Theorem hashgadd 13166

Description: G maps ordinal addition to integer addition. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)

Hypothesis

Ref	Expression
hashgadd.1	|- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` om )

Assertion

Ref	Expression
hashgadd	|- ( ( A e. om /\ B e. om ) -> ( G ` ( A +o B ) ) = ( ( G ` A ) + ( G ` B ) ) )

Proof of Theorem hashgadd

Dummy variables n z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6658	. . . . . 6 |- ( n = (/) -> ( A +o n ) = ( A +o (/) ) )
2	1	fveq2d 6195	. . . . 5 |- ( n = (/) -> ( G ` ( A +o n ) ) = ( G ` ( A +o (/) ) ) )
3		fveq2 6191	. . . . . 6 |- ( n = (/) -> ( G ` n ) = ( G ` (/) ) )
4	3	oveq2d 6666	. . . . 5 |- ( n = (/) -> ( ( G ` A ) + ( G ` n ) ) = ( ( G ` A ) + ( G ` (/) ) ) )
5	2, 4	eqeq12d 2637	. . . 4 |- ( n = (/) -> ( ( G ` ( A +o n ) ) = ( ( G ` A ) + ( G ` n ) ) <-> ( G ` ( A +o (/) ) ) = ( ( G ` A ) + ( G ` (/) ) ) ) )
6	5	imbi2d 330	. . 3 |- ( n = (/) -> ( ( A e. om -> ( G ` ( A +o n ) ) = ( ( G ` A ) + ( G ` n ) ) ) <-> ( A e. om -> ( G ` ( A +o (/) ) ) = ( ( G ` A ) + ( G ` (/) ) ) ) ) )
7		oveq2 6658	. . . . . 6 |- ( n = z -> ( A +o n ) = ( A +o z ) )
8	7	fveq2d 6195	. . . . 5 |- ( n = z -> ( G ` ( A +o n ) ) = ( G ` ( A +o z ) ) )
9		fveq2 6191	. . . . . 6 |- ( n = z -> ( G ` n ) = ( G ` z ) )
10	9	oveq2d 6666	. . . . 5 |- ( n = z -> ( ( G ` A ) + ( G ` n ) ) = ( ( G ` A ) + ( G ` z ) ) )
11	8, 10	eqeq12d 2637	. . . 4 |- ( n = z -> ( ( G ` ( A +o n ) ) = ( ( G ` A ) + ( G ` n ) ) <-> ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) )
12	11	imbi2d 330	. . 3 |- ( n = z -> ( ( A e. om -> ( G ` ( A +o n ) ) = ( ( G ` A ) + ( G ` n ) ) ) <-> ( A e. om -> ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) ) )
13		oveq2 6658	. . . . . 6 |- ( n = suc z -> ( A +o n ) = ( A +o suc z ) )
14	13	fveq2d 6195	. . . . 5 |- ( n = suc z -> ( G ` ( A +o n ) ) = ( G ` ( A +o suc z ) ) )
15		fveq2 6191	. . . . . 6 |- ( n = suc z -> ( G ` n ) = ( G ` suc z ) )
16	15	oveq2d 6666	. . . . 5 |- ( n = suc z -> ( ( G ` A ) + ( G ` n ) ) = ( ( G ` A ) + ( G ` suc z ) ) )
17	14, 16	eqeq12d 2637	. . . 4 |- ( n = suc z -> ( ( G ` ( A +o n ) ) = ( ( G ` A ) + ( G ` n ) ) <-> ( G ` ( A +o suc z ) ) = ( ( G ` A ) + ( G ` suc z ) ) ) )
18	17	imbi2d 330	. . 3 |- ( n = suc z -> ( ( A e. om -> ( G ` ( A +o n ) ) = ( ( G ` A ) + ( G ` n ) ) ) <-> ( A e. om -> ( G ` ( A +o suc z ) ) = ( ( G ` A ) + ( G ` suc z ) ) ) ) )
19		oveq2 6658	. . . . . 6 |- ( n = B -> ( A +o n ) = ( A +o B ) )
20	19	fveq2d 6195	. . . . 5 |- ( n = B -> ( G ` ( A +o n ) ) = ( G ` ( A +o B ) ) )
21		fveq2 6191	. . . . . 6 |- ( n = B -> ( G ` n ) = ( G ` B ) )
22	21	oveq2d 6666	. . . . 5 |- ( n = B -> ( ( G ` A ) + ( G ` n ) ) = ( ( G ` A ) + ( G ` B ) ) )
23	20, 22	eqeq12d 2637	. . . 4 |- ( n = B -> ( ( G ` ( A +o n ) ) = ( ( G ` A ) + ( G ` n ) ) <-> ( G ` ( A +o B ) ) = ( ( G ` A ) + ( G ` B ) ) ) )
24	23	imbi2d 330	. . 3 |- ( n = B -> ( ( A e. om -> ( G ` ( A +o n ) ) = ( ( G ` A ) + ( G ` n ) ) ) <-> ( A e. om -> ( G ` ( A +o B ) ) = ( ( G ` A ) + ( G ` B ) ) ) ) )
25		hashgadd.1	. . . . . . . . 9 |- G = ( rec ( ( x e. _V |-> ( x + 1 ) ) , 0 ) |` om )
26	25	hashgf1o 12770	. . . . . . . 8 |- G : om -1-1-onto-> NN0
27		f1of 6137	. . . . . . . 8 |- ( G : om -1-1-onto-> NN0 -> G : om --> NN0 )
28	26, 27	ax-mp 5	. . . . . . 7 |- G : om --> NN0
29	28	ffvelrni 6358	. . . . . 6 |- ( A e. om -> ( G ` A ) e. NN0 )
30	29	nn0cnd 11353	. . . . 5 |- ( A e. om -> ( G ` A ) e. CC )
31	30	addid1d 10236	. . . 4 |- ( A e. om -> ( ( G ` A ) + 0 ) = ( G ` A ) )
32		0z 11388	. . . . . . 7 |- 0 e. ZZ
33	32, 25	om2uz0i 12746	. . . . . 6 |- ( G ` (/) ) = 0
34	33	oveq2i 6661	. . . . 5 |- ( ( G ` A ) + ( G ` (/) ) ) = ( ( G ` A ) + 0 )
35	34	a1i 11	. . . 4 |- ( A e. om -> ( ( G ` A ) + ( G ` (/) ) ) = ( ( G ` A ) + 0 ) )
36		nna0 7684	. . . . 5 |- ( A e. om -> ( A +o (/) ) = A )
37	36	fveq2d 6195	. . . 4 |- ( A e. om -> ( G ` ( A +o (/) ) ) = ( G ` A ) )
38	31, 35, 37	3eqtr4rd 2667	. . 3 |- ( A e. om -> ( G ` ( A +o (/) ) ) = ( ( G ` A ) + ( G ` (/) ) ) )
39		nnasuc 7686	. . . . . . . . . 10 |- ( ( A e. om /\ z e. om ) -> ( A +o suc z ) = suc ( A +o z ) )
40	39	fveq2d 6195	. . . . . . . . 9 |- ( ( A e. om /\ z e. om ) -> ( G ` ( A +o suc z ) ) = ( G ` suc ( A +o z ) ) )
41		nnacl 7691	. . . . . . . . . 10 |- ( ( A e. om /\ z e. om ) -> ( A +o z ) e. om )
42	32, 25	om2uzsuci 12747	. . . . . . . . . 10 |- ( ( A +o z ) e. om -> ( G ` suc ( A +o z ) ) = ( ( G ` ( A +o z ) ) + 1 ) )
43	41, 42	syl 17	. . . . . . . . 9 |- ( ( A e. om /\ z e. om ) -> ( G ` suc ( A +o z ) ) = ( ( G ` ( A +o z ) ) + 1 ) )
44	40, 43	eqtrd 2656	. . . . . . . 8 |- ( ( A e. om /\ z e. om ) -> ( G ` ( A +o suc z ) ) = ( ( G ` ( A +o z ) ) + 1 ) )
45	44	3adant3 1081	. . . . . . 7 |- ( ( A e. om /\ z e. om /\ ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) -> ( G ` ( A +o suc z ) ) = ( ( G ` ( A +o z ) ) + 1 ) )
46	28	ffvelrni 6358	. . . . . . . . . . 11 |- ( z e. om -> ( G ` z ) e. NN0 )
47	46	nn0cnd 11353	. . . . . . . . . 10 |- ( z e. om -> ( G ` z ) e. CC )
48		ax-1cn 9994	. . . . . . . . . . 11 |- 1 e. CC
49		addass 10023	. . . . . . . . . . 11 |- ( ( ( G ` A ) e. CC /\ ( G ` z ) e. CC /\ 1 e. CC ) -> ( ( ( G ` A ) + ( G ` z ) ) + 1 ) = ( ( G ` A ) + ( ( G ` z ) + 1 ) ) )
50	48, 49	mp3an3 1413	. . . . . . . . . 10 |- ( ( ( G ` A ) e. CC /\ ( G ` z ) e. CC ) -> ( ( ( G ` A ) + ( G ` z ) ) + 1 ) = ( ( G ` A ) + ( ( G ` z ) + 1 ) ) )
51	30, 47, 50	syl2an 494	. . . . . . . . 9 |- ( ( A e. om /\ z e. om ) -> ( ( ( G ` A ) + ( G ` z ) ) + 1 ) = ( ( G ` A ) + ( ( G ` z ) + 1 ) ) )
52	51	3adant3 1081	. . . . . . . 8 |- ( ( A e. om /\ z e. om /\ ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) -> ( ( ( G ` A ) + ( G ` z ) ) + 1 ) = ( ( G ` A ) + ( ( G ` z ) + 1 ) ) )
53		oveq1 6657	. . . . . . . . 9 |- ( ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) -> ( ( G ` ( A +o z ) ) + 1 ) = ( ( ( G ` A ) + ( G ` z ) ) + 1 ) )
54	53	3ad2ant3 1084	. . . . . . . 8 |- ( ( A e. om /\ z e. om /\ ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) -> ( ( G ` ( A +o z ) ) + 1 ) = ( ( ( G ` A ) + ( G ` z ) ) + 1 ) )
55	32, 25	om2uzsuci 12747	. . . . . . . . . 10 |- ( z e. om -> ( G ` suc z ) = ( ( G ` z ) + 1 ) )
56	55	oveq2d 6666	. . . . . . . . 9 |- ( z e. om -> ( ( G ` A ) + ( G ` suc z ) ) = ( ( G ` A ) + ( ( G ` z ) + 1 ) ) )
57	56	3ad2ant2 1083	. . . . . . . 8 |- ( ( A e. om /\ z e. om /\ ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) -> ( ( G ` A ) + ( G ` suc z ) ) = ( ( G ` A ) + ( ( G ` z ) + 1 ) ) )
58	52, 54, 57	3eqtr4d 2666	. . . . . . 7 |- ( ( A e. om /\ z e. om /\ ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) -> ( ( G ` ( A +o z ) ) + 1 ) = ( ( G ` A ) + ( G ` suc z ) ) )
59	45, 58	eqtrd 2656	. . . . . 6 |- ( ( A e. om /\ z e. om /\ ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) -> ( G ` ( A +o suc z ) ) = ( ( G ` A ) + ( G ` suc z ) ) )
60	59	3expia 1267	. . . . 5 |- ( ( A e. om /\ z e. om ) -> ( ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) -> ( G ` ( A +o suc z ) ) = ( ( G ` A ) + ( G ` suc z ) ) ) )
61	60	expcom 451	. . . 4 |- ( z e. om -> ( A e. om -> ( ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) -> ( G ` ( A +o suc z ) ) = ( ( G ` A ) + ( G ` suc z ) ) ) ) )
62	61	a2d 29	. . 3 |- ( z e. om -> ( ( A e. om -> ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) -> ( A e. om -> ( G ` ( A +o suc z ) ) = ( ( G ` A ) + ( G ` suc z ) ) ) ) )
63	6, 12, 18, 24, 38, 62	finds 7092	. 2 |- ( B e. om -> ( A e. om -> ( G ` ( A +o B ) ) = ( ( G ` A ) + ( G ` B ) ) ) )
64	63	impcom 446	1 |- ( ( A e. om /\ B e. om ) -> ( G ` ( A +o B ) ) = ( ( G ` A ) + ( G ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   |-> cmpt 4729   |` cres 5116   suc csuc 5725   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   omcom 7065   reccrdg 7505   +o coa 7557   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   NN0cn0 11292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688

This theorem is referenced by:  hashdom  13168  hashun  13171