Theorem ex-fl 16739

Description: Example for df-fl 10705. 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 8325	. . . 4 |- 1 e. RR
2		3re 9378	. . . . 5 |- 3 e. RR
3	2	rehalfcli 9554	. . . 4 |- ( 3 / 2 ) e. RR
4		2cn 9375	. . . . . . 7 |- 2 e. CC
5	4	mullidi 8329	. . . . . 6 |- ( 1 x. 2 ) = 2
6		2lt3 9475	. . . . . 6 |- 2 < 3
7	5, 6	eqbrtri 4151	. . . . 5 |- ( 1 x. 2 ) < 3
8		2pos 9395	. . . . . 6 |- 0 < 2
9		2re 9374	. . . . . . 7 |- 2 e. RR
10	1, 2, 9	ltmuldivi 9252	. . . . . 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 8428	. . 3 |- 1 <_ ( 3 / 2 )
14		3lt4 9477	. . . . . 6 |- 3 < 4
15		2t2e4 9459	. . . . . 6 |- ( 2 x. 2 ) = 4
16	14, 15	breqtrri 4157	. . . . 5 |- 3 < ( 2 x. 2 )
17	9, 8	pm3.2i 272	. . . . . 6 |- ( 2 e. RR /\ 0 < 2 )
18		ltdivmul 9206	. . . . . 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 1378	. . . . 5 |- ( ( 3 / 2 ) < 2 <-> 3 < ( 2 x. 2 ) )
20	16, 19	mpbir 146	. . . 4 |- ( 3 / 2 ) < 2
21		df-2 9363	. . . 4 |- 2 = ( 1 + 1 )
22	20, 21	breqtri 4155	. . 3 |- ( 3 / 2 ) < ( 1 + 1 )
23		3z 9673	. . . . 5 |- 3 e. ZZ
24		2nn 9466	. . . . 5 |- 2 e. NN
25		znq 10024	. . . . 5 |- ( ( 3 e. ZZ /\ 2 e. NN ) -> ( 3 / 2 ) e. QQ )
26	23, 24, 25	mp2an 430	. . . 4 |- ( 3 / 2 ) e. QQ
27		1z 9670	. . . 4 |- 1 e. ZZ
28		flqbi 10725	. . . 4 |- ( ( ( 3 / 2 ) e. QQ /\ 1 e. ZZ ) -> ( ( |_ ` ( 3 / 2 ) ) = 1 <-> ( 1 <_ ( 3 / 2 ) /\ ( 3 / 2 ) < ( 1 + 1 ) ) ) )
29	26, 27, 28	mp2an 430	. . 3 |- ( ( |_ ` ( 3 / 2 ) ) = 1 <-> ( 1 <_ ( 3 / 2 ) /\ ( 3 / 2 ) < ( 1 + 1 ) ) )
30	13, 22, 29	mpbir2an 955	. 2 |- ( |_ ` ( 3 / 2 ) ) = 1
31	9	renegcli 8588	. . . 4 |- -u 2 e. RR
32	3	renegcli 8588	. . . 4 |- -u ( 3 / 2 ) e. RR
33	3, 9	ltnegi 8821	. . . . 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 8428	. . 3 |- -u 2 <_ -u ( 3 / 2 )
36	4	negcli 8594	. . . . . . 7 |- -u 2 e. CC
37		ax-1cn 8272	. . . . . . 7 |- 1 e. CC
38		negdi2 8584	. . . . . . 7 |- ( ( -u 2 e. CC /\ 1 e. CC ) -> -u ( -u 2 + 1 ) = ( -u -u 2 - 1 ) )
39	36, 37, 38	mp2an 430	. . . . . 6 |- -u ( -u 2 + 1 ) = ( -u -u 2 - 1 )
40	4	negnegi 8596	. . . . . . 7 |- -u -u 2 = 2
41	40	oveq1i 6095	. . . . . 6 |- ( -u -u 2 - 1 ) = ( 2 - 1 )
42	39, 41	eqtri 2259	. . . . 5 |- -u ( -u 2 + 1 ) = ( 2 - 1 )
43		2m1e1 9422	. . . . . 6 |- ( 2 - 1 ) = 1
44	43, 12	eqbrtri 4151	. . . . 5 |- ( 2 - 1 ) < ( 3 / 2 )
45	42, 44	eqbrtri 4151	. . . 4 |- -u ( -u 2 + 1 ) < ( 3 / 2 )
46	31, 1	readdcli 8339	. . . . 5 |- ( -u 2 + 1 ) e. RR
47	46, 3	ltnegcon1i 8827	. . . 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 10036	. . . . 5 |- ( ( 3 / 2 ) e. QQ -> -u ( 3 / 2 ) e. QQ )
50	26, 49	ax-mp 5	. . . 4 |- -u ( 3 / 2 ) e. QQ
51		2z 9672	. . . . 5 |- 2 e. ZZ
52		znegcl 9675	. . . . 5 |- ( 2 e. ZZ -> -u 2 e. ZZ )
53	51, 52	ax-mp 5	. . . 4 |- -u 2 e. ZZ
54		flqbi 10725	. . . 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 430	. . 3 |- ( ( |_ ` -u ( 3 / 2 ) ) = -u 2 <-> ( -u 2 <_ -u ( 3 / 2 ) /\ -u ( 3 / 2 ) < ( -u 2 + 1 ) ) )
56	35, 48, 55	mpbir2an 955	. 2 |- ( |_ ` -u ( 3 / 2 ) ) = -u 2
57	30, 56	pm3.2i 272	1 |- ( ( |_ ` ( 3 / 2 ) ) = 1 /\ ( |_ ` -u ( 3 / 2 ) ) = -u 2 )

This proof depends on syntax axioms:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4130   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   < clt 8360   <_ cle 8361   - cmin 8497   -ucneg 8498   / cdiv 9002   NNcn 9304   2c2 9355   3c3 9356   4c4 9357   ZZcz 9644   QQcq 10019   |_cfl 10703

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-q 10020  df-rp 10055  df-fl 10705

This theorem is used by:  ex-ceil  16740