Theorem ex-fl 16653

Description: Example for df-fl 10683. 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 8315	. . . 4 |- 1 e. RR
2		3re 9357	. . . . 5 |- 3 e. RR
3	2	rehalfcli 9533	. . . 4 |- ( 3 / 2 ) e. RR
4		2cn 9354	. . . . . . 7 |- 2 e. CC
5	4	mullidi 8319	. . . . . 6 |- ( 1 x. 2 ) = 2
6		2lt3 9454	. . . . . 6 |- 2 < 3
7	5, 6	eqbrtri 4146	. . . . 5 |- ( 1 x. 2 ) < 3
8		2pos 9374	. . . . . 6 |- 0 < 2
9		2re 9353	. . . . . . 7 |- 2 e. RR
10	1, 2, 9	ltmuldivi 9242	. . . . . 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 8418	. . 3 |- 1 <_ ( 3 / 2 )
14		3lt4 9456	. . . . . 6 |- 3 < 4
15		2t2e4 9438	. . . . . 6 |- ( 2 x. 2 ) = 4
16	14, 15	breqtrri 4152	. . . . 5 |- 3 < ( 2 x. 2 )
17	9, 8	pm3.2i 272	. . . . . 6 |- ( 2 e. RR /\ 0 < 2 )
18		ltdivmul 9196	. . . . . 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 9342	. . . 4 |- 2 = ( 1 + 1 )
22	20, 21	breqtri 4150	. . 3 |- ( 3 / 2 ) < ( 1 + 1 )
23		3z 9652	. . . . 5 |- 3 e. ZZ
24		2nn 9445	. . . . 5 |- 2 e. NN
25		znq 10003	. . . . 5 |- ( ( 3 e. ZZ /\ 2 e. NN ) -> ( 3 / 2 ) e. QQ )
26	23, 24, 25	mp2an 430	. . . 4 |- ( 3 / 2 ) e. QQ
27		1z 9649	. . . 4 |- 1 e. ZZ
28		flqbi 10703	. . . 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 8578	. . . 4 |- -u 2 e. RR
32	3	renegcli 8578	. . . 4 |- -u ( 3 / 2 ) e. RR
33	3, 9	ltnegi 8811	. . . . 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 8418	. . 3 |- -u 2 <_ -u ( 3 / 2 )
36	4	negcli 8584	. . . . . . 7 |- -u 2 e. CC
37		ax-1cn 8262	. . . . . . 7 |- 1 e. CC
38		negdi2 8574	. . . . . . 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 8586	. . . . . . 7 |- -u -u 2 = 2
41	40	oveq1i 6085	. . . . . 6 |- ( -u -u 2 - 1 ) = ( 2 - 1 )
42	39, 41	eqtri 2259	. . . . 5 |- -u ( -u 2 + 1 ) = ( 2 - 1 )
43		2m1e1 9401	. . . . . 6 |- ( 2 - 1 ) = 1
44	43, 12	eqbrtri 4146	. . . . 5 |- ( 2 - 1 ) < ( 3 / 2 )
45	42, 44	eqbrtri 4146	. . . 4 |- -u ( -u 2 + 1 ) < ( 3 / 2 )
46	31, 1	readdcli 8329	. . . . 5 |- ( -u 2 + 1 ) e. RR
47	46, 3	ltnegcon1i 8817	. . . 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 10015	. . . . 5 |- ( ( 3 / 2 ) e. QQ -> -u ( 3 / 2 ) e. QQ )
50	26, 49	ax-mp 5	. . . 4 |- -u ( 3 / 2 ) e. QQ
51		2z 9651	. . . . 5 |- 2 e. ZZ
52		znegcl 9654	. . . . 5 |- ( 2 e. ZZ -> -u 2 e. ZZ )
53	51, 52	ax-mp 5	. . . 4 |- -u 2 e. ZZ
54		flqbi 10703	. . . 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 )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   CCcc 8167   RRcr 8168   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   < clt 8350   <_ cle 8351   - cmin 8487   -ucneg 8488   / cdiv 8992   NNcn 9283   2c2 9334   3c3 9335   4c4 9336   ZZcz 9623   QQcq 9998   |_cfl 10681

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 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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288

This theorem 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 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-q 9999  df-rp 10034  df-fl 10683

This theorem is referenced by:  ex-ceil  16654