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Theorem halfnqq 7242

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 7198	. . . . . . . . 9
2		addclnq 7207	. . . . . . . . 9 $/\$
3	1, 1, 2	mp2an 423	. . . . . . . 8
4		recclnq 7224	. . . . . . . . 9
5	3, 4	ax-mp 5	. . . . . . . 8
6		distrnqg 7219	. . . . . . . 8 $/\$ $/\$
7	3, 5, 5, 6	mp3an 1316	. . . . . . 7
8		recidnq 7225	. . . . . . . . 9
9	3, 8	ax-mp 5	. . . . . . . 8
10	9, 9	oveq12i 5794	. . . . . . 7
11	7, 10	eqtri 2161	. . . . . 6
12	11	oveq1i 5792	. . . . 5
13	9	oveq2i 5793	. . . . . 6
14		addclnq 7207	. . . . . . . . 9 $/\$
15	5, 5, 14	mp2an 423	. . . . . . . 8
16		mulassnqg 7216	. . . . . . . 8 $/\$ $/\$
17	15, 3, 5, 16	mp3an 1316	. . . . . . 7
18		mulcomnqg 7215	. . . . . . . . 9 $/\$
19	15, 3, 18	mp2an 423	. . . . . . . 8
20	19	oveq1i 5792	. . . . . . 7
21	17, 20	eqtr3i 2163	. . . . . 6
22	4, 4, 14	syl2anc 409	. . . . . . 7
23		mulidnq 7221	. . . . . . 7
24	3, 22, 23	mp2b 8	. . . . . 6
25	13, 21, 24	3eqtr3i 2169	. . . . 5
26	12, 25, 9	3eqtr3i 2169	. . . 4
27	26	oveq2i 5793	. . 3
28		distrnqg 7219	. . . 4 $/\$ $/\$
29	5, 5, 28	mp3an23 1308	. . 3
30		mulidnq 7221	. . 3
31	27, 29, 30	3eqtr3a 2197	. 2
32		mulclnq 7208	. . . 4 $/\$
33	5, 32	mpan2 422	. . 3
34		id 19	. . . . . 6
35	34, 34	oveq12d 5800	. . . . 5
36	35	eqeq1d 2149	. . . 4
37	36	adantl 275	. . 3 $/\$
38	33, 37	rspcedv 2797	. 2
39	31, 38	mpd 13	1

Colors of variables: wff set class

Syntax hints:

wi 4

wb 104

wceq 1332

wcel 1481

wrex 2418

cfv 5131 (class class class)co 5782

cnq 7112

c1q 7113

cplq 7114

cmq 7115

crq 7116

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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510

This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-irdg 6275 df-1o 6321 df-oadd 6325 df-omul 6326 df-er 6437 df-ec 6439 df-qs 6443 df-ni 7136 df-pli 7137 df-mi 7138 df-plpq 7176 df-mpq 7177 df-enq 7179 df-nqqs 7180 df-plqqs 7181 df-mqqs 7182 df-1nqqs 7183 df-rq 7184

This theorem is referenced by: halfnq 7243 nsmallnqq 7244 subhalfnqq 7246 addlocpr 7368 addcanprleml 7446 addcanprlemu 7447 cauappcvgprlemm 7477 cauappcvgprlem1 7491 caucvgprlemm 7500