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24 | 23 | eleq1d 2258 |
. . . . . 6
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25 | 22, 24 | rspcedv 2860 |
. . . . 5
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26 | 14, 25 | mpd 13 |
. . . 4
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27 | 26 | anassrs 400 |
. . 3
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28 | 10, 27 | exlimddv 1910 |
. 2
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29 | 5, 28 | exlimddv 1910 |
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 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 |
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 4308 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5234 df-fn 5235 df-f 5236 df-f1 5237 df-fo 5238 df-f1o 5239 df-fv 5240 df-ov 5895 df-oprab 5896 df-mpo 5897 df-1st 6160 df-2nd 6161 df-qs 6560 df-ni 7328 df-nqqs 7372 df-inp 7490 |
This theorem is referenced by: addclpr 7561 mulclpr 7596 |
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