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Theorem halfnqq 7690

Description: One-half of any positive fraction is a fraction. (Contributed by Jim Kingdon, 23-Sep-2019.)

**Assertion**
Ref	Expression
halfnqq

**Proof of Theorem halfnqq**
Step	Hyp	Ref	Expression
1		1nq 7646	. . . . . . . . 9
2		addclnq 7655	. . . . . . . . 9 $/\$
3	1, 1, 2	mp2an 426	. . . . . . . 8
4		recclnq 7672	. . . . . . . . 9
5	3, 4	ax-mp 5	. . . . . . . 8
6		distrnqg 7667	. . . . . . . 8 $/\$ $/\$
7	3, 5, 5, 6	mp3an 1374	. . . . . . 7
8		recidnq 7673	. . . . . . . . 9
9	3, 8	ax-mp 5	. . . . . . . 8
10	9, 9	oveq12i 6040	. . . . . . 7
11	7, 10	eqtri 2252	. . . . . 6
12	11	oveq1i 6038	. . . . 5
13	9	oveq2i 6039	. . . . . 6
14		addclnq 7655	. . . . . . . . 9 $/\$
15	5, 5, 14	mp2an 426	. . . . . . . 8
16		mulassnqg 7664	. . . . . . . 8 $/\$ $/\$
17	15, 3, 5, 16	mp3an 1374	. . . . . . 7
18		mulcomnqg 7663	. . . . . . . . 9 $/\$
19	15, 3, 18	mp2an 426	. . . . . . . 8
20	19	oveq1i 6038	. . . . . . 7
21	17, 20	eqtr3i 2254	. . . . . 6
22	4, 4, 14	syl2anc 411	. . . . . . 7
23		mulidnq 7669	. . . . . . 7
24	3, 22, 23	mp2b 8	. . . . . 6
25	13, 21, 24	3eqtr3i 2260	. . . . 5
26	12, 25, 9	3eqtr3i 2260	. . . 4
27	26	oveq2i 6039	. . 3
28		distrnqg 7667	. . . 4 $/\$ $/\$
29	5, 5, 28	mp3an23 1366	. . 3
30		mulidnq 7669	. . 3
31	27, 29, 30	3eqtr3a 2288	. 2
32		mulclnq 7656	. . . 4 $/\$
33	5, 32	mpan2 425	. . 3
34		id 19	. . . . . 6
35	34, 34	oveq12d 6046	. . . . 5
36	35	eqeq1d 2240	. . . 4
37	36	adantl 277	. . 3 $/\$
38	33, 37	rspcedv 2915	. 2
39	31, 38	mpd 13	1

Colors of variables: wff set class

Syntax hints:

wi 4

wb 105

wceq 1398

wcel 2202

wrex 2512

cfv 5333 (class class class)co 6028

cnq 7560

c1q 7561

cplq 7562

cmq 7563

crq 7564

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4209 ax-sep 4212 ax-nul 4220 ax-pow 4270 ax-pr 4305 ax-un 4536 ax-setind 4641 ax-iinf 4692

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-csb 3129 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-nul 3497 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-iun 3977 df-br 4094 df-opab 4156 df-mpt 4157 df-tr 4193 df-id 4396 df-iord 4469 df-on 4471 df-suc 4474 df-iom 4695 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-rn 4742 df-res 4743 df-ima 4744 df-iota 5293 df-fun 5335 df-fn 5336 df-f 5337 df-f1 5338 df-fo 5339 df-f1o 5340 df-fv 5341 df-ov 6031 df-oprab 6032 df-mpo 6033 df-1st 6312 df-2nd 6313 df-recs 6514 df-irdg 6579 df-1o 6625 df-oadd 6629 df-omul 6630 df-er 6745 df-ec 6747 df-qs 6751 df-ni 7584 df-pli 7585 df-mi 7586 df-plpq 7624 df-mpq 7625 df-enq 7627 df-nqqs 7628 df-plqqs 7629 df-mqqs 7630 df-1nqqs 7631 df-rq 7632

This theorem is referenced by: halfnq 7691 nsmallnqq 7692 subhalfnqq 7694 addlocpr 7816 addcanprleml 7894 addcanprlemu 7895 cauappcvgprlemm 7925 cauappcvgprlem1 7939 caucvgprlemm 7948