Theorem ex-fl 15217

Description: Example for df-fl 10339. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

Assertion

Ref	Expression
ex-fl	|- ( ( |_ ` ( 3 / 2 ) ) = 1 /\ ( |_ ` -u ( 3 / 2 ) ) = -u 2 )

Proof of Theorem ex-fl

Step	Hyp	Ref	Expression
1		1re 8018	. . . 4 |- 1 e. RR
2		3re 9056	. . . . 5 |- 3 e. RR
3	2	rehalfcli 9231	. . . 4 |- ( 3 / 2 ) e. RR
4		2cn 9053	. . . . . . 7 |- 2 e. CC
5	4	mullidi 8022	. . . . . 6 |- ( 1 x. 2 ) = 2
6		2lt3 9152	. . . . . 6 |- 2 < 3
7	5, 6	eqbrtri 4050	. . . . 5 |- ( 1 x. 2 ) < 3
8		2pos 9073	. . . . . 6 |- 0 < 2
9		2re 9052	. . . . . . 7 |- 2 e. RR
10	1, 2, 9	ltmuldivi 8941	. . . . . 6 |- ( 0 < 2 -> ( ( 1 x. 2 ) < 3 <-> 1 < ( 3 / 2 ) ) )
11	8, 10	ax-mp 5	. . . . 5 |- ( ( 1 x. 2 ) < 3 <-> 1 < ( 3 / 2 ) )
12	7, 11	mpbi 145	. . . 4 |- 1 < ( 3 / 2 )
13	1, 3, 12	ltleii 8122	. . 3 |- 1 <_ ( 3 / 2 )
14		3lt4 9154	. . . . . 6 |- 3 < 4
15		2t2e4 9136	. . . . . 6 |- ( 2 x. 2 ) = 4
16	14, 15	breqtrri 4056	. . . . 5 |- 3 < ( 2 x. 2 )
17	9, 8	pm3.2i 272	. . . . . 6 |- ( 2 e. RR /\ 0 < 2 )
18		ltdivmul 8895	. . . . . 6 |- ( ( 3 e. RR /\ 2 e. RR /\ ( 2 e. RR /\ 0 < 2 ) ) -> ( ( 3 / 2 ) < 2 <-> 3 < ( 2 x. 2 ) ) )
19	2, 9, 17, 18	mp3an 1348	. . . . 5 |- ( ( 3 / 2 ) < 2 <-> 3 < ( 2 x. 2 ) )
20	16, 19	mpbir 146	. . . 4 |- ( 3 / 2 ) < 2
21		df-2 9041	. . . 4 |- 2 = ( 1 + 1 )
22	20, 21	breqtri 4054	. . 3 |- ( 3 / 2 ) < ( 1 + 1 )
23		3z 9346	. . . . 5 |- 3 e. ZZ
24		2nn 9143	. . . . 5 |- 2 e. NN
25		znq 9689	. . . . 5 |- ( ( 3 e. ZZ /\ 2 e. NN ) -> ( 3 / 2 ) e. QQ )
26	23, 24, 25	mp2an 426	. . . 4 |- ( 3 / 2 ) e. QQ
27		1z 9343	. . . 4 |- 1 e. ZZ
28		flqbi 10359	. . . 4 |- ( ( ( 3 / 2 ) e. QQ /\ 1 e. ZZ ) -> ( ( |_ ` ( 3 / 2 ) ) = 1 <-> ( 1 <_ ( 3 / 2 ) /\ ( 3 / 2 ) < ( 1 + 1 ) ) ) )
29	26, 27, 28	mp2an 426	. . 3 |- ( ( |_ ` ( 3 / 2 ) ) = 1 <-> ( 1 <_ ( 3 / 2 ) /\ ( 3 / 2 ) < ( 1 + 1 ) ) )
30	13, 22, 29	mpbir2an 944	. 2 |- ( |_ ` ( 3 / 2 ) ) = 1
31	9	renegcli 8281	. . . 4 |- -u 2 e. RR
32	3	renegcli 8281	. . . 4 |- -u ( 3 / 2 ) e. RR
33	3, 9	ltnegi 8512	. . . . 5 |- ( ( 3 / 2 ) < 2 <-> -u 2 < -u ( 3 / 2 ) )
34	20, 33	mpbi 145	. . . 4 |- -u 2 < -u ( 3 / 2 )
35	31, 32, 34	ltleii 8122	. . 3 |- -u 2 <_ -u ( 3 / 2 )
36	4	negcli 8287	. . . . . . 7 |- -u 2 e. CC
37		ax-1cn 7965	. . . . . . 7 |- 1 e. CC
38		negdi2 8277	. . . . . . 7 |- ( ( -u 2 e. CC /\ 1 e. CC ) -> -u ( -u 2 + 1 ) = ( -u -u 2 - 1 ) )
39	36, 37, 38	mp2an 426	. . . . . 6 |- -u ( -u 2 + 1 ) = ( -u -u 2 - 1 )
40	4	negnegi 8289	. . . . . . 7 |- -u -u 2 = 2
41	40	oveq1i 5928	. . . . . 6 |- ( -u -u 2 - 1 ) = ( 2 - 1 )
42	39, 41	eqtri 2214	. . . . 5 |- -u ( -u 2 + 1 ) = ( 2 - 1 )
43		2m1e1 9100	. . . . . 6 |- ( 2 - 1 ) = 1
44	43, 12	eqbrtri 4050	. . . . 5 |- ( 2 - 1 ) < ( 3 / 2 )
45	42, 44	eqbrtri 4050	. . . 4 |- -u ( -u 2 + 1 ) < ( 3 / 2 )
46	31, 1	readdcli 8032	. . . . 5 |- ( -u 2 + 1 ) e. RR
47	46, 3	ltnegcon1i 8518	. . . 4 |- ( -u ( -u 2 + 1 ) < ( 3 / 2 ) <-> -u ( 3 / 2 ) < ( -u 2 + 1 ) )
48	45, 47	mpbi 145	. . 3 |- -u ( 3 / 2 ) < ( -u 2 + 1 )
49		qnegcl 9701	. . . . 5 |- ( ( 3 / 2 ) e. QQ -> -u ( 3 / 2 ) e. QQ )
50	26, 49	ax-mp 5	. . . 4 |- -u ( 3 / 2 ) e. QQ
51		2z 9345	. . . . 5 |- 2 e. ZZ
52		znegcl 9348	. . . . 5 |- ( 2 e. ZZ -> -u 2 e. ZZ )
53	51, 52	ax-mp 5	. . . 4 |- -u 2 e. ZZ
54		flqbi 10359	. . . 4 |- ( ( -u ( 3 / 2 ) e. QQ /\ -u 2 e. ZZ ) -> ( ( |_ ` -u ( 3 / 2 ) ) = -u 2 <-> ( -u 2 <_ -u ( 3 / 2 ) /\ -u ( 3 / 2 ) < ( -u 2 + 1 ) ) ) )
55	50, 53, 54	mp2an 426	. . 3 |- ( ( |_ ` -u ( 3 / 2 ) ) = -u 2 <-> ( -u 2 <_ -u ( 3 / 2 ) /\ -u ( 3 / 2 ) < ( -u 2 + 1 ) ) )
56	35, 48, 55	mpbir2an 944	. 2 |- ( |_ ` -u ( 3 / 2 ) ) = -u 2
57	30, 56	pm3.2i 272	1 |- ( ( |_ ` ( 3 / 2 ) ) = 1 /\ ( |_ ` -u ( 3 / 2 ) ) = -u 2 )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1364   e. wcel 2164   class class class wbr 4029   ` cfv 5254  (class class class)co 5918   CCcc 7870   RRcr 7871   0cc0 7872   1c1 7873   + caddc 7875   x. cmul 7877   < clt 8054   <_ cle 8055   - cmin 8190   -ucneg 8191   / cdiv 8691   NNcn 8982   2c2 9033   3c3 9034   4c4 9035   ZZcz 9317   QQcq 9684   |_cfl 10337

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  ax-arch 7991

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-2 9041  df-3 9042  df-4 9043  df-n0 9241  df-z 9318  df-q 9685  df-rp 9720  df-fl 10339

This theorem is referenced by:  ex-ceil  15218