Theorem omgadd 10784

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 5885	. . . . . 6 |- ( n = (/) -> ( A +o n ) = ( A +o (/) ) )
2	1	fveq2d 5521	. . . . 5 |- ( n = (/) -> ( G ` ( A +o n ) ) = ( G ` ( A +o (/) ) ) )
3		fveq2 5517	. . . . . 6 |- ( n = (/) -> ( G ` n ) = ( G ` (/) ) )
4	3	oveq2d 5893	. . . . 5 |- ( n = (/) -> ( ( G ` A ) + ( G ` n ) ) = ( ( G ` A ) + ( G ` (/) ) ) )
5	2, 4	eqeq12d 2192	. . . 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 5885	. . . . . 6 |- ( n = z -> ( A +o n ) = ( A +o z ) )
8	7	fveq2d 5521	. . . . 5 |- ( n = z -> ( G ` ( A +o n ) ) = ( G ` ( A +o z ) ) )
9		fveq2 5517	. . . . . 6 |- ( n = z -> ( G ` n ) = ( G ` z ) )
10	9	oveq2d 5893	. . . . 5 |- ( n = z -> ( ( G ` A ) + ( G ` n ) ) = ( ( G ` A ) + ( G ` z ) ) )
11	8, 10	eqeq12d 2192	. . . 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 5885	. . . . . 6 |- ( n = suc z -> ( A +o n ) = ( A +o suc z ) )
14	13	fveq2d 5521	. . . . 5 |- ( n = suc z -> ( G ` ( A +o n ) ) = ( G ` ( A +o suc z ) ) )
15		fveq2 5517	. . . . . 6 |- ( n = suc z -> ( G ` n ) = ( G ` suc z ) )
16	15	oveq2d 5893	. . . . 5 |- ( n = suc z -> ( ( G ` A ) + ( G ` n ) ) = ( ( G ` A ) + ( G ` suc z ) ) )
17	14, 16	eqeq12d 2192	. . . 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 5885	. . . . . 6 |- ( n = B -> ( A +o n ) = ( A +o B ) )
20	19	fveq2d 5521	. . . . 5 |- ( n = B -> ( G ` ( A +o n ) ) = ( G ` ( A +o B ) ) )
21		fveq2 5517	. . . . . 6 |- ( n = B -> ( G ` n ) = ( G ` B ) )
22	21	oveq2d 5893	. . . . 5 |- ( n = B -> ( ( G ` A ) + ( G ` n ) ) = ( ( G ` A ) + ( G ` B ) ) )
23	20, 22	eqeq12d 2192	. . . 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 10430	. . . . . . . 8 |- G : om -1-1-onto-> NN0
27		f1of 5463	. . . . . . . 8 |- ( G : om -1-1-onto-> NN0 -> G : om --> NN0 )
28	26, 27	ax-mp 5	. . . . . . 7 |- G : om --> NN0
29	28	ffvelcdmi 5652	. . . . . 6 |- ( A e. om -> ( G ` A ) e. NN0 )
30	29	nn0cnd 9233	. . . . 5 |- ( A e. om -> ( G ` A ) e. CC )
31	30	addid1d 8108	. . . 4 |- ( A e. om -> ( ( G ` A ) + 0 ) = ( G ` A ) )
32		0zd 9267	. . . . . 6 |- ( A e. om -> 0 e. ZZ )
33	32, 25	frec2uz0d 10401	. . . . 5 |- ( A e. om -> ( G ` (/) ) = 0 )
34	33	oveq2d 5893	. . . 4 |- ( A e. om -> ( ( G ` A ) + ( G ` (/) ) ) = ( ( G ` A ) + 0 ) )
35		nna0 6477	. . . . 5 |- ( A e. om -> ( A +o (/) ) = A )
36	35	fveq2d 5521	. . . 4 |- ( A e. om -> ( G ` ( A +o (/) ) ) = ( G ` A ) )
37	31, 34, 36	3eqtr4rd 2221	. . 3 |- ( A e. om -> ( G ` ( A +o (/) ) ) = ( ( G ` A ) + ( G ` (/) ) ) )
38		nnasuc 6479	. . . . . . . . . 10 |- ( ( A e. om /\ z e. om ) -> ( A +o suc z ) = suc ( A +o z ) )
39	38	fveq2d 5521	. . . . . . . . 9 |- ( ( A e. om /\ z e. om ) -> ( G ` ( A +o suc z ) ) = ( G ` suc ( A +o z ) ) )
40		0zd 9267	. . . . . . . . . 10 |- ( ( A e. om /\ z e. om ) -> 0 e. ZZ )
41		nnacl 6483	. . . . . . . . . 10 |- ( ( A e. om /\ z e. om ) -> ( A +o z ) e. om )
42	40, 25, 41	frec2uzsucd 10403	. . . . . . . . 9 |- ( ( A e. om /\ z e. om ) -> ( G ` suc ( A +o z ) ) = ( ( G ` ( A +o z ) ) + 1 ) )
43	39, 42	eqtrd 2210	. . . . . . . 8 |- ( ( A e. om /\ z e. om ) -> ( G ` ( A +o suc z ) ) = ( ( G ` ( A +o z ) ) + 1 ) )
44	43	3adant3 1017	. . . . . . 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 1018	. . . . . . . . 9 |- ( ( A e. om /\ z e. om /\ ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) -> ( G ` A ) e. CC )
46	28	ffvelcdmi 5652	. . . . . . . . . . 11 |- ( z e. om -> ( G ` z ) e. NN0 )
47	46	nn0cnd 9233	. . . . . . . . . 10 |- ( z e. om -> ( G ` z ) e. CC )
48	47	3ad2ant2 1019	. . . . . . . . 9 |- ( ( A e. om /\ z e. om /\ ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) -> ( G ` z ) e. CC )
49		1cnd 7975	. . . . . . . . 9 |- ( ( A e. om /\ z e. om /\ ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) -> 1 e. CC )
50	45, 48, 49	addassd 7982	. . . . . . . 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 5884	. . . . . . . . 9 |- ( ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) -> ( ( G ` ( A +o z ) ) + 1 ) = ( ( ( G ` A ) + ( G ` z ) ) + 1 ) )
52	51	3ad2ant3 1020	. . . . . . . 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 9267	. . . . . . . . . . 11 |- ( z e. om -> 0 e. ZZ )
54		id 19	. . . . . . . . . . 11 |- ( z e. om -> z e. om )
55	53, 25, 54	frec2uzsucd 10403	. . . . . . . . . 10 |- ( z e. om -> ( G ` suc z ) = ( ( G ` z ) + 1 ) )
56	55	oveq2d 5893	. . . . . . . . 9 |- ( z e. om -> ( ( G ` A ) + ( G ` suc z ) ) = ( ( G ` A ) + ( ( G ` z ) + 1 ) ) )
57	56	3ad2ant2 1019	. . . . . . . 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 2220	. . . . . . 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 2210	. . . . . 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 1205	. . . . 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 4601	. 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 978   = wceq 1353   e. wcel 2148   (/)c0 3424   |-> cmpt 4066   suc csuc 4367   omcom 4591   -->wf 5214   -1-1-onto->wf1o 5217   ` cfv 5218  (class class class)co 5877  freccfrec 6393   +o coa 6416   CCcc 7811   0cc0 7813   1c1 7814   + caddc 7816   NN0cn0 9178   ZZcz 9255

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-addcom 7913  ax-addass 7915  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-0id 7921  ax-rnegex 7922  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-ltadd 7929

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-frec 6394  df-oadd 6423  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-inn 8922  df-n0 9179  df-z 9256  df-uz 9531

This theorem is referenced by:  hashun  10787