qreccl - Intuitionistic Logic Explorer

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Theorem qreccl 9707

Description: Closure of reciprocal of rationals. (Contributed by NM, 3-Aug-2004.)

**Assertion**
Ref	Expression
qreccl	$/\$

Proof of Theorem qreccl Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ax-1cn 7965	. . . . . 6
2		1ap0 8609	. . . . . 6 #
3	1, 2	div0api 8765	. . . . 5
4		0z 9328	. . . . . 6
5		1nn 8993	. . . . . 6
6		znq 9689	. . . . . 6 $/\$
7	4, 5, 6	mp2an 426	. . . . 5
8	3, 7	eqeltrri 2267	. . . 4
9		qapne 9704	. . . 4 $/\$ #
10	8, 9	mpan2 425	. . 3 #
11	10	biimpar 297	. 2 $/\$ #
12		elq 9687	. . . 4
13		nnne0 9010	. . . . . . . 8
14	13	ancli 323	. . . . . . 7 $/\$
15		nnz 9336	. . . . . . . . . . . . . . . 16
16		zapne 9391	. . . . . . . . . . . . . . . 16 $/\$ #
17	15, 4, 16	sylancl 413	. . . . . . . . . . . . . . 15 #
18	17	adantl 277	. . . . . . . . . . . . . 14 $/\$ #
19	18	pm5.32i 454	. . . . . . . . . . . . 13 $/\$ $/\$ # $/\$ $/\$
20	19	anbi1i 458	. . . . . . . . . . . 12 $/\$ $/\$ # $/\$ $/\$ $/\$ $/\$
21		breq1 4032	. . . . . . . . . . . . 13 # #
22		zcn 9322	. . . . . . . . . . . . . . . 16
23		nncn 8990	. . . . . . . . . . . . . . . 16
24	22, 23	anim12i 338	. . . . . . . . . . . . . . 15 $/\$ $/\$
25		divap0b 8702	. . . . . . . . . . . . . . . 16 $/\$ $/\$ # # #
26	25	3expa 1205	. . . . . . . . . . . . . . 15 $/\$ $/\$ # # #
27	24, 26	sylan 283	. . . . . . . . . . . . . 14 $/\$ $/\$ # # #
28	27	bicomd 141	. . . . . . . . . . . . 13 $/\$ $/\$ # # #
29	21, 28	sylan9bbr 463	. . . . . . . . . . . 12 $/\$ $/\$ # $/\$ # #
30	20, 29	sylbir 135	. . . . . . . . . . 11 $/\$ $/\$ $/\$ # #
31		simplll 533	. . . . . . . . . . . 12 $/\$ $/\$ $/\$
32		zapne 9391	. . . . . . . . . . . 12 $/\$ #
33	31, 4, 32	sylancl 413	. . . . . . . . . . 11 $/\$ $/\$ $/\$ #
34	30, 33	bitrd 188	. . . . . . . . . 10 $/\$ $/\$ $/\$ #
35		zmulcl 9370	. . . . . . . . . . . . . . . . 17 $/\$
36	15, 35	sylan2 286	. . . . . . . . . . . . . . . 16 $/\$
37	36	adantr 276	. . . . . . . . . . . . . . 15 $/\$ $/\$
38		msqznn 9417	. . . . . . . . . . . . . . . 16 $/\$
39	38	adantlr 477	. . . . . . . . . . . . . . 15 $/\$ $/\$
40	37, 39	jca 306	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$
41	40	adantlr 477	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
42	41	adantlr 477	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
43	20	anbi1i 458	. . . . . . . . . . . . . 14 $/\$ $/\$ # $/\$ $/\$ # $/\$ $/\$ $/\$ $/\$ #
44	33	pm5.32i 454	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ # $/\$ $/\$ $/\$ $/\$
45	43, 44	bitri 184	. . . . . . . . . . . . 13 $/\$ $/\$ # $/\$ $/\$ # $/\$ $/\$ $/\$ $/\$
46		oveq2 5926	. . . . . . . . . . . . . . 15
47		dividap 8720	. . . . . . . . . . . . . . . . . . . . 21 $/\$ #
48	47	adantr 276	. . . . . . . . . . . . . . . . . . . 20 $/\$ # $/\$ $/\$ #
49	48	oveq1d 5933	. . . . . . . . . . . . . . . . . . 19 $/\$ # $/\$ $/\$ #
50		simpll 527	. . . . . . . . . . . . . . . . . . . 20 $/\$ # $/\$ $/\$ #
51		simpl 109	. . . . . . . . . . . . . . . . . . . 20 $/\$ # $/\$ $/\$ # $/\$ #
52		simpr 110	. . . . . . . . . . . . . . . . . . . 20 $/\$ # $/\$ $/\$ # $/\$ #
53		divdivdivap 8732	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$ # $/\$ $/\$ # $/\$ $/\$ #
54	50, 51, 51, 52, 53	syl22anc 1250	. . . . . . . . . . . . . . . . . . 19 $/\$ # $/\$ $/\$ #
55	49, 54	eqtr3d 2228	. . . . . . . . . . . . . . . . . 18 $/\$ # $/\$ $/\$ #
56	55	an4s 588	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ # $/\$ #
57	24, 56	sylan 283	. . . . . . . . . . . . . . . 16 $/\$ $/\$ # $/\$ #
58	57	anass1rs 571	. . . . . . . . . . . . . . 15 $/\$ $/\$ # $/\$ #
59	46, 58	sylan9eqr 2248	. . . . . . . . . . . . . 14 $/\$ $/\$ # $/\$ # $/\$
60	59	an32s 568	. . . . . . . . . . . . 13 $/\$ $/\$ # $/\$ $/\$ #
61	45, 60	sylbir 135	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
62	42, 61	jca 306	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
63	62	ex 115	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
64	34, 63	sylbid 150	. . . . . . . . 9 $/\$ $/\$ $/\$ # $/\$ $/\$
65	64	ex 115	. . . . . . . 8 $/\$ $/\$ # $/\$ $/\$
66	65	anasss 399	. . . . . . 7 $/\$ $/\$ # $/\$ $/\$
67	14, 66	sylan2 286	. . . . . 6 $/\$ # $/\$ $/\$
68		rspceov 5960	. . . . . . . 8 $/\$ $/\$
69	68	3expa 1205	. . . . . . 7 $/\$ $/\$
70		elq 9687	. . . . . . 7
71	69, 70	sylibr 134	. . . . . 6 $/\$ $/\$
72	67, 71	syl8 71	. . . . 5 $/\$ #
73	72	rexlimivv 2617	. . . 4 #
74	12, 73	sylbi 121	. . 3 #
75	74	imp 124	. 2 $/\$ #
76	11, 75	syldan 282	1 $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105

wceq 1364

wcel 2164

wne 2364

wrex 2473 class class class wbr 4029 (class class class)co 5918

cc 7870

cc0 7872

c1 7873

cmul 7877 # cap 8600

cdiv 8691

cn 8982

cz 9317

cq 9684

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-po 4327 df-iso 4328 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-fv 5262 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-n0 9241 df-z 9318 df-q 9685

This theorem is referenced by: qdivcl 9708 qexpclz 10631

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