Theorem nna0r 6446

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 5850	. . 3 |- ( x = (/) -> ( (/) +o x ) = ( (/) +o (/) ) )
2		id 19	. . 3 |- ( x = (/) -> x = (/) )
3	1, 2	eqeq12d 2180	. 2 |- ( x = (/) -> ( ( (/) +o x ) = x <-> ( (/) +o (/) ) = (/) ) )
4		oveq2 5850	. . 3 |- ( x = y -> ( (/) +o x ) = ( (/) +o y ) )
5		id 19	. . 3 |- ( x = y -> x = y )
6	4, 5	eqeq12d 2180	. 2 |- ( x = y -> ( ( (/) +o x ) = x <-> ( (/) +o y ) = y ) )
7		oveq2 5850	. . 3 |- ( x = suc y -> ( (/) +o x ) = ( (/) +o suc y ) )
8		id 19	. . 3 |- ( x = suc y -> x = suc y )
9	7, 8	eqeq12d 2180	. 2 |- ( x = suc y -> ( ( (/) +o x ) = x <-> ( (/) +o suc y ) = suc y ) )
10		oveq2 5850	. . 3 |- ( x = A -> ( (/) +o x ) = ( (/) +o A ) )
11		id 19	. . 3 |- ( x = A -> x = A )
12	10, 11	eqeq12d 2180	. 2 |- ( x = A -> ( ( (/) +o x ) = x <-> ( (/) +o A ) = A ) )
13		0elon 4370	. . 3 |- (/) e. On
14		oa0 6425	. . 3 |- ( (/) e. On -> ( (/) +o (/) ) = (/) )
15	13, 14	ax-mp 5	. 2 |- ( (/) +o (/) ) = (/)
16		peano1 4571	. . . 4 |- (/) e. om
17		nnasuc 6444	. . . 4 |- ( ( (/) e. om /\ y e. om ) -> ( (/) +o suc y ) = suc ( (/) +o y ) )
18	16, 17	mpan 421	. . 3 |- ( y e. om -> ( (/) +o suc y ) = suc ( (/) +o y ) )
19		suceq 4380	. . . 4 |- ( ( (/) +o y ) = y -> suc ( (/) +o y ) = suc y )
20	19	eqeq2d 2177	. . 3 |- ( ( (/) +o y ) = y -> ( ( (/) +o suc y ) = suc ( (/) +o y ) <-> ( (/) +o suc y ) = suc y ) )
21	18, 20	syl5ibcom 154	. 2 |- ( y e. om -> ( ( (/) +o y ) = y -> ( (/) +o suc y ) = suc y ) )
22	3, 6, 9, 12, 15, 21	finds 4577	1 |- ( A e. om -> ( (/) +o A ) = A )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   (/)c0 3409   Oncon0 4341   suc csuc 4343   omcom 4567  (class class class)co 5842   +o coa 6381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-oadd 6388

This theorem is referenced by:  nnacom  6452  nnaword  6479  nnm1  6492  prarloclem5  7441