Theorem ex-fl 16619

Description: Example for df-fl 10654. 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 8289	. . . 4 |- 1 e. RR
2		3re 9328	. . . . 5 |- 3 e. RR
3	2	rehalfcli 9504	. . . 4 |- ( 3 / 2 ) e. RR
4		2cn 9325	. . . . . . 7 |- 2 e. CC
5	4	mullidi 8293	. . . . . 6 |- ( 1 x. 2 ) = 2
6		2lt3 9425	. . . . . 6 |- 2 < 3
7	5, 6	eqbrtri 4135	. . . . 5 |- ( 1 x. 2 ) < 3
8		2pos 9345	. . . . . 6 |- 0 < 2
9		2re 9324	. . . . . . 7 |- 2 e. RR
10	1, 2, 9	ltmuldivi 9213	. . . . . 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 8392	. . 3 |- 1 <_ ( 3 / 2 )
14		3lt4 9427	. . . . . 6 |- 3 < 4
15		2t2e4 9409	. . . . . 6 |- ( 2 x. 2 ) = 4
16	14, 15	breqtrri 4141	. . . . 5 |- 3 < ( 2 x. 2 )
17	9, 8	pm3.2i 272	. . . . . 6 |- ( 2 e. RR /\ 0 < 2 )
18		ltdivmul 9167	. . . . . 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 1374	. . . . 5 |- ( ( 3 / 2 ) < 2 <-> 3 < ( 2 x. 2 ) )
20	16, 19	mpbir 146	. . . 4 |- ( 3 / 2 ) < 2
21		df-2 9313	. . . 4 |- 2 = ( 1 + 1 )
22	20, 21	breqtri 4139	. . 3 |- ( 3 / 2 ) < ( 1 + 1 )
23		3z 9623	. . . . 5 |- 3 e. ZZ
24		2nn 9416	. . . . 5 |- 2 e. NN
25		znq 9974	. . . . 5 |- ( ( 3 e. ZZ /\ 2 e. NN ) -> ( 3 / 2 ) e. QQ )
26	23, 24, 25	mp2an 426	. . . 4 |- ( 3 / 2 ) e. QQ
27		1z 9620	. . . 4 |- 1 e. ZZ
28		flqbi 10674	. . . 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 951	. 2 |- ( |_ ` ( 3 / 2 ) ) = 1
31	9	renegcli 8551	. . . 4 |- -u 2 e. RR
32	3	renegcli 8551	. . . 4 |- -u ( 3 / 2 ) e. RR
33	3, 9	ltnegi 8784	. . . . 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 8392	. . 3 |- -u 2 <_ -u ( 3 / 2 )
36	4	negcli 8557	. . . . . . 7 |- -u 2 e. CC
37		ax-1cn 8236	. . . . . . 7 |- 1 e. CC
38		negdi2 8547	. . . . . . 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 8559	. . . . . . 7 |- -u -u 2 = 2
41	40	oveq1i 6068	. . . . . 6 |- ( -u -u 2 - 1 ) = ( 2 - 1 )
42	39, 41	eqtri 2255	. . . . 5 |- -u ( -u 2 + 1 ) = ( 2 - 1 )
43		2m1e1 9372	. . . . . 6 |- ( 2 - 1 ) = 1
44	43, 12	eqbrtri 4135	. . . . 5 |- ( 2 - 1 ) < ( 3 / 2 )
45	42, 44	eqbrtri 4135	. . . 4 |- -u ( -u 2 + 1 ) < ( 3 / 2 )
46	31, 1	readdcli 8303	. . . . 5 |- ( -u 2 + 1 ) e. RR
47	46, 3	ltnegcon1i 8790	. . . 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 9986	. . . . 5 |- ( ( 3 / 2 ) e. QQ -> -u ( 3 / 2 ) e. QQ )
50	26, 49	ax-mp 5	. . . 4 |- -u ( 3 / 2 ) e. QQ
51		2z 9622	. . . . 5 |- 2 e. ZZ
52		znegcl 9625	. . . . 5 |- ( 2 e. ZZ -> -u 2 e. ZZ )
53	51, 52	ax-mp 5	. . . 4 |- -u 2 e. ZZ
54		flqbi 10674	. . . 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 951	. 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 1398   e. wcel 2205   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   CCcc 8141   RRcr 8142   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   < clt 8324   <_ cle 8325   - cmin 8460   -ucneg 8461   / cdiv 8963   NNcn 9254   2c2 9305   3c3 9306   4c4 9307   ZZcz 9594   QQcq 9969   |_cfl 10652

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261  ax-arch 8262

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-po 4422  df-iso 4423  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-n0 9514  df-z 9595  df-q 9970  df-rp 10005  df-fl 10654

This theorem is referenced by:  ex-ceil  16620