Theorem ex-ceil 16322

Description: Example for df-ceil 10530. (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 16321	. 2 |- ( ( |_ ` ( 3 / 2 ) ) = 1 /\ ( |_ ` -u ( 3 / 2 ) ) = -u 2 )
2		3z 9507	. . . . . . 7 |- 3 e. ZZ
3		2nn 9304	. . . . . . 7 |- 2 e. NN
4		znq 9857	. . . . . . 7 |- ( ( 3 e. ZZ /\ 2 e. NN ) -> ( 3 / 2 ) e. QQ )
5	2, 3, 4	mp2an 426	. . . . . 6 |- ( 3 / 2 ) e. QQ
6		qnegcl 9869	. . . . . 6 |- ( ( 3 / 2 ) e. QQ -> -u ( 3 / 2 ) e. QQ )
7	5, 6	ax-mp 5	. . . . 5 |- -u ( 3 / 2 ) e. QQ
8		ceilqval 10567	. . . . 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 9867	. . . . . . . . . . 11 |- ( ( 3 / 2 ) e. QQ -> ( 3 / 2 ) e. CC )
11	5, 10	ax-mp 5	. . . . . . . . . 10 |- ( 3 / 2 ) e. CC
12	11	negnegi 8448	. . . . . . . . 9 |- -u -u ( 3 / 2 ) = ( 3 / 2 )
13	12	eqcomi 2235	. . . . . . . 8 |- ( 3 / 2 ) = -u -u ( 3 / 2 )
14	13	fveq2i 5642	. . . . . . 7 |- ( |_ ` ( 3 / 2 ) ) = ( |_ ` -u -u ( 3 / 2 ) )
15	14	eqeq1i 2239	. . . . . 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 8373	. . . 4 |- ( ( |_ ` ( 3 / 2 ) ) = 1 -> -u ( |_ ` -u -u ( 3 / 2 ) ) = -u 1 )
18	9, 17	eqtrid 2276	. . 3 |- ( ( |_ ` ( 3 / 2 ) ) = 1 -> (⌈ ` -u ( 3 / 2 ) ) = -u 1 )
19		ceilqval 10567	. . . . 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 8371	. . . . 5 |- ( ( |_ ` -u ( 3 / 2 ) ) = -u 2 -> -u ( |_ ` -u ( 3 / 2 ) ) = -u -u 2 )
22		2cn 9213	. . . . . 6 |- 2 e. CC
23	22	negnegi 8448	. . . . 5 |- -u -u 2 = 2
24	21, 23	eqtrdi 2280	. . . 4 |- ( ( |_ ` -u ( 3 / 2 ) ) = -u 2 -> -u ( |_ ` -u ( 3 / 2 ) ) = 2 )
25	20, 24	eqtrid 2276	. . 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 1397   e. wcel 2202   ` cfv 5326  (class class class)co 6017   CCcc 8029   1c1 8032   -ucneg 8350   / cdiv 8851   NNcn 9142   2c2 9193   3c3 9194   ZZcz 9478   QQcq 9852   |_cfl 10527  ⌈cceil 10528

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-q 9853  df-rp 9888  df-fl 10529  df-ceil 10530

This theorem is referenced by: (None)