Theorem nna0r 6751

Description: Addition to zero. Remark in proof of Theorem 4K(2) of [Enderton] p. 81. (Contributed by NM, 20-Sep-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Assertion

Ref	Expression
nna0r	|- ( A e. om -> ( (/) +o A ) = A )

Proof of Theorem nna0r

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6093	. . 3 |- ( x = (/) -> ( (/) +o x ) = ( (/) +o (/) ) )
2		id 19	. . 3 |- ( x = (/) -> x = (/) )
3	1, 2	eqeq12d 2253	. 2 |- ( x = (/) -> ( ( (/) +o x ) = x <-> ( (/) +o (/) ) = (/) ) )
4		oveq2 6093	. . 3 |- ( x = y -> ( (/) +o x ) = ( (/) +o y ) )
5		id 19	. . 3 |- ( x = y -> x = y )
6	4, 5	eqeq12d 2253	. 2 |- ( x = y -> ( ( (/) +o x ) = x <-> ( (/) +o y ) = y ) )
7		oveq2 6093	. . 3 |- ( x = suc y -> ( (/) +o x ) = ( (/) +o suc y ) )
8		id 19	. . 3 |- ( x = suc y -> x = suc y )
9	7, 8	eqeq12d 2253	. 2 |- ( x = suc y -> ( ( (/) +o x ) = x <-> ( (/) +o suc y ) = suc y ) )
10		oveq2 6093	. . 3 |- ( x = A -> ( (/) +o x ) = ( (/) +o A ) )
11		id 19	. . 3 |- ( x = A -> x = A )
12	10, 11	eqeq12d 2253	. 2 |- ( x = A -> ( ( (/) +o x ) = x <-> ( (/) +o A ) = A ) )
13		0elon 4537	. . 3 |- (/) e. On
14		oa0 6730	. . 3 |- ( (/) e. On -> ( (/) +o (/) ) = (/) )
15	13, 14	ax-mp 5	. 2 |- ( (/) +o (/) ) = (/)
16		peano1 4741	. . . 4 |- (/) e. om
17		nnasuc 6749	. . . 4 |- ( ( (/) e. om /\ y e. om ) -> ( (/) +o suc y ) = suc ( (/) +o y ) )
18	16, 17	mpan 428	. . 3 |- ( y e. om -> ( (/) +o suc y ) = suc ( (/) +o y ) )
19		suceq 4547	. . . 4 |- ( ( (/) +o y ) = y -> suc ( (/) +o y ) = suc y )
20	19	eqeq2d 2250	. . 3 |- ( ( (/) +o y ) = y -> ( ( (/) +o suc y ) = suc ( (/) +o y ) <-> ( (/) +o suc y ) = suc y ) )
21	18, 20	syl5ibcom 155	. 2 |- ( y e. om -> ( ( (/) +o y ) = y -> ( (/) +o suc y ) = suc y ) )
22	3, 6, 9, 12, 15, 21	finds 4747	1 |- ( A e. om -> ( (/) +o A ) = A )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   (/)c0 3520   Oncon0 4508   suc csuc 4510   omcom 4737  (class class class)co 6085   +o coa 6684

This proof depends on 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-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  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-ral 2533  df-rex 2534  df-reu 2535  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-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-oadd 6691

This theorem is used by:  nnacom  6757  nnaword  6784  nnm1  6798  prarloclem5  7867