Theorem ex-fl 16944

Description: Example for df-fl 10722. 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 8331	. . . 4 |- 1 e. RR
2		3re 9387	. . . . 5 |- 3 e. RR
3	2	rehalfcli 9565	. . . 4 |- ( 3 / 2 ) e. RR
4		2cn 9384	. . . . . . 7 |- 2 e. CC
5	4	mullidi 8335	. . . . . 6 |- ( 1 x. 2 ) = 2
6		2lt3 9486	. . . . . 6 |- 2 < 3
7	5, 6	eqbrtri 4152	. . . . 5 |- ( 1 x. 2 ) < 3
8		2pos 9404	. . . . . 6 |- 0 < 2
9		2re 9383	. . . . . . 7 |- 2 e. RR
10	1, 2, 9	ltmuldivi 9260	. . . . . 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 8435	. . 3 |- 1 <_ ( 3 / 2 )
14		3lt4 9488	. . . . . 6 |- 3 < 4
15		2t2e4 9468	. . . . . 6 |- ( 2 x. 2 ) = 4
16	14, 15	breqtrri 4158	. . . . 5 |- 3 < ( 2 x. 2 )
17	9, 8	pm3.2i 272	. . . . . 6 |- ( 2 e. RR /\ 0 < 2 )
18		ltdivmul 9214	. . . . . 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 9372	. . . 4 |- 2 = ( 1 + 1 )
22	20, 21	breqtri 4156	. . 3 |- ( 3 / 2 ) < ( 1 + 1 )
23		3z 9684	. . . . 5 |- 3 e. ZZ
24		2nn 9477	. . . . 5 |- 2 e. NN
25		znq 10040	. . . . 5 |- ( ( 3 e. ZZ /\ 2 e. NN ) -> ( 3 / 2 ) e. QQ )
26	23, 24, 25	mp2an 430	. . . 4 |- ( 3 / 2 ) e. QQ
27		1z 9681	. . . 4 |- 1 e. ZZ
28		flqbi 10746	. . . 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 8595	. . . 4 |- -u 2 e. RR
32	3	renegcli 8595	. . . 4 |- -u ( 3 / 2 ) e. RR
33	3, 9	ltnegi 8828	. . . . 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 8435	. . 3 |- -u 2 <_ -u ( 3 / 2 )
36	4	negcli 8601	. . . . . . 7 |- -u 2 e. CC
37		ax-1cn 8278	. . . . . . 7 |- 1 e. CC
38		negdi2 8591	. . . . . . 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 8603	. . . . . . 7 |- -u -u 2 = 2
41	40	oveq1i 6098	. . . . . 6 |- ( -u -u 2 - 1 ) = ( 2 - 1 )
42	39, 41	eqtri 2259	. . . . 5 |- -u ( -u 2 + 1 ) = ( 2 - 1 )
43		2m1e1 9431	. . . . . 6 |- ( 2 - 1 ) = 1
44	43, 12	eqbrtri 4152	. . . . 5 |- ( 2 - 1 ) < ( 3 / 2 )
45	42, 44	eqbrtri 4152	. . . 4 |- -u ( -u 2 + 1 ) < ( 3 / 2 )
46	31, 1	readdcli 8345	. . . . 5 |- ( -u 2 + 1 ) e. RR
47	46, 3	ltnegcon1i 8834	. . . 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 10052	. . . . 5 |- ( ( 3 / 2 ) e. QQ -> -u ( 3 / 2 ) e. QQ )
50	26, 49	ax-mp 5	. . . 4 |- -u ( 3 / 2 ) e. QQ
51		2z 9683	. . . . 5 |- 2 e. ZZ
52		znegcl 9686	. . . . 5 |- ( 2 e. ZZ -> -u 2 e. ZZ )
53	51, 52	ax-mp 5	. . . 4 |- -u 2 e. ZZ
54		flqbi 10746	. . . 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 4131   ` cfv 5380  (class class class)co 6088   CCcc 8183   RRcr 8184   0cc0 8185   1c1 8186   + caddc 8188   x. cmul 8190   < clt 8366   <_ cle 8367   - cmin 8504   -ucneg 8505   / cdiv 9010   NNcn 9313   2c2 9364   3c3 9365   4c4 9366   ZZcz 9655   QQcq 10035   |_cfl 10720

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 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303  ax-arch 8304

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 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-po 4443  df-iso 4444  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-n0 9575  df-z 9656  df-q 10036  df-rp 10072  df-fl 10722

This theorem is used by:  ex-ceil  16945