Theorem qreccl 7453

Description: Closure of reciprocal of rationals.

Assertion

Ref	Expression
qreccl	|- ((A e. QQ /\ A =/= 0) -> (1 / A) e. QQ)

Proof of Theorem qreccl

Step	Hyp	Ref	Expression
1		elq 7437	. . 3 |- (A e. QQ <-> E.x e. ZZ E.y e. NN A = (x / y))
2		neeq1 2024	. . . . . . . . . 10 |- (A = (x / y) -> (A =/= 0 <-> (x / y) =/= 0))
3		divne0b 6911	. . . . . . . . . . . . 13 |- ((x e. CC /\ y e. CC /\ y =/= 0) -> (x =/= 0 <-> (x / y) =/= 0))
4	3	3expa 1067	. . . . . . . . . . . 12 |- (((x e. CC /\ y e. CC) /\ y =/= 0) -> (x =/= 0 <-> (x / y) =/= 0))
5		zcn 7349	. . . . . . . . . . . . 13 |- (x e. ZZ -> x e. CC)
6		nncn 7113	. . . . . . . . . . . . 13 |- (y e. NN -> y e. CC)
7	5, 6	anim12i 360	. . . . . . . . . . . 12 |- ((x e. ZZ /\ y e. NN) -> (x e. CC /\ y e. CC))
8	4, 7	sylan 497	. . . . . . . . . . 11 |- (((x e. ZZ /\ y e. NN) /\ y =/= 0) -> (x =/= 0 <-> (x / y) =/= 0))
9	8	bicomd 580	. . . . . . . . . 10 |- (((x e. ZZ /\ y e. NN) /\ y =/= 0) -> ((x / y) =/= 0 <-> x =/= 0))
10	2, 9	sylan9bbr 600	. . . . . . . . 9 |- ((((x e. ZZ /\ y e. NN) /\ y =/= 0) /\ A = (x / y)) -> (A =/= 0 <-> x =/= 0))
11		zmulcl 7389	. . . . . . . . . . . . . . . 16 |- ((x e. ZZ /\ y e. ZZ) -> (x x. y) e. ZZ)
12		nnz 7362	. . . . . . . . . . . . . . . 16 |- (y e. NN -> y e. ZZ)
13	11, 12	sylan2 500	. . . . . . . . . . . . . . 15 |- ((x e. ZZ /\ y e. NN) -> (x x. y) e. ZZ)
14	13	adantr 425	. . . . . . . . . . . . . 14 |- (((x e. ZZ /\ y e. NN) /\ x =/= 0) -> (x x. y) e. ZZ)
15		msqznn 7408	. . . . . . . . . . . . . . 15 |- ((x e. ZZ /\ x =/= 0) -> (x x. x) e. NN)
16	15	adantlr 429	. . . . . . . . . . . . . 14 |- (((x e. ZZ /\ y e. NN) /\ x =/= 0) -> (x x. x) e. NN)
17	14, 16	jca 310	. . . . . . . . . . . . 13 |- (((x e. ZZ /\ y e. NN) /\ x =/= 0) -> ((x x. y) e. ZZ /\ (x x. x) e. NN))
18	17	adantlr 429	. . . . . . . . . . . 12 |- ((((x e. ZZ /\ y e. NN) /\ y =/= 0) /\ x =/= 0) -> ((x x. y) e. ZZ /\ (x x. x) e. NN))
19	18	adantlr 429	. . . . . . . . . . 11 |- (((((x e. ZZ /\ y e. NN) /\ y =/= 0) /\ A = (x / y)) /\ x =/= 0) -> ((x x. y) e. ZZ /\ (x x. x) e. NN))
20		opreq2 4890	. . . . . . . . . . . . 13 |- (A = (x / y) -> (1 / A) = (1 / (x / y)))
21		divid 6942	. . . . . . . . . . . . . . . . . . . 20 |- ((x e. CC /\ x =/= 0) -> (x / x) = 1)
22	21	adantr 425	. . . . . . . . . . . . . . . . . . 19 |- (((x e. CC /\ x =/= 0) /\ (y e. CC /\ y =/= 0)) -> (x / x) = 1)
23	22	opreq1d 4897	. . . . . . . . . . . . . . . . . 18 |- (((x e. CC /\ x =/= 0) /\ (y e. CC /\ y =/= 0)) -> ((x / x) / (x / y)) = (1 / (x / y)))
24		simpll 448	. . . . . . . . . . . . . . . . . . 19 |- (((x e. CC /\ x =/= 0) /\ (y e. CC /\ y =/= 0)) -> x e. CC)
25		simpl 346	. . . . . . . . . . . . . . . . . . 19 |- (((x e. CC /\ x =/= 0) /\ (y e. CC /\ y =/= 0)) -> (x e. CC /\ x =/= 0))
26		simpr 350	. . . . . . . . . . . . . . . . . . 19 |- (((x e. CC /\ x =/= 0) /\ (y e. CC /\ y =/= 0)) -> (y e. CC /\ y =/= 0))
27		divdivdiv 6961	. . . . . . . . . . . . . . . . . . 19 |- (((x e. CC /\ (x e. CC /\ x =/= 0)) /\ ((x e. CC /\ x =/= 0) /\ (y e. CC /\ y =/= 0))) -> ((x / x) / (x / y)) = ((x x. y) / (x x. x)))
28	24, 25, 25, 26, 27	syl22anc 1101	. . . . . . . . . . . . . . . . . 18 |- (((x e. CC /\ x =/= 0) /\ (y e. CC /\ y =/= 0)) -> ((x / x) / (x / y)) = ((x x. y) / (x x. x)))
29	23, 28	eqtr3d 1927	. . . . . . . . . . . . . . . . 17 |- (((x e. CC /\ x =/= 0) /\ (y e. CC /\ y =/= 0)) -> (1 / (x / y)) = ((x x. y) / (x x. x)))
30	29	an4s 566	. . . . . . . . . . . . . . . 16 |- (((x e. CC /\ y e. CC) /\ (x =/= 0 /\ y =/= 0)) -> (1 / (x / y)) = ((x x. y) / (x x. x)))
31	30, 7	sylan 497	. . . . . . . . . . . . . . 15 |- (((x e. ZZ /\ y e. NN) /\ (x =/= 0 /\ y =/= 0)) -> (1 / (x / y)) = ((x x. y) / (x x. x)))
32	31	ancom2s 545	. . . . . . . . . . . . . 14 |- (((x e. ZZ /\ y e. NN) /\ (y =/= 0 /\ x =/= 0)) -> (1 / (x / y)) = ((x x. y) / (x x. x)))
33	32	anassrs 489	. . . . . . . . . . . . 13 |- ((((x e. ZZ /\ y e. NN) /\ y =/= 0) /\ x =/= 0) -> (1 / (x / y)) = ((x x. y) / (x x. x)))
34	20, 33	sylan9eqr 1951	. . . . . . . . . . . 12 |- (((((x e. ZZ /\ y e. NN) /\ y =/= 0) /\ x =/= 0) /\ A = (x / y)) -> (1 / A) = ((x x. y) / (x x. x)))
35	34	an1rs 547	. . . . . . . . . . 11 |- (((((x e. ZZ /\ y e. NN) /\ y =/= 0) /\ A = (x / y)) /\ x =/= 0) -> (1 / A) = ((x x. y) / (x x. x)))
36	19, 35	jca 310	. . . . . . . . . 10 |- (((((x e. ZZ /\ y e. NN) /\ y =/= 0) /\ A = (x / y)) /\ x =/= 0) -> (((x x. y) e. ZZ /\ (x x. x) e. NN) /\ (1 / A) = ((x x. y) / (x x. x))))
37	36	ex 402	. . . . . . . . 9 |- ((((x e. ZZ /\ y e. NN) /\ y =/= 0) /\ A = (x / y)) -> (x =/= 0 -> (((x x. y) e. ZZ /\ (x x. x) e. NN) /\ (1 / A) = ((x x. y) / (x x. x)))))
38	10, 37	sylbid 220	. . . . . . . 8 |- ((((x e. ZZ /\ y e. NN) /\ y =/= 0) /\ A = (x / y)) -> (A =/= 0 -> (((x x. y) e. ZZ /\ (x x. x) e. NN) /\ (1 / A) = ((x x. y) / (x x. x)))))
39	38	ex 402	. . . . . . 7 |- (((x e. ZZ /\ y e. NN) /\ y =/= 0) -> (A = (x / y) -> (A =/= 0 -> (((x x. y) e. ZZ /\ (x x. x) e. NN) /\ (1 / A) = ((x x. y) / (x x. x))))))
40	39	anasss 488	. . . . . 6 |- ((x e. ZZ /\ (y e. NN /\ y =/= 0)) -> (A = (x / y) -> (A =/= 0 -> (((x x. y) e. ZZ /\ (x x. x) e. NN) /\ (1 / A) = ((x x. y) / (x x. x))))))
41		nnne0 7132	. . . . . . 7 |- (y e. NN -> y =/= 0)
42	41	ancli 320	. . . . . 6 |- (y e. NN -> (y e. NN /\ y =/= 0))
43	40, 42	sylan2 500	. . . . 5 |- ((x e. ZZ /\ y e. NN) -> (A = (x / y) -> (A =/= 0 -> (((x x. y) e. ZZ /\ (x x. x) e. NN) /\ (1 / A) = ((x x. y) / (x x. x))))))
44		rcla4eopr 4915	. . . . . . 7 |- (((x x. y) e. ZZ /\ (x x. x) e. NN /\ (1 / A) = ((x x. y) / (x x. x))) -> E.z e. ZZ E.w e. NN (1 / A) = (z / w))
45	44	3expa 1067	. . . . . 6 |- ((((x x. y) e. ZZ /\ (x x. x) e. NN) /\ (1 / A) = ((x x. y) / (x x. x))) -> E.z e. ZZ E.w e. NN (1 / A) = (z / w))
46		elq 7437	. . . . . 6 |- ((1 / A) e. QQ <-> E.z e. ZZ E.w e. NN (1 / A) = (z / w))
47	45, 46	sylibr 217	. . . . 5 |- ((((x x. y) e. ZZ /\ (x x. x) e. NN) /\ (1 / A) = ((x x. y) / (x x. x))) -> (1 / A) e. QQ)
48	43, 47	syl8 27	. . . 4 |- ((x e. ZZ /\ y e. NN) -> (A = (x / y) -> (A =/= 0 -> (1 / A) e. QQ)))
49	48	r19.23aivv 2217	. . 3 |- (E.x e. ZZ E.y e. NN A = (x / y) -> (A =/= 0 -> (1 / A) e. QQ))
50	1, 49	sylbi 216	. 2 |- (A e. QQ -> (A =/= 0 -> (1 / A) e. QQ))
51	50	imp 377	1 |- ((A e. QQ /\ A =/= 0) -> (1 / A) e. QQ)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  E.wrex 2106  (class class class)co 4884  CCcc 6384  0cc0 6386  1c1 6387   x. cmul 6391   / cdiv 6447  NNcn 6449  ZZcz 6451  QQcq 6452

This theorem is referenced by:  qdivcl 7454

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-n0 7309  df-z 7345  df-q 7436

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