Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > subhalfnqq | Unicode version |
Description: There is a number which is less than half of any positive fraction. The case where is one is Lemma 11.4 of [BauerTaylor], p. 50, and they use the word "approximate half" for such a number (since there may be constructions, for some structures other than the rationals themselves, which rely on such an approximate half but do not require division by two as seen at halfnqq 7342). (Contributed by Jim Kingdon, 25-Nov-2019.) |
Ref | Expression |
---|---|
subhalfnqq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | halfnqq 7342 | . . . . . 6 | |
2 | df-rex 2448 | . . . . . . 7 | |
3 | halfnqq 7342 | . . . . . . . . . 10 | |
4 | 3 | adantr 274 | . . . . . . . . 9 |
5 | 4 | ancli 321 | . . . . . . . 8 |
6 | 5 | eximi 1587 | . . . . . . 7 |
7 | 2, 6 | sylbi 120 | . . . . . 6 |
8 | 1, 7 | syl 14 | . . . . 5 |
9 | df-rex 2448 | . . . . . . 7 | |
10 | 9 | anbi2i 453 | . . . . . 6 |
11 | 10 | exbii 1592 | . . . . 5 |
12 | 8, 11 | sylib 121 | . . . 4 |
13 | exdistr 1896 | . . . 4 | |
14 | 12, 13 | sylibr 133 | . . 3 |
15 | simprl 521 | . . . . . 6 | |
16 | simpll 519 | . . . . . . . . 9 | |
17 | ltaddnq 7339 | . . . . . . . . 9 | |
18 | 16, 16, 17 | syl2anc 409 | . . . . . . . 8 |
19 | breq2 3980 | . . . . . . . . 9 | |
20 | 19 | ad2antlr 481 | . . . . . . . 8 |
21 | 18, 20 | mpbid 146 | . . . . . . 7 |
22 | breq1 3979 | . . . . . . . 8 | |
23 | 22 | ad2antll 483 | . . . . . . 7 |
24 | 21, 23 | mpbird 166 | . . . . . 6 |
25 | 15, 24 | jca 304 | . . . . 5 |
26 | 25 | eximi 1587 | . . . 4 |
27 | 26 | exlimiv 1585 | . . 3 |
28 | 14, 27 | syl 14 | . 2 |
29 | df-rex 2448 | . 2 | |
30 | 28, 29 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1342 wex 1479 wcel 2135 wrex 2443 class class class wbr 3976 (class class class)co 5836 cnq 7212 cplq 7214 cltq 7217 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-eprel 4261 df-id 4265 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-irdg 6329 df-1o 6375 df-oadd 6379 df-omul 6380 df-er 6492 df-ec 6494 df-qs 6498 df-ni 7236 df-pli 7237 df-mi 7238 df-lti 7239 df-plpq 7276 df-mpq 7277 df-enq 7279 df-nqqs 7280 df-plqqs 7281 df-mqqs 7282 df-1nqqs 7283 df-rq 7284 df-ltnqqs 7285 |
This theorem is referenced by: prarloc 7435 cauappcvgprlemloc 7584 caucvgprlemloc 7607 caucvgprprlemml 7626 caucvgprprlemloc 7635 |
Copyright terms: Public domain | W3C validator |