Theorem nna0r 6563

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 5951	. . 3 |- ( x = (/) -> ( (/) +o x ) = ( (/) +o (/) ) )
2		id 19	. . 3 |- ( x = (/) -> x = (/) )
3	1, 2	eqeq12d 2219	. 2 |- ( x = (/) -> ( ( (/) +o x ) = x <-> ( (/) +o (/) ) = (/) ) )
4		oveq2 5951	. . 3 |- ( x = y -> ( (/) +o x ) = ( (/) +o y ) )
5		id 19	. . 3 |- ( x = y -> x = y )
6	4, 5	eqeq12d 2219	. 2 |- ( x = y -> ( ( (/) +o x ) = x <-> ( (/) +o y ) = y ) )
7		oveq2 5951	. . 3 |- ( x = suc y -> ( (/) +o x ) = ( (/) +o suc y ) )
8		id 19	. . 3 |- ( x = suc y -> x = suc y )
9	7, 8	eqeq12d 2219	. 2 |- ( x = suc y -> ( ( (/) +o x ) = x <-> ( (/) +o suc y ) = suc y ) )
10		oveq2 5951	. . 3 |- ( x = A -> ( (/) +o x ) = ( (/) +o A ) )
11		id 19	. . 3 |- ( x = A -> x = A )
12	10, 11	eqeq12d 2219	. 2 |- ( x = A -> ( ( (/) +o x ) = x <-> ( (/) +o A ) = A ) )
13		0elon 4438	. . 3 |- (/) e. On
14		oa0 6542	. . 3 |- ( (/) e. On -> ( (/) +o (/) ) = (/) )
15	13, 14	ax-mp 5	. 2 |- ( (/) +o (/) ) = (/)
16		peano1 4641	. . . 4 |- (/) e. om
17		nnasuc 6561	. . . 4 |- ( ( (/) e. om /\ y e. om ) -> ( (/) +o suc y ) = suc ( (/) +o y ) )
18	16, 17	mpan 424	. . 3 |- ( y e. om -> ( (/) +o suc y ) = suc ( (/) +o y ) )
19		suceq 4448	. . . 4 |- ( ( (/) +o y ) = y -> suc ( (/) +o y ) = suc y )
20	19	eqeq2d 2216	. . 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 4647	1 |- ( A e. om -> ( (/) +o A ) = A )

Syntax hints:   -> wi 4   = wceq 1372   e. wcel 2175   (/)c0 3459   Oncon0 4409   suc csuc 4411   omcom 4637  (class class class)co 5943   +o coa 6498

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-iord 4412  df-on 4414  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-oadd 6505

This theorem is referenced by:  nnacom  6569  nnaword  6596  nnm1  6610  prarloclem5  7612