Theorem ex-ceil 15862

Description: Example for df-ceil 10451. (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 15861	. 2 |- ( ( |_ ` ( 3 / 2 ) ) = 1 /\ ( |_ ` -u ( 3 / 2 ) ) = -u 2 )
2		3z 9436	. . . . . . 7 |- 3 e. ZZ
3		2nn 9233	. . . . . . 7 |- 2 e. NN
4		znq 9780	. . . . . . 7 |- ( ( 3 e. ZZ /\ 2 e. NN ) -> ( 3 / 2 ) e. QQ )
5	2, 3, 4	mp2an 426	. . . . . 6 |- ( 3 / 2 ) e. QQ
6		qnegcl 9792	. . . . . 6 |- ( ( 3 / 2 ) e. QQ -> -u ( 3 / 2 ) e. QQ )
7	5, 6	ax-mp 5	. . . . 5 |- -u ( 3 / 2 ) e. QQ
8		ceilqval 10488	. . . . 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 9790	. . . . . . . . . . 11 |- ( ( 3 / 2 ) e. QQ -> ( 3 / 2 ) e. CC )
11	5, 10	ax-mp 5	. . . . . . . . . 10 |- ( 3 / 2 ) e. CC
12	11	negnegi 8377	. . . . . . . . 9 |- -u -u ( 3 / 2 ) = ( 3 / 2 )
13	12	eqcomi 2211	. . . . . . . 8 |- ( 3 / 2 ) = -u -u ( 3 / 2 )
14	13	fveq2i 5602	. . . . . . 7 |- ( |_ ` ( 3 / 2 ) ) = ( |_ ` -u -u ( 3 / 2 ) )
15	14	eqeq1i 2215	. . . . . 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 8302	. . . 4 |- ( ( |_ ` ( 3 / 2 ) ) = 1 -> -u ( |_ ` -u -u ( 3 / 2 ) ) = -u 1 )
18	9, 17	eqtrid 2252	. . 3 |- ( ( |_ ` ( 3 / 2 ) ) = 1 -> (⌈ ` -u ( 3 / 2 ) ) = -u 1 )
19		ceilqval 10488	. . . . 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 8300	. . . . 5 |- ( ( |_ ` -u ( 3 / 2 ) ) = -u 2 -> -u ( |_ ` -u ( 3 / 2 ) ) = -u -u 2 )
22		2cn 9142	. . . . . 6 |- 2 e. CC
23	22	negnegi 8377	. . . . 5 |- -u -u 2 = 2
24	21, 23	eqtrdi 2256	. . . 4 |- ( ( |_ ` -u ( 3 / 2 ) ) = -u 2 -> -u ( |_ ` -u ( 3 / 2 ) ) = 2 )
25	20, 24	eqtrid 2252	. . 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 )

Syntax hints:   /\ wa 104   = wceq 1373   e. wcel 2178   ` cfv 5290  (class class class)co 5967   CCcc 7958   1c1 7961   -ucneg 8279   / cdiv 8780   NNcn 9071   2c2 9122   3c3 9123   ZZcz 9407   QQcq 9775   |_cfl 10448  ⌈cceil 10449

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-po 4361  df-iso 4362  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-q 9776  df-rp 9811  df-fl 10450  df-ceil 10451

This theorem is referenced by: (None)