Theorem isodd 41542

Description: The predicate "is an odd number". An odd number is an integer which is not divisible by 2, i.e. the result of dividing the odd integer increased by 1 and then divided by 2 is still an integer. (Contributed by AV, 14-Jun-2020.)

Assertion

Ref	Expression
isodd	|- ( Z e. Odd <-> ( Z e. ZZ /\ ( ( Z + 1 ) / 2 ) e. ZZ ) )

Proof of Theorem isodd

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . 4 |- ( z = Z -> ( z + 1 ) = ( Z + 1 ) )
2	1	oveq1d 6665	. . 3 |- ( z = Z -> ( ( z + 1 ) / 2 ) = ( ( Z + 1 ) / 2 ) )
3	2	eleq1d 2686	. 2 |- ( z = Z -> ( ( ( z + 1 ) / 2 ) e. ZZ <-> ( ( Z + 1 ) / 2 ) e. ZZ ) )
4		df-odd 41540	. 2 |- Odd = { z e. ZZ | ( ( z + 1 ) / 2 ) e. ZZ }
5	3, 4	elrab2 3366	1 |- ( Z e. Odd <-> ( Z e. ZZ /\ ( ( Z + 1 ) / 2 ) e. ZZ ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990  (class class class)co 6650   1c1 9937   + caddc 9939   / cdiv 10684   2c2 11070   ZZcz 11377   Odd codd 41538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-odd 41540

This theorem is referenced by:  oddz  41544  oddp1div2z  41546  isodd2  41548  evenm1odd  41552  evennodd  41556  oddneven  41557  onego  41559  zeoALTV  41581  oddp1evenALTV  41587  1oddALTV  41601