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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 | |
genpelvl.2 |
Ref | Expression |
---|---|
genpml |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prop 7430 | . . . 4 | |
2 | prml 7432 | . . . 4 | |
3 | rexex 2516 | . . . 4 | |
4 | 1, 2, 3 | 3syl 17 | . . 3 |
5 | 4 | adantr 274 | . 2 |
6 | prop 7430 | . . . . 5 | |
7 | prml 7432 | . . . . 5 | |
8 | rexex 2516 | . . . . 5 | |
9 | 6, 7, 8 | 3syl 17 | . . . 4 |
10 | 9 | ad2antlr 486 | . . 3 |
11 | genpelvl.1 | . . . . . . 7 | |
12 | genpelvl.2 | . . . . . . 7 | |
13 | 11, 12 | genpprecll 7469 | . . . . . 6 |
14 | 13 | imp 123 | . . . . 5 |
15 | elprnql 7436 | . . . . . . . . . 10 | |
16 | 1, 15 | sylan 281 | . . . . . . . . 9 |
17 | elprnql 7436 | . . . . . . . . . 10 | |
18 | 6, 17 | sylan 281 | . . . . . . . . 9 |
19 | 16, 18 | anim12i 336 | . . . . . . . 8 |
20 | 19 | an4s 583 | . . . . . . 7 |
21 | 12 | caovcl 6005 | . . . . . . 7 |
22 | 20, 21 | syl 14 | . . . . . 6 |
23 | simpr 109 | . . . . . . 7 | |
24 | 23 | eleq1d 2239 | . . . . . 6 |
25 | 22, 24 | rspcedv 2838 | . . . . 5 |
26 | 14, 25 | mpd 13 | . . . 4 |
27 | 26 | anassrs 398 | . . 3 |
28 | 10, 27 | exlimddv 1891 | . 2 |
29 | 5, 28 | exlimddv 1891 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 973 wceq 1348 wex 1485 wcel 2141 wrex 2449 crab 2452 cop 3584 cfv 5196 (class class class)co 5851 cmpo 5853 c1st 6115 c2nd 6116 cnq 7235 cnp 7246 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-qs 6517 df-ni 7259 df-nqqs 7303 df-inp 7421 |
This theorem is referenced by: addclpr 7492 mulclpr 7527 |
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