Theorem nna0r 6531

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 5926	. . 3 |- ( x = (/) -> ( (/) +o x ) = ( (/) +o (/) ) )
2		id 19	. . 3 |- ( x = (/) -> x = (/) )
3	1, 2	eqeq12d 2208	. 2 |- ( x = (/) -> ( ( (/) +o x ) = x <-> ( (/) +o (/) ) = (/) ) )
4		oveq2 5926	. . 3 |- ( x = y -> ( (/) +o x ) = ( (/) +o y ) )
5		id 19	. . 3 |- ( x = y -> x = y )
6	4, 5	eqeq12d 2208	. 2 |- ( x = y -> ( ( (/) +o x ) = x <-> ( (/) +o y ) = y ) )
7		oveq2 5926	. . 3 |- ( x = suc y -> ( (/) +o x ) = ( (/) +o suc y ) )
8		id 19	. . 3 |- ( x = suc y -> x = suc y )
9	7, 8	eqeq12d 2208	. 2 |- ( x = suc y -> ( ( (/) +o x ) = x <-> ( (/) +o suc y ) = suc y ) )
10		oveq2 5926	. . 3 |- ( x = A -> ( (/) +o x ) = ( (/) +o A ) )
11		id 19	. . 3 |- ( x = A -> x = A )
12	10, 11	eqeq12d 2208	. 2 |- ( x = A -> ( ( (/) +o x ) = x <-> ( (/) +o A ) = A ) )
13		0elon 4423	. . 3 |- (/) e. On
14		oa0 6510	. . 3 |- ( (/) e. On -> ( (/) +o (/) ) = (/) )
15	13, 14	ax-mp 5	. 2 |- ( (/) +o (/) ) = (/)
16		peano1 4626	. . . 4 |- (/) e. om
17		nnasuc 6529	. . . 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 4433	. . . 4 |- ( ( (/) +o y ) = y -> suc ( (/) +o y ) = suc y )
20	19	eqeq2d 2205	. . 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 4632	1 |- ( A e. om -> ( (/) +o A ) = A )

Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   (/)c0 3446   Oncon0 4394   suc csuc 4396   omcom 4622  (class class class)co 5918   +o coa 6466

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-recs 6358  df-irdg 6423  df-oadd 6473

This theorem is referenced by:  nnacom  6537  nnaword  6564  nnm1  6578  prarloclem5  7560