Theorem ex-ceil 16654

Description: Example for df-ceil 10684. (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 16653	. 2 |- ( ( |_ ` ( 3 / 2 ) ) = 1 /\ ( |_ ` -u ( 3 / 2 ) ) = -u 2 )
2		3z 9652	. . . . . . 7 |- 3 e. ZZ
3		2nn 9445	. . . . . . 7 |- 2 e. NN
4		znq 10003	. . . . . . 7 |- ( ( 3 e. ZZ /\ 2 e. NN ) -> ( 3 / 2 ) e. QQ )
5	2, 3, 4	mp2an 430	. . . . . 6 |- ( 3 / 2 ) e. QQ
6		qnegcl 10015	. . . . . 6 |- ( ( 3 / 2 ) e. QQ -> -u ( 3 / 2 ) e. QQ )
7	5, 6	ax-mp 5	. . . . 5 |- -u ( 3 / 2 ) e. QQ
8		ceilqval 10721	. . . . 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 10013	. . . . . . . . . . 11 |- ( ( 3 / 2 ) e. QQ -> ( 3 / 2 ) e. CC )
11	5, 10	ax-mp 5	. . . . . . . . . 10 |- ( 3 / 2 ) e. CC
12	11	negnegi 8586	. . . . . . . . 9 |- -u -u ( 3 / 2 ) = ( 3 / 2 )
13	12	eqcomi 2242	. . . . . . . 8 |- ( 3 / 2 ) = -u -u ( 3 / 2 )
14	13	fveq2i 5693	. . . . . . 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 8511	. . . 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 10721	. . . . 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 8509	. . . . 5 |- ( ( |_ ` -u ( 3 / 2 ) ) = -u 2 -> -u ( |_ ` -u ( 3 / 2 ) ) = -u -u 2 )
22		2cn 9354	. . . . . 6 |- 2 e. CC
23	22	negnegi 8586	. . . . 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 )

Syntax hints:   /\ wa 104   = wceq 1402   e. wcel 2209   ` cfv 5372  (class class class)co 6075   CCcc 8167   1c1 8170   -ucneg 8488   / cdiv 8992   NNcn 9283   2c2 9334   3c3 9335   ZZcz 9623   QQcq 9998   |_cfl 10681  ⌈cceil 10682

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  df-ceil 10684

This theorem is referenced by: (None)