Theorem ex-ceil 16740

Description: Example for df-ceil 10706. (Contributed by AV, 4-Sep-2021.)

Assertion

Ref	Expression
ex-ceil	|- ( (⌈ ` ( 3 / 2 ) ) = 2 /\ (⌈ ` -u ( 3 / 2 ) ) = -u 1 )

Proof of Theorem ex-ceil

Step	Hyp	Ref	Expression
1		ex-fl 16739	. 2 |- ( ( |_ ` ( 3 / 2 ) ) = 1 /\ ( |_ ` -u ( 3 / 2 ) ) = -u 2 )
2		3z 9673	. . . . . . 7 |- 3 e. ZZ
3		2nn 9466	. . . . . . 7 |- 2 e. NN
4		znq 10024	. . . . . . 7 |- ( ( 3 e. ZZ /\ 2 e. NN ) -> ( 3 / 2 ) e. QQ )
5	2, 3, 4	mp2an 430	. . . . . 6 |- ( 3 / 2 ) e. QQ
6		qnegcl 10036	. . . . . 6 |- ( ( 3 / 2 ) e. QQ -> -u ( 3 / 2 ) e. QQ )
7	5, 6	ax-mp 5	. . . . 5 |- -u ( 3 / 2 ) e. QQ
8		ceilqval 10743	. . . . 5 |- ( -u ( 3 / 2 ) e. QQ -> (⌈ ` -u ( 3 / 2 ) ) = -u ( |_ ` -u -u ( 3 / 2 ) ) )
9	7, 8	ax-mp 5	. . . 4 |- (⌈ ` -u ( 3 / 2 ) ) = -u ( |_ ` -u -u ( 3 / 2 ) )
10		qcn 10034	. . . . . . . . . . 11 |- ( ( 3 / 2 ) e. QQ -> ( 3 / 2 ) e. CC )
11	5, 10	ax-mp 5	. . . . . . . . . 10 |- ( 3 / 2 ) e. CC
12	11	negnegi 8596	. . . . . . . . 9 |- -u -u ( 3 / 2 ) = ( 3 / 2 )
13	12	eqcomi 2242	. . . . . . . 8 |- ( 3 / 2 ) = -u -u ( 3 / 2 )
14	13	fveq2i 5698	. . . . . . 7 |- ( |_ ` ( 3 / 2 ) ) = ( |_ ` -u -u ( 3 / 2 ) )
15	14	eqeq1i 2246	. . . . . 6 |- ( ( |_ ` ( 3 / 2 ) ) = 1 <-> ( |_ ` -u -u ( 3 / 2 ) ) = 1 )
16	15	biimpi 120	. . . . 5 |- ( ( |_ ` ( 3 / 2 ) ) = 1 -> ( |_ ` -u -u ( 3 / 2 ) ) = 1 )
17	16	negeqd 8521	. . . 4 |- ( ( |_ ` ( 3 / 2 ) ) = 1 -> -u ( |_ ` -u -u ( 3 / 2 ) ) = -u 1 )
18	9, 17	eqtrid 2283	. . 3 |- ( ( |_ ` ( 3 / 2 ) ) = 1 -> (⌈ ` -u ( 3 / 2 ) ) = -u 1 )
19		ceilqval 10743	. . . . 5 |- ( ( 3 / 2 ) e. QQ -> (⌈ ` ( 3 / 2 ) ) = -u ( |_ ` -u ( 3 / 2 ) ) )
20	5, 19	ax-mp 5	. . . 4 |- (⌈ ` ( 3 / 2 ) ) = -u ( |_ ` -u ( 3 / 2 ) )
21		negeq 8519	. . . . 5 |- ( ( |_ ` -u ( 3 / 2 ) ) = -u 2 -> -u ( |_ ` -u ( 3 / 2 ) ) = -u -u 2 )
22		2cn 9375	. . . . . 6 |- 2 e. CC
23	22	negnegi 8596	. . . . 5 |- -u -u 2 = 2
24	21, 23	eqtrdi 2287	. . . 4 |- ( ( |_ ` -u ( 3 / 2 ) ) = -u 2 -> -u ( |_ ` -u ( 3 / 2 ) ) = 2 )
25	20, 24	eqtrid 2283	. . 3 |- ( ( |_ ` -u ( 3 / 2 ) ) = -u 2 -> (⌈ ` ( 3 / 2 ) ) = 2 )
26	18, 25	anim12ci 339	. 2 |- ( ( ( |_ ` ( 3 / 2 ) ) = 1 /\ ( |_ ` -u ( 3 / 2 ) ) = -u 2 ) -> ( (⌈ ` ( 3 / 2 ) ) = 2 /\ (⌈ ` -u ( 3 / 2 ) ) = -u 1 ) )
27	1, 26	ax-mp 5	1 |- ( (⌈ ` ( 3 / 2 ) ) = 2 /\ (⌈ ` -u ( 3 / 2 ) ) = -u 1 )

This proof depends on syntax axioms:   /\ wa 104   = wceq 1402   e. wcel 2209   ` cfv 5377  (class class class)co 6085   CCcc 8177   1c1 8180   -ucneg 8498   / cdiv 9002   NNcn 9304   2c2 9355   3c3 9356   ZZcz 9644   QQcq 10019   |_cfl 10703  ⌈cceil 10704

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  df-ceil 10706

This theorem is used by: (None)