Theorem halfleoddlt 12639

Description: An integer is greater than half of an odd number iff it is greater than or equal to the half of the odd number. (Contributed by AV, 1-Jul-2021.)

Assertion

Ref	Expression
halfleoddlt	|- ( ( N e. ZZ /\ -. 2 || N /\ M e. ZZ ) -> ( ( N / 2 ) <_ M <-> ( N / 2 ) < M ) )

Proof of Theorem halfleoddlt

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		odd2np1 12618	. . 3 |- ( N e. ZZ -> ( -. 2 || N <-> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
2		0xr 8362	. . . . . . . . . . . 12 |- 0 e. RR*
3		1re 8315	. . . . . . . . . . . . 13 |- 1 e. RR
4	3	rexri 8373	. . . . . . . . . . . 12 |- 1 e. RR*
5		halfre 9497	. . . . . . . . . . . . 13 |- ( 1 / 2 ) e. RR
6	5	rexri 8373	. . . . . . . . . . . 12 |- ( 1 / 2 ) e. RR*
7	2, 4, 6	3pm3.2i 1206	. . . . . . . . . . 11 |- ( 0 e. RR* /\ 1 e. RR* /\ ( 1 / 2 ) e. RR* )
8		halfgt0 9499	. . . . . . . . . . . 12 |- 0 < ( 1 / 2 )
9		halflt1 9501	. . . . . . . . . . . 12 |- ( 1 / 2 ) < 1
10	8, 9	pm3.2i 272	. . . . . . . . . . 11 |- ( 0 < ( 1 / 2 ) /\ ( 1 / 2 ) < 1 )
11		elioo3g 10291	. . . . . . . . . . 11 |- ( ( 1 / 2 ) e. ( 0 (,) 1 ) <-> ( ( 0 e. RR* /\ 1 e. RR* /\ ( 1 / 2 ) e. RR* ) /\ ( 0 < ( 1 / 2 ) /\ ( 1 / 2 ) < 1 ) ) )
12	7, 10, 11	mpbir2an 955	. . . . . . . . . 10 |- ( 1 / 2 ) e. ( 0 (,) 1 )
13		zltaddlt1le 10389	. . . . . . . . . 10 |- ( ( n e. ZZ /\ M e. ZZ /\ ( 1 / 2 ) e. ( 0 (,) 1 ) ) -> ( ( n + ( 1 / 2 ) ) < M <-> ( n + ( 1 / 2 ) ) <_ M ) )
14	12, 13	mp3an3 1367	. . . . . . . . 9 |- ( ( n e. ZZ /\ M e. ZZ ) -> ( ( n + ( 1 / 2 ) ) < M <-> ( n + ( 1 / 2 ) ) <_ M ) )
15		zcn 9628	. . . . . . . . . . . 12 |- ( n e. ZZ -> n e. CC )
16	15	adantr 276	. . . . . . . . . . 11 |- ( ( n e. ZZ /\ M e. ZZ ) -> n e. CC )
17		1cnd 8332	. . . . . . . . . . 11 |- ( ( n e. ZZ /\ M e. ZZ ) -> 1 e. CC )
18		2cn 9354	. . . . . . . . . . . . 13 |- 2 e. CC
19		2ap0 9376	. . . . . . . . . . . . 13 |- 2 # 0
20	18, 19	pm3.2i 272	. . . . . . . . . . . 12 |- ( 2 e. CC /\ 2 # 0 )
21	20	a1i 9	. . . . . . . . . . 11 |- ( ( n e. ZZ /\ M e. ZZ ) -> ( 2 e. CC /\ 2 # 0 ) )
22		muldivdirap 9027	. . . . . . . . . . 11 |- ( ( n e. CC /\ 1 e. CC /\ ( 2 e. CC /\ 2 # 0 ) ) -> ( ( ( 2 x. n ) + 1 ) / 2 ) = ( n + ( 1 / 2 ) ) )
23	16, 17, 21, 22	syl3anc 1278	. . . . . . . . . 10 |- ( ( n e. ZZ /\ M e. ZZ ) -> ( ( ( 2 x. n ) + 1 ) / 2 ) = ( n + ( 1 / 2 ) ) )
24	23	breq1d 4135	. . . . . . . . 9 |- ( ( n e. ZZ /\ M e. ZZ ) -> ( ( ( ( 2 x. n ) + 1 ) / 2 ) < M <-> ( n + ( 1 / 2 ) ) < M ) )
25	23	breq1d 4135	. . . . . . . . 9 |- ( ( n e. ZZ /\ M e. ZZ ) -> ( ( ( ( 2 x. n ) + 1 ) / 2 ) <_ M <-> ( n + ( 1 / 2 ) ) <_ M ) )
26	14, 24, 25	3bitr4rd 221	. . . . . . . 8 |- ( ( n e. ZZ /\ M e. ZZ ) -> ( ( ( ( 2 x. n ) + 1 ) / 2 ) <_ M <-> ( ( ( 2 x. n ) + 1 ) / 2 ) < M ) )
27		oveq1 6082	. . . . . . . . . 10 |- ( ( ( 2 x. n ) + 1 ) = N -> ( ( ( 2 x. n ) + 1 ) / 2 ) = ( N / 2 ) )
28	27	breq1d 4135	. . . . . . . . 9 |- ( ( ( 2 x. n ) + 1 ) = N -> ( ( ( ( 2 x. n ) + 1 ) / 2 ) <_ M <-> ( N / 2 ) <_ M ) )
29	27	breq1d 4135	. . . . . . . . 9 |- ( ( ( 2 x. n ) + 1 ) = N -> ( ( ( ( 2 x. n ) + 1 ) / 2 ) < M <-> ( N / 2 ) < M ) )
30	28, 29	bibi12d 235	. . . . . . . 8 |- ( ( ( 2 x. n ) + 1 ) = N -> ( ( ( ( ( 2 x. n ) + 1 ) / 2 ) <_ M <-> ( ( ( 2 x. n ) + 1 ) / 2 ) < M ) <-> ( ( N / 2 ) <_ M <-> ( N / 2 ) < M ) ) )
31	26, 30	syl5ibcom 155	. . . . . . 7 |- ( ( n e. ZZ /\ M e. ZZ ) -> ( ( ( 2 x. n ) + 1 ) = N -> ( ( N / 2 ) <_ M <-> ( N / 2 ) < M ) ) )
32	31	ex 115	. . . . . 6 |- ( n e. ZZ -> ( M e. ZZ -> ( ( ( 2 x. n ) + 1 ) = N -> ( ( N / 2 ) <_ M <-> ( N / 2 ) < M ) ) ) )
33	32	adantl 277	. . . . 5 |- ( ( N e. ZZ /\ n e. ZZ ) -> ( M e. ZZ -> ( ( ( 2 x. n ) + 1 ) = N -> ( ( N / 2 ) <_ M <-> ( N / 2 ) < M ) ) ) )
34	33	com23 78	. . . 4 |- ( ( N e. ZZ /\ n e. ZZ ) -> ( ( ( 2 x. n ) + 1 ) = N -> ( M e. ZZ -> ( ( N / 2 ) <_ M <-> ( N / 2 ) < M ) ) ) )
35	34	rexlimdva 2668	. . 3 |- ( N e. ZZ -> ( E. n e. ZZ ( ( 2 x. n ) + 1 ) = N -> ( M e. ZZ -> ( ( N / 2 ) <_ M <-> ( N / 2 ) < M ) ) ) )
36	1, 35	sylbid 150	. 2 |- ( N e. ZZ -> ( -. 2 || N -> ( M e. ZZ -> ( ( N / 2 ) <_ M <-> ( N / 2 ) < M ) ) ) )
37	36	3imp 1224	1 |- ( ( N e. ZZ /\ -. 2 || N /\ M e. ZZ ) -> ( ( N / 2 ) <_ M <-> ( N / 2 ) < M ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4125  (class class class)co 6075   CCcc 8167   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   RR*cxr 8349   < clt 8350   <_ cle 8351   # cap 8899   / cdiv 8992   2c2 9334   ZZcz 9623   (,)cioo 10269   || cdvds 12532

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

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-xor 1425  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-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-br 4126  df-opab 4188  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-iota 5332  df-fun 5374  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  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-n0 9543  df-z 9624  df-rp 10034  df-ioo 10273  df-dvds 12533

This theorem is referenced by:  gausslemma2dlem1a  16091