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Theorem halfnqq 7477

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 7433	. . . . . . . . 9
2		addclnq 7442	. . . . . . . . 9 $/\$
3	1, 1, 2	mp2an 426	. . . . . . . 8
4		recclnq 7459	. . . . . . . . 9
5	3, 4	ax-mp 5	. . . . . . . 8
6		distrnqg 7454	. . . . . . . 8 $/\$ $/\$
7	3, 5, 5, 6	mp3an 1348	. . . . . . 7
8		recidnq 7460	. . . . . . . . 9
9	3, 8	ax-mp 5	. . . . . . . 8
10	9, 9	oveq12i 5934	. . . . . . 7
11	7, 10	eqtri 2217	. . . . . 6
12	11	oveq1i 5932	. . . . 5
13	9	oveq2i 5933	. . . . . 6
14		addclnq 7442	. . . . . . . . 9 $/\$
15	5, 5, 14	mp2an 426	. . . . . . . 8
16		mulassnqg 7451	. . . . . . . 8 $/\$ $/\$
17	15, 3, 5, 16	mp3an 1348	. . . . . . 7
18		mulcomnqg 7450	. . . . . . . . 9 $/\$
19	15, 3, 18	mp2an 426	. . . . . . . 8
20	19	oveq1i 5932	. . . . . . 7
21	17, 20	eqtr3i 2219	. . . . . 6
22	4, 4, 14	syl2anc 411	. . . . . . 7
23		mulidnq 7456	. . . . . . 7
24	3, 22, 23	mp2b 8	. . . . . 6
25	13, 21, 24	3eqtr3i 2225	. . . . 5
26	12, 25, 9	3eqtr3i 2225	. . . 4
27	26	oveq2i 5933	. . 3
28		distrnqg 7454	. . . 4 $/\$ $/\$
29	5, 5, 28	mp3an23 1340	. . 3
30		mulidnq 7456	. . 3
31	27, 29, 30	3eqtr3a 2253	. 2
32		mulclnq 7443	. . . 4 $/\$
33	5, 32	mpan2 425	. . 3
34		id 19	. . . . . 6
35	34, 34	oveq12d 5940	. . . . 5
36	35	eqeq1d 2205	. . . 4
37	36	adantl 277	. . 3 $/\$
38	33, 37	rspcedv 2872	. 2
39	31, 38	mpd 13	1

Colors of variables: wff set class

Syntax hints:

wi 4

wb 105

wceq 1364

wcel 2167

wrex 2476

cfv 5258 (class class class)co 5922

cnq 7347

c1q 7348

cplq 7349

cmq 7350

crq 7351

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-iord 4401 df-on 4403 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-irdg 6428 df-1o 6474 df-oadd 6478 df-omul 6479 df-er 6592 df-ec 6594 df-qs 6598 df-ni 7371 df-pli 7372 df-mi 7373 df-plpq 7411 df-mpq 7412 df-enq 7414 df-nqqs 7415 df-plqqs 7416 df-mqqs 7417 df-1nqqs 7418 df-rq 7419

This theorem is referenced by: halfnq 7478 nsmallnqq 7479 subhalfnqq 7481 addlocpr 7603 addcanprleml 7681 addcanprlemu 7682 cauappcvgprlemm 7712 cauappcvgprlem1 7726 caucvgprlemm 7735