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Theorem halfnqq 7630

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 7586	. . . . . . . . 9
2		addclnq 7595	. . . . . . . . 9 $/\$
3	1, 1, 2	mp2an 426	. . . . . . . 8
4		recclnq 7612	. . . . . . . . 9
5	3, 4	ax-mp 5	. . . . . . . 8
6		distrnqg 7607	. . . . . . . 8 $/\$ $/\$
7	3, 5, 5, 6	mp3an 1373	. . . . . . 7
8		recidnq 7613	. . . . . . . . 9
9	3, 8	ax-mp 5	. . . . . . . 8
10	9, 9	oveq12i 6030	. . . . . . 7
11	7, 10	eqtri 2252	. . . . . 6
12	11	oveq1i 6028	. . . . 5
13	9	oveq2i 6029	. . . . . 6
14		addclnq 7595	. . . . . . . . 9 $/\$
15	5, 5, 14	mp2an 426	. . . . . . . 8
16		mulassnqg 7604	. . . . . . . 8 $/\$ $/\$
17	15, 3, 5, 16	mp3an 1373	. . . . . . 7
18		mulcomnqg 7603	. . . . . . . . 9 $/\$
19	15, 3, 18	mp2an 426	. . . . . . . 8
20	19	oveq1i 6028	. . . . . . 7
21	17, 20	eqtr3i 2254	. . . . . 6
22	4, 4, 14	syl2anc 411	. . . . . . 7
23		mulidnq 7609	. . . . . . 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 6029	. . 3
28		distrnqg 7607	. . . 4 $/\$ $/\$
29	5, 5, 28	mp3an23 1365	. . 3
30		mulidnq 7609	. . 3
31	27, 29, 30	3eqtr3a 2288	. 2
32		mulclnq 7596	. . . 4 $/\$
33	5, 32	mpan2 425	. . 3
34		id 19	. . . . . 6
35	34, 34	oveq12d 6036	. . . . 5
36	35	eqeq1d 2240	. . . 4
37	36	adantl 277	. . 3 $/\$
38	33, 37	rspcedv 2914	. 2
39	31, 38	mpd 13	1

Colors of variables: wff set class

Syntax hints:

wi 4

wb 105

wceq 1397

wcel 2202

wrex 2511

cfv 5326 (class class class)co 6018

cnq 7500

c1q 7501

cplq 7502

cmq 7503

crq 7504

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-iord 4463 df-on 4465 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-recs 6471 df-irdg 6536 df-1o 6582 df-oadd 6586 df-omul 6587 df-er 6702 df-ec 6704 df-qs 6708 df-ni 7524 df-pli 7525 df-mi 7526 df-plpq 7564 df-mpq 7565 df-enq 7567 df-nqqs 7568 df-plqqs 7569 df-mqqs 7570 df-1nqqs 7571 df-rq 7572

This theorem is referenced by: halfnq 7631 nsmallnqq 7632 subhalfnqq 7634 addlocpr 7756 addcanprleml 7834 addcanprlemu 7835 cauappcvgprlemm 7865 cauappcvgprlem1 7879 caucvgprlemm 7888