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Mirrors > Home > ILE Home > Th. List > genpml | Unicode version |
Description: The lower cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Oct-2019.) |
Ref | Expression |
---|---|
genpelvl.1 |
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genpelvl.2 |
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Ref | Expression |
---|---|
genpml |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prop 7504 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | prml 7506 |
. . . 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 7504 |
. . . . 5
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7 | prml 7506 |
. . . . 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 | genpprecll 7543 |
. . . . . 6
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14 | 13 | imp 124 |
. . . . 5
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15 | elprnql 7510 |
. . . . . . . . . 10
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16 | 1, 15 | sylan 283 |
. . . . . . . . 9
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17 | elprnql 7510 |
. . . . . . . . . 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 6051 |
. . . . . . 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 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 5899 df-oprab 5900 df-mpo 5901 df-1st 6165 df-2nd 6166 df-qs 6565 df-ni 7333 df-nqqs 7377 df-inp 7495 |
This theorem is referenced by: addclpr 7566 mulclpr 7601 |
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