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Mirrors > Home > MPE Home > Th. List > halfnq | Structured version Visualization version Unicode version |
Description: One-half of any positive fraction exists. Lemma for Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 16-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
halfnq |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | distrnq 9783 | . . . 4 | |
2 | distrnq 9783 | . . . . . . . 8 | |
3 | 1nq 9750 | . . . . . . . . . . 11 | |
4 | addclnq 9767 | . . . . . . . . . . 11 | |
5 | 3, 3, 4 | mp2an 708 | . . . . . . . . . 10 |
6 | recidnq 9787 | . . . . . . . . . 10 | |
7 | 5, 6 | ax-mp 5 | . . . . . . . . 9 |
8 | 7, 7 | oveq12i 6662 | . . . . . . . 8 |
9 | 2, 8 | eqtri 2644 | . . . . . . 7 |
10 | 9 | oveq1i 6660 | . . . . . 6 |
11 | 7 | oveq2i 6661 | . . . . . . 7 |
12 | mulassnq 9781 | . . . . . . . 8 | |
13 | mulcomnq 9775 | . . . . . . . . 9 | |
14 | 13 | oveq1i 6660 | . . . . . . . 8 |
15 | 12, 14 | eqtr3i 2646 | . . . . . . 7 |
16 | recclnq 9788 | . . . . . . . . 9 | |
17 | addclnq 9767 | . . . . . . . . 9 | |
18 | 16, 16, 17 | syl2anc 693 | . . . . . . . 8 |
19 | mulidnq 9785 | . . . . . . . 8 | |
20 | 5, 18, 19 | mp2b 10 | . . . . . . 7 |
21 | 11, 15, 20 | 3eqtr3i 2652 | . . . . . 6 |
22 | 10, 21, 7 | 3eqtr3i 2652 | . . . . 5 |
23 | 22 | oveq2i 6661 | . . . 4 |
24 | 1, 23 | eqtr3i 2646 | . . 3 |
25 | mulidnq 9785 | . . 3 | |
26 | 24, 25 | syl5eq 2668 | . 2 |
27 | ovex 6678 | . . 3 | |
28 | oveq12 6659 | . . . . 5 | |
29 | 28 | anidms 677 | . . . 4 |
30 | 29 | eqeq1d 2624 | . . 3 |
31 | 27, 30 | spcev 3300 | . 2 |
32 | 26, 31 | syl 17 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wex 1704 wcel 1990 cfv 5888 (class class class)co 6650 cnq 9674 c1q 9675 cplq 9677 cmq 9678 crq 9679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-omul 7565 df-er 7742 df-ni 9694 df-pli 9695 df-mi 9696 df-lti 9697 df-plpq 9730 df-mpq 9731 df-enq 9733 df-nq 9734 df-erq 9735 df-plq 9736 df-mq 9737 df-1nq 9738 df-rq 9739 |
This theorem is referenced by: nsmallnq 9799 |
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