Theorem omgadd 11220

Description: Mapping ordinal addition to integer addition. (Contributed by Jim Kingdon, 24-Feb-2022.)

Hypothesis

Ref	Expression
omgadd.1	|- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )

Assertion

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

Proof of Theorem omgadd

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

Step	Hyp	Ref	Expression
1		oveq2 6083	. . . . . 6 |- ( n = (/) -> ( A +o n ) = ( A +o (/) ) )
2	1	fveq2d 5694	. . . . 5 |- ( n = (/) -> ( G ` ( A +o n ) ) = ( G ` ( A +o (/) ) ) )
3		fveq2 5690	. . . . . 6 |- ( n = (/) -> ( G ` n ) = ( G ` (/) ) )
4	3	oveq2d 6091	. . . . 5 |- ( n = (/) -> ( ( G ` A ) + ( G ` n ) ) = ( ( G ` A ) + ( G ` (/) ) ) )
5	2, 4	eqeq12d 2253	. . . 4 |- ( n = (/) -> ( ( G ` ( A +o n ) ) = ( ( G ` A ) + ( G ` n ) ) <-> ( G ` ( A +o (/) ) ) = ( ( G ` A ) + ( G ` (/) ) ) ) )
6	5	imbi2d 230	. . 3 |- ( n = (/) -> ( ( A e. om -> ( G ` ( A +o n ) ) = ( ( G ` A ) + ( G ` n ) ) ) <-> ( A e. om -> ( G ` ( A +o (/) ) ) = ( ( G ` A ) + ( G ` (/) ) ) ) ) )
7		oveq2 6083	. . . . . 6 |- ( n = z -> ( A +o n ) = ( A +o z ) )
8	7	fveq2d 5694	. . . . 5 |- ( n = z -> ( G ` ( A +o n ) ) = ( G ` ( A +o z ) ) )
9		fveq2 5690	. . . . . 6 |- ( n = z -> ( G ` n ) = ( G ` z ) )
10	9	oveq2d 6091	. . . . 5 |- ( n = z -> ( ( G ` A ) + ( G ` n ) ) = ( ( G ` A ) + ( G ` z ) ) )
11	8, 10	eqeq12d 2253	. . . 4 |- ( n = z -> ( ( G ` ( A +o n ) ) = ( ( G ` A ) + ( G ` n ) ) <-> ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) )
12	11	imbi2d 230	. . 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 6083	. . . . . 6 |- ( n = suc z -> ( A +o n ) = ( A +o suc z ) )
14	13	fveq2d 5694	. . . . 5 |- ( n = suc z -> ( G ` ( A +o n ) ) = ( G ` ( A +o suc z ) ) )
15		fveq2 5690	. . . . . 6 |- ( n = suc z -> ( G ` n ) = ( G ` suc z ) )
16	15	oveq2d 6091	. . . . 5 |- ( n = suc z -> ( ( G ` A ) + ( G ` n ) ) = ( ( G ` A ) + ( G ` suc z ) ) )
17	14, 16	eqeq12d 2253	. . . 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 230	. . 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 6083	. . . . . 6 |- ( n = B -> ( A +o n ) = ( A +o B ) )
20	19	fveq2d 5694	. . . . 5 |- ( n = B -> ( G ` ( A +o n ) ) = ( G ` ( A +o B ) ) )
21		fveq2 5690	. . . . . 6 |- ( n = B -> ( G ` n ) = ( G ` B ) )
22	21	oveq2d 6091	. . . . 5 |- ( n = B -> ( ( G ` A ) + ( G ` n ) ) = ( ( G ` A ) + ( G ` B ) ) )
23	20, 22	eqeq12d 2253	. . . 4 |- ( n = B -> ( ( G ` ( A +o n ) ) = ( ( G ` A ) + ( G ` n ) ) <-> ( G ` ( A +o B ) ) = ( ( G ` A ) + ( G ` B ) ) ) )
24	23	imbi2d 230	. . 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		omgadd.1	. . . . . . . . 9 |- G = frec ( ( x e. ZZ |-> ( x + 1 ) ) , 0 )
26	25	frechashgf1o 10843	. . . . . . . 8 |- G : om -1-1-onto-> NN0
27		f1of 5634	. . . . . . . 8 |- ( G : om -1-1-onto-> NN0 -> G : om --> NN0 )
28	26, 27	ax-mp 5	. . . . . . 7 |- G : om --> NN0
29	28	ffvelcdmi 5833	. . . . . 6 |- ( A e. om -> ( G ` A ) e. NN0 )
30	29	nn0cnd 9601	. . . . 5 |- ( A e. om -> ( G ` A ) e. CC )
31	30	addridd 8465	. . . 4 |- ( A e. om -> ( ( G ` A ) + 0 ) = ( G ` A ) )
32		0zd 9635	. . . . . 6 |- ( A e. om -> 0 e. ZZ )
33	32, 25	frec2uz0d 10814	. . . . 5 |- ( A e. om -> ( G ` (/) ) = 0 )
34	33	oveq2d 6091	. . . 4 |- ( A e. om -> ( ( G ` A ) + ( G ` (/) ) ) = ( ( G ` A ) + 0 ) )
35		nna0 6737	. . . . 5 |- ( A e. om -> ( A +o (/) ) = A )
36	35	fveq2d 5694	. . . 4 |- ( A e. om -> ( G ` ( A +o (/) ) ) = ( G ` A ) )
37	31, 34, 36	3eqtr4rd 2282	. . 3 |- ( A e. om -> ( G ` ( A +o (/) ) ) = ( ( G ` A ) + ( G ` (/) ) ) )
38		nnasuc 6739	. . . . . . . . . 10 |- ( ( A e. om /\ z e. om ) -> ( A +o suc z ) = suc ( A +o z ) )
39	38	fveq2d 5694	. . . . . . . . 9 |- ( ( A e. om /\ z e. om ) -> ( G ` ( A +o suc z ) ) = ( G ` suc ( A +o z ) ) )
40		0zd 9635	. . . . . . . . . 10 |- ( ( A e. om /\ z e. om ) -> 0 e. ZZ )
41		nnacl 6743	. . . . . . . . . 10 |- ( ( A e. om /\ z e. om ) -> ( A +o z ) e. om )
42	40, 25, 41	frec2uzsucd 10816	. . . . . . . . 9 |- ( ( A e. om /\ z e. om ) -> ( G ` suc ( A +o z ) ) = ( ( G ` ( A +o z ) ) + 1 ) )
43	39, 42	eqtrd 2271	. . . . . . . 8 |- ( ( A e. om /\ z e. om ) -> ( G ` ( A +o suc z ) ) = ( ( G ` ( A +o z ) ) + 1 ) )
44	43	3adant3 1048	. . . . . . 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 ) )
45	30	3ad2ant1 1049	. . . . . . . . 9 |- ( ( A e. om /\ z e. om /\ ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) -> ( G ` A ) e. CC )
46	28	ffvelcdmi 5833	. . . . . . . . . . 11 |- ( z e. om -> ( G ` z ) e. NN0 )
47	46	nn0cnd 9601	. . . . . . . . . 10 |- ( z e. om -> ( G ` z ) e. CC )
48	47	3ad2ant2 1050	. . . . . . . . 9 |- ( ( A e. om /\ z e. om /\ ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) -> ( G ` z ) e. CC )
49		1cnd 8332	. . . . . . . . 9 |- ( ( A e. om /\ z e. om /\ ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) -> 1 e. CC )
50	45, 48, 49	addassd 8338	. . . . . . . 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 ) ) )
51		oveq1 6082	. . . . . . . . 9 |- ( ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) -> ( ( G ` ( A +o z ) ) + 1 ) = ( ( ( G ` A ) + ( G ` z ) ) + 1 ) )
52	51	3ad2ant3 1051	. . . . . . . 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 ) )
53		0zd 9635	. . . . . . . . . . 11 |- ( z e. om -> 0 e. ZZ )
54		id 19	. . . . . . . . . . 11 |- ( z e. om -> z e. om )
55	53, 25, 54	frec2uzsucd 10816	. . . . . . . . . 10 |- ( z e. om -> ( G ` suc z ) = ( ( G ` z ) + 1 ) )
56	55	oveq2d 6091	. . . . . . . . 9 |- ( z e. om -> ( ( G ` A ) + ( G ` suc z ) ) = ( ( G ` A ) + ( ( G ` z ) + 1 ) ) )
57	56	3ad2ant2 1050	. . . . . . . 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	50, 52, 57	3eqtr4d 2281	. . . . . . 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	44, 58	eqtrd 2271	. . . . . 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 1236	. . . . 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 116	. . . 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 26	. . 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, 37, 62	finds 4742	. 2 |- ( B e. om -> ( A e. om -> ( G ` ( A +o B ) ) = ( ( G ` A ) + ( G ` B ) ) ) )
64	63	impcom 125	1 |- ( ( A e. om /\ B e. om ) -> ( G ` ( A +o B ) ) = ( ( G ` A ) + ( G ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   (/)c0 3520   |-> cmpt 4187   suc csuc 4505   omcom 4732   -->wf 5368   -1-1-onto->wf1o 5371   ` cfv 5372  (class class class)co 6075  freccfrec 6651   +o coa 6674   CCcc 8167   0cc0 8169   1c1 8170   + caddc 8172   NN0cn0 9542   ZZcz 9623

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-oadd 6681  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901

This theorem is referenced by:  hashun  11223