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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 | |
genpelvl.2 |
Ref | Expression |
---|---|
genpmu |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prop 7251 | . . . 4 | |
2 | prmu 7254 | . . . 4 | |
3 | rexex 2456 | . . . 4 | |
4 | 1, 2, 3 | 3syl 17 | . . 3 |
5 | 4 | adantr 274 | . 2 |
6 | prop 7251 | . . . . 5 | |
7 | prmu 7254 | . . . . 5 | |
8 | rexex 2456 | . . . . 5 | |
9 | 6, 7, 8 | 3syl 17 | . . . 4 |
10 | 9 | ad2antlr 480 | . . 3 |
11 | genpelvl.1 | . . . . . . 7 | |
12 | genpelvl.2 | . . . . . . 7 | |
13 | 11, 12 | genppreclu 7291 | . . . . . 6 |
14 | 13 | imp 123 | . . . . 5 |
15 | elprnqu 7258 | . . . . . . . . . 10 | |
16 | 1, 15 | sylan 281 | . . . . . . . . 9 |
17 | elprnqu 7258 | . . . . . . . . . 10 | |
18 | 6, 17 | sylan 281 | . . . . . . . . 9 |
19 | 16, 18 | anim12i 336 | . . . . . . . 8 |
20 | 19 | an4s 562 | . . . . . . 7 |
21 | 12 | caovcl 5893 | . . . . . . 7 |
22 | 20, 21 | syl 14 | . . . . . 6 |
23 | simpr 109 | . . . . . . 7 | |
24 | 23 | eleq1d 2186 | . . . . . 6 |
25 | 22, 24 | rspcedv 2767 | . . . . 5 |
26 | 14, 25 | mpd 13 | . . . 4 |
27 | 26 | anassrs 397 | . . 3 |
28 | 10, 27 | exlimddv 1854 | . 2 |
29 | 5, 28 | exlimddv 1854 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wceq 1316 wex 1453 wcel 1465 wrex 2394 crab 2397 cop 3500 cfv 5093 (class class class)co 5742 cmpo 5744 c1st 6004 c2nd 6005 cnq 7056 cnp 7067 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-qs 6403 df-ni 7080 df-nqqs 7124 df-inp 7242 |
This theorem is referenced by: addclpr 7313 mulclpr 7348 |
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