Theorem nna0r 6711

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 6058	. . 3 |- ( x = (/) -> ( (/) +o x ) = ( (/) +o (/) ) )
2		id 19	. . 3 |- ( x = (/) -> x = (/) )
3	1, 2	eqeq12d 2247	. 2 |- ( x = (/) -> ( ( (/) +o x ) = x <-> ( (/) +o (/) ) = (/) ) )
4		oveq2 6058	. . 3 |- ( x = y -> ( (/) +o x ) = ( (/) +o y ) )
5		id 19	. . 3 |- ( x = y -> x = y )
6	4, 5	eqeq12d 2247	. 2 |- ( x = y -> ( ( (/) +o x ) = x <-> ( (/) +o y ) = y ) )
7		oveq2 6058	. . 3 |- ( x = suc y -> ( (/) +o x ) = ( (/) +o suc y ) )
8		id 19	. . 3 |- ( x = suc y -> x = suc y )
9	7, 8	eqeq12d 2247	. 2 |- ( x = suc y -> ( ( (/) +o x ) = x <-> ( (/) +o suc y ) = suc y ) )
10		oveq2 6058	. . 3 |- ( x = A -> ( (/) +o x ) = ( (/) +o A ) )
11		id 19	. . 3 |- ( x = A -> x = A )
12	10, 11	eqeq12d 2247	. 2 |- ( x = A -> ( ( (/) +o x ) = x <-> ( (/) +o A ) = A ) )
13		0elon 4513	. . 3 |- (/) e. On
14		oa0 6690	. . 3 |- ( (/) e. On -> ( (/) +o (/) ) = (/) )
15	13, 14	ax-mp 5	. 2 |- ( (/) +o (/) ) = (/)
16		peano1 4716	. . . 4 |- (/) e. om
17		nnasuc 6709	. . . 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 4523	. . . 4 |- ( ( (/) +o y ) = y -> suc ( (/) +o y ) = suc y )
20	19	eqeq2d 2244	. . 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 4722	1 |- ( A e. om -> ( (/) +o A ) = A )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   (/)c0 3508   Oncon0 4484   suc csuc 4486   omcom 4712  (class class class)co 6050   +o coa 6644

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-oadd 6651

This theorem is referenced by:  nnacom  6717  nnaword  6744  nnm1  6758  prarloclem5  7815