Theorem omgadd 9878

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 5571	. . . . . 6 |- ( n = (/) -> ( A +o n ) = ( A +o (/) ) )
2	1	fveq2d 5233	. . . . 5 |- ( n = (/) -> ( G ` ( A +o n ) ) = ( G ` ( A +o (/) ) ) )
3		fveq2 5229	. . . . . 6 |- ( n = (/) -> ( G ` n ) = ( G ` (/) ) )
4	3	oveq2d 5579	. . . . 5 |- ( n = (/) -> ( ( G ` A ) + ( G ` n ) ) = ( ( G ` A ) + ( G ` (/) ) ) )
5	2, 4	eqeq12d 2097	. . . 4 |- ( n = (/) -> ( ( G ` ( A +o n ) ) = ( ( G ` A ) + ( G ` n ) ) <-> ( G ` ( A +o (/) ) ) = ( ( G ` A ) + ( G ` (/) ) ) ) )
6	5	imbi2d 228	. . 3 |- ( n = (/) -> ( ( A e. om -> ( G ` ( A +o n ) ) = ( ( G ` A ) + ( G ` n ) ) ) <-> ( A e. om -> ( G ` ( A +o (/) ) ) = ( ( G ` A ) + ( G ` (/) ) ) ) ) )
7		oveq2 5571	. . . . . 6 |- ( n = z -> ( A +o n ) = ( A +o z ) )
8	7	fveq2d 5233	. . . . 5 |- ( n = z -> ( G ` ( A +o n ) ) = ( G ` ( A +o z ) ) )
9		fveq2 5229	. . . . . 6 |- ( n = z -> ( G ` n ) = ( G ` z ) )
10	9	oveq2d 5579	. . . . 5 |- ( n = z -> ( ( G ` A ) + ( G ` n ) ) = ( ( G ` A ) + ( G ` z ) ) )
11	8, 10	eqeq12d 2097	. . . 4 |- ( n = z -> ( ( G ` ( A +o n ) ) = ( ( G ` A ) + ( G ` n ) ) <-> ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) )
12	11	imbi2d 228	. . 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 5571	. . . . . 6 |- ( n = suc z -> ( A +o n ) = ( A +o suc z ) )
14	13	fveq2d 5233	. . . . 5 |- ( n = suc z -> ( G ` ( A +o n ) ) = ( G ` ( A +o suc z ) ) )
15		fveq2 5229	. . . . . 6 |- ( n = suc z -> ( G ` n ) = ( G ` suc z ) )
16	15	oveq2d 5579	. . . . 5 |- ( n = suc z -> ( ( G ` A ) + ( G ` n ) ) = ( ( G ` A ) + ( G ` suc z ) ) )
17	14, 16	eqeq12d 2097	. . . 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 228	. . 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 5571	. . . . . 6 |- ( n = B -> ( A +o n ) = ( A +o B ) )
20	19	fveq2d 5233	. . . . 5 |- ( n = B -> ( G ` ( A +o n ) ) = ( G ` ( A +o B ) ) )
21		fveq2 5229	. . . . . 6 |- ( n = B -> ( G ` n ) = ( G ` B ) )
22	21	oveq2d 5579	. . . . 5 |- ( n = B -> ( ( G ` A ) + ( G ` n ) ) = ( ( G ` A ) + ( G ` B ) ) )
23	20, 22	eqeq12d 2097	. . . 4 |- ( n = B -> ( ( G ` ( A +o n ) ) = ( ( G ` A ) + ( G ` n ) ) <-> ( G ` ( A +o B ) ) = ( ( G ` A ) + ( G ` B ) ) ) )
24	23	imbi2d 228	. . 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 9562	. . . . . . . 8 |- G : om -1-1-onto-> NN0
27		f1of 5177	. . . . . . . 8 |- ( G : om -1-1-onto-> NN0 -> G : om --> NN0 )
28	26, 27	ax-mp 7	. . . . . . 7 |- G : om --> NN0
29	28	ffvelrni 5353	. . . . . 6 |- ( A e. om -> ( G ` A ) e. NN0 )
30	29	nn0cnd 8462	. . . . 5 |- ( A e. om -> ( G ` A ) e. CC )
31	30	addid1d 7376	. . . 4 |- ( A e. om -> ( ( G ` A ) + 0 ) = ( G ` A ) )
32		0zd 8496	. . . . . 6 |- ( A e. om -> 0 e. ZZ )
33	32, 25	frec2uz0d 9533	. . . . 5 |- ( A e. om -> ( G ` (/) ) = 0 )
34	33	oveq2d 5579	. . . 4 |- ( A e. om -> ( ( G ` A ) + ( G ` (/) ) ) = ( ( G ` A ) + 0 ) )
35		nna0 6138	. . . . 5 |- ( A e. om -> ( A +o (/) ) = A )
36	35	fveq2d 5233	. . . 4 |- ( A e. om -> ( G ` ( A +o (/) ) ) = ( G ` A ) )
37	31, 34, 36	3eqtr4rd 2126	. . 3 |- ( A e. om -> ( G ` ( A +o (/) ) ) = ( ( G ` A ) + ( G ` (/) ) ) )
38		nnasuc 6140	. . . . . . . . . 10 |- ( ( A e. om /\ z e. om ) -> ( A +o suc z ) = suc ( A +o z ) )
39	38	fveq2d 5233	. . . . . . . . 9 |- ( ( A e. om /\ z e. om ) -> ( G ` ( A +o suc z ) ) = ( G ` suc ( A +o z ) ) )
40		0zd 8496	. . . . . . . . . 10 |- ( ( A e. om /\ z e. om ) -> 0 e. ZZ )
41		nnacl 6144	. . . . . . . . . 10 |- ( ( A e. om /\ z e. om ) -> ( A +o z ) e. om )
42	40, 25, 41	frec2uzsucd 9535	. . . . . . . . 9 |- ( ( A e. om /\ z e. om ) -> ( G ` suc ( A +o z ) ) = ( ( G ` ( A +o z ) ) + 1 ) )
43	39, 42	eqtrd 2115	. . . . . . . 8 |- ( ( A e. om /\ z e. om ) -> ( G ` ( A +o suc z ) ) = ( ( G ` ( A +o z ) ) + 1 ) )
44	43	3adant3 959	. . . . . . 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 960	. . . . . . . . 9 |- ( ( A e. om /\ z e. om /\ ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) -> ( G ` A ) e. CC )
46	28	ffvelrni 5353	. . . . . . . . . . 11 |- ( z e. om -> ( G ` z ) e. NN0 )
47	46	nn0cnd 8462	. . . . . . . . . 10 |- ( z e. om -> ( G ` z ) e. CC )
48	47	3ad2ant2 961	. . . . . . . . 9 |- ( ( A e. om /\ z e. om /\ ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) -> ( G ` z ) e. CC )
49		1cnd 7249	. . . . . . . . 9 |- ( ( A e. om /\ z e. om /\ ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) -> 1 e. CC )
50	45, 48, 49	addassd 7255	. . . . . . . 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 5570	. . . . . . . . 9 |- ( ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) -> ( ( G ` ( A +o z ) ) + 1 ) = ( ( ( G ` A ) + ( G ` z ) ) + 1 ) )
52	51	3ad2ant3 962	. . . . . . . 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 8496	. . . . . . . . . . 11 |- ( z e. om -> 0 e. ZZ )
54		id 19	. . . . . . . . . . 11 |- ( z e. om -> z e. om )
55	53, 25, 54	frec2uzsucd 9535	. . . . . . . . . 10 |- ( z e. om -> ( G ` suc z ) = ( ( G ` z ) + 1 ) )
56	55	oveq2d 5579	. . . . . . . . 9 |- ( z e. om -> ( ( G ` A ) + ( G ` suc z ) ) = ( ( G ` A ) + ( ( G ` z ) + 1 ) ) )
57	56	3ad2ant2 961	. . . . . . . 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 2125	. . . . . . 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 2115	. . . . . 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 1141	. . . . 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 114	. . . 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 4369	. 2 |- ( B e. om -> ( A e. om -> ( G ` ( A +o B ) ) = ( ( G ` A ) + ( G ` B ) ) ) )
64	63	impcom 123	1 |- ( ( A e. om /\ B e. om ) -> ( G ` ( A +o B ) ) = ( ( G ` A ) + ( G ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 102   /\ w3a 920   = wceq 1285   e. wcel 1434   (/)c0 3267   |-> cmpt 3859   suc csuc 4148   omcom 4359   -->wf 4948   -1-1-onto->wf1o 4951   ` cfv 4952  (class class class)co 5563  freccfrec 6059   +o coa 6082   CCcc 7093   0cc0 7095   1c1 7096   + caddc 7098   NN0cn0 8407   ZZcz 8484

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-ltadd 7206

This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-1st 5818  df-2nd 5819  df-recs 5974  df-irdg 6039  df-frec 6060  df-oadd 6089  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-inn 8159  df-n0 8408  df-z 8485  df-uz 8753

This theorem is referenced by:  hashun  9881