Theorem ex-fl 12360

Description: Example for df-fl 9826. 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 7584	. . . 4 |- 1 e. RR
2		3re 8594	. . . . 5 |- 3 e. RR
3	2	rehalfcli 8762	. . . 4 |- ( 3 / 2 ) e. RR
4		2cn 8591	. . . . . . 7 |- 2 e. CC
5	4	mulid2i 7588	. . . . . 6 |- ( 1 x. 2 ) = 2
6		2lt3 8684	. . . . . 6 |- 2 < 3
7	5, 6	eqbrtri 3886	. . . . 5 |- ( 1 x. 2 ) < 3
8		2pos 8611	. . . . . 6 |- 0 < 2
9		2re 8590	. . . . . . 7 |- 2 e. RR
10	1, 2, 9	ltmuldivi 8480	. . . . . 6 |- ( 0 < 2 -> ( ( 1 x. 2 ) < 3 <-> 1 < ( 3 / 2 ) ) )
11	8, 10	ax-mp 7	. . . . 5 |- ( ( 1 x. 2 ) < 3 <-> 1 < ( 3 / 2 ) )
12	7, 11	mpbi 144	. . . 4 |- 1 < ( 3 / 2 )
13	1, 3, 12	ltleii 7684	. . 3 |- 1 <_ ( 3 / 2 )
14		3lt4 8686	. . . . . 6 |- 3 < 4
15		2t2e4 8668	. . . . . 6 |- ( 2 x. 2 ) = 4
16	14, 15	breqtrri 3892	. . . . 5 |- 3 < ( 2 x. 2 )
17	9, 8	pm3.2i 267	. . . . . 6 |- ( 2 e. RR /\ 0 < 2 )
18		ltdivmul 8434	. . . . . 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 1280	. . . . 5 |- ( ( 3 / 2 ) < 2 <-> 3 < ( 2 x. 2 ) )
20	16, 19	mpbir 145	. . . 4 |- ( 3 / 2 ) < 2
21		df-2 8579	. . . 4 |- 2 = ( 1 + 1 )
22	20, 21	breqtri 3890	. . 3 |- ( 3 / 2 ) < ( 1 + 1 )
23		3z 8877	. . . . 5 |- 3 e. ZZ
24		2nn 8675	. . . . 5 |- 2 e. NN
25		znq 9208	. . . . 5 |- ( ( 3 e. ZZ /\ 2 e. NN ) -> ( 3 / 2 ) e. QQ )
26	23, 24, 25	mp2an 418	. . . 4 |- ( 3 / 2 ) e. QQ
27		1z 8874	. . . 4 |- 1 e. ZZ
28		flqbi 9846	. . . 4 |- ( ( ( 3 / 2 ) e. QQ /\ 1 e. ZZ ) -> ( ( |_ ` ( 3 / 2 ) ) = 1 <-> ( 1 <_ ( 3 / 2 ) /\ ( 3 / 2 ) < ( 1 + 1 ) ) ) )
29	26, 27, 28	mp2an 418	. . 3 |- ( ( |_ ` ( 3 / 2 ) ) = 1 <-> ( 1 <_ ( 3 / 2 ) /\ ( 3 / 2 ) < ( 1 + 1 ) ) )
30	13, 22, 29	mpbir2an 891	. 2 |- ( |_ ` ( 3 / 2 ) ) = 1
31	9	renegcli 7841	. . . 4 |- -u 2 e. RR
32	3	renegcli 7841	. . . 4 |- -u ( 3 / 2 ) e. RR
33	3, 9	ltnegi 8068	. . . . 5 |- ( ( 3 / 2 ) < 2 <-> -u 2 < -u ( 3 / 2 ) )
34	20, 33	mpbi 144	. . . 4 |- -u 2 < -u ( 3 / 2 )
35	31, 32, 34	ltleii 7684	. . 3 |- -u 2 <_ -u ( 3 / 2 )
36	4	negcli 7847	. . . . . . 7 |- -u 2 e. CC
37		ax-1cn 7535	. . . . . . 7 |- 1 e. CC
38		negdi2 7837	. . . . . . 7 |- ( ( -u 2 e. CC /\ 1 e. CC ) -> -u ( -u 2 + 1 ) = ( -u -u 2 - 1 ) )
39	36, 37, 38	mp2an 418	. . . . . 6 |- -u ( -u 2 + 1 ) = ( -u -u 2 - 1 )
40	4	negnegi 7849	. . . . . . 7 |- -u -u 2 = 2
41	40	oveq1i 5700	. . . . . 6 |- ( -u -u 2 - 1 ) = ( 2 - 1 )
42	39, 41	eqtri 2115	. . . . 5 |- -u ( -u 2 + 1 ) = ( 2 - 1 )
43		2m1e1 8638	. . . . . 6 |- ( 2 - 1 ) = 1
44	43, 12	eqbrtri 3886	. . . . 5 |- ( 2 - 1 ) < ( 3 / 2 )
45	42, 44	eqbrtri 3886	. . . 4 |- -u ( -u 2 + 1 ) < ( 3 / 2 )
46	31, 1	readdcli 7598	. . . . 5 |- ( -u 2 + 1 ) e. RR
47	46, 3	ltnegcon1i 8074	. . . 4 |- ( -u ( -u 2 + 1 ) < ( 3 / 2 ) <-> -u ( 3 / 2 ) < ( -u 2 + 1 ) )
48	45, 47	mpbi 144	. . 3 |- -u ( 3 / 2 ) < ( -u 2 + 1 )
49		qnegcl 9220	. . . . 5 |- ( ( 3 / 2 ) e. QQ -> -u ( 3 / 2 ) e. QQ )
50	26, 49	ax-mp 7	. . . 4 |- -u ( 3 / 2 ) e. QQ
51		2z 8876	. . . . 5 |- 2 e. ZZ
52		znegcl 8879	. . . . 5 |- ( 2 e. ZZ -> -u 2 e. ZZ )
53	51, 52	ax-mp 7	. . . 4 |- -u 2 e. ZZ
54		flqbi 9846	. . . 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 418	. . 3 |- ( ( |_ ` -u ( 3 / 2 ) ) = -u 2 <-> ( -u 2 <_ -u ( 3 / 2 ) /\ -u ( 3 / 2 ) < ( -u 2 + 1 ) ) )
56	35, 48, 55	mpbir2an 891	. 2 |- ( |_ ` -u ( 3 / 2 ) ) = -u 2
57	30, 56	pm3.2i 267	1 |- ( ( |_ ` ( 3 / 2 ) ) = 1 /\ ( |_ ` -u ( 3 / 2 ) ) = -u 2 )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1296   e. wcel 1445   class class class wbr 3867   ` cfv 5049  (class class class)co 5690   CCcc 7445   RRcr 7446   0cc0 7447   1c1 7448   + caddc 7450   x. cmul 7452   < clt 7619   <_ cle 7620   - cmin 7750   -ucneg 7751   / cdiv 8236   NNcn 8520   2c2 8571   3c3 8572   4c4 8573   ZZcz 8848   QQcq 9203   |_cfl 9824

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-un 4284  ax-setind 4381  ax-cnex 7533  ax-resscn 7534  ax-1cn 7535  ax-1re 7536  ax-icn 7537  ax-addcl 7538  ax-addrcl 7539  ax-mulcl 7540  ax-mulrcl 7541  ax-addcom 7542  ax-mulcom 7543  ax-addass 7544  ax-mulass 7545  ax-distr 7546  ax-i2m1 7547  ax-0lt1 7548  ax-1rid 7549  ax-0id 7550  ax-rnegex 7551  ax-precex 7552  ax-cnre 7553  ax-pre-ltirr 7554  ax-pre-ltwlin 7555  ax-pre-lttrn 7556  ax-pre-apti 7557  ax-pre-ltadd 7558  ax-pre-mulgt0 7559  ax-pre-mulext 7560  ax-arch 7561

This theorem depends on definitions:  df-bi 116  df-3or 928  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-nel 2358  df-ral 2375  df-rex 2376  df-reu 2377  df-rmo 2378  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-int 3711  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-po 4147  df-iso 4148  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-fv 5057  df-riota 5646  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-1st 5949  df-2nd 5950  df-pnf 7621  df-mnf 7622  df-xr 7623  df-ltxr 7624  df-le 7625  df-sub 7752  df-neg 7753  df-reap 8149  df-ap 8156  df-div 8237  df-inn 8521  df-2 8579  df-3 8580  df-4 8581  df-n0 8772  df-z 8849  df-q 9204  df-rp 9234  df-fl 9826

This theorem is referenced by:  ex-ceil  12361