Theorem omgadd 10704

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 5844	. . . . . 6 |- ( n = (/) -> ( A +o n ) = ( A +o (/) ) )
2	1	fveq2d 5484	. . . . 5 |- ( n = (/) -> ( G ` ( A +o n ) ) = ( G ` ( A +o (/) ) ) )
3		fveq2 5480	. . . . . 6 |- ( n = (/) -> ( G ` n ) = ( G ` (/) ) )
4	3	oveq2d 5852	. . . . 5 |- ( n = (/) -> ( ( G ` A ) + ( G ` n ) ) = ( ( G ` A ) + ( G ` (/) ) ) )
5	2, 4	eqeq12d 2179	. . . 4 |- ( n = (/) -> ( ( G ` ( A +o n ) ) = ( ( G ` A ) + ( G ` n ) ) <-> ( G ` ( A +o (/) ) ) = ( ( G ` A ) + ( G ` (/) ) ) ) )
6	5	imbi2d 229	. . 3 |- ( n = (/) -> ( ( A e. om -> ( G ` ( A +o n ) ) = ( ( G ` A ) + ( G ` n ) ) ) <-> ( A e. om -> ( G ` ( A +o (/) ) ) = ( ( G ` A ) + ( G ` (/) ) ) ) ) )
7		oveq2 5844	. . . . . 6 |- ( n = z -> ( A +o n ) = ( A +o z ) )
8	7	fveq2d 5484	. . . . 5 |- ( n = z -> ( G ` ( A +o n ) ) = ( G ` ( A +o z ) ) )
9		fveq2 5480	. . . . . 6 |- ( n = z -> ( G ` n ) = ( G ` z ) )
10	9	oveq2d 5852	. . . . 5 |- ( n = z -> ( ( G ` A ) + ( G ` n ) ) = ( ( G ` A ) + ( G ` z ) ) )
11	8, 10	eqeq12d 2179	. . . 4 |- ( n = z -> ( ( G ` ( A +o n ) ) = ( ( G ` A ) + ( G ` n ) ) <-> ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) )
12	11	imbi2d 229	. . 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 5844	. . . . . 6 |- ( n = suc z -> ( A +o n ) = ( A +o suc z ) )
14	13	fveq2d 5484	. . . . 5 |- ( n = suc z -> ( G ` ( A +o n ) ) = ( G ` ( A +o suc z ) ) )
15		fveq2 5480	. . . . . 6 |- ( n = suc z -> ( G ` n ) = ( G ` suc z ) )
16	15	oveq2d 5852	. . . . 5 |- ( n = suc z -> ( ( G ` A ) + ( G ` n ) ) = ( ( G ` A ) + ( G ` suc z ) ) )
17	14, 16	eqeq12d 2179	. . . 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 229	. . 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 5844	. . . . . 6 |- ( n = B -> ( A +o n ) = ( A +o B ) )
20	19	fveq2d 5484	. . . . 5 |- ( n = B -> ( G ` ( A +o n ) ) = ( G ` ( A +o B ) ) )
21		fveq2 5480	. . . . . 6 |- ( n = B -> ( G ` n ) = ( G ` B ) )
22	21	oveq2d 5852	. . . . 5 |- ( n = B -> ( ( G ` A ) + ( G ` n ) ) = ( ( G ` A ) + ( G ` B ) ) )
23	20, 22	eqeq12d 2179	. . . 4 |- ( n = B -> ( ( G ` ( A +o n ) ) = ( ( G ` A ) + ( G ` n ) ) <-> ( G ` ( A +o B ) ) = ( ( G ` A ) + ( G ` B ) ) ) )
24	23	imbi2d 229	. . 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 10353	. . . . . . . 8 |- G : om -1-1-onto-> NN0
27		f1of 5426	. . . . . . . 8 |- ( G : om -1-1-onto-> NN0 -> G : om --> NN0 )
28	26, 27	ax-mp 5	. . . . . . 7 |- G : om --> NN0
29	28	ffvelrni 5613	. . . . . 6 |- ( A e. om -> ( G ` A ) e. NN0 )
30	29	nn0cnd 9160	. . . . 5 |- ( A e. om -> ( G ` A ) e. CC )
31	30	addid1d 8038	. . . 4 |- ( A e. om -> ( ( G ` A ) + 0 ) = ( G ` A ) )
32		0zd 9194	. . . . . 6 |- ( A e. om -> 0 e. ZZ )
33	32, 25	frec2uz0d 10324	. . . . 5 |- ( A e. om -> ( G ` (/) ) = 0 )
34	33	oveq2d 5852	. . . 4 |- ( A e. om -> ( ( G ` A ) + ( G ` (/) ) ) = ( ( G ` A ) + 0 ) )
35		nna0 6433	. . . . 5 |- ( A e. om -> ( A +o (/) ) = A )
36	35	fveq2d 5484	. . . 4 |- ( A e. om -> ( G ` ( A +o (/) ) ) = ( G ` A ) )
37	31, 34, 36	3eqtr4rd 2208	. . 3 |- ( A e. om -> ( G ` ( A +o (/) ) ) = ( ( G ` A ) + ( G ` (/) ) ) )
38		nnasuc 6435	. . . . . . . . . 10 |- ( ( A e. om /\ z e. om ) -> ( A +o suc z ) = suc ( A +o z ) )
39	38	fveq2d 5484	. . . . . . . . 9 |- ( ( A e. om /\ z e. om ) -> ( G ` ( A +o suc z ) ) = ( G ` suc ( A +o z ) ) )
40		0zd 9194	. . . . . . . . . 10 |- ( ( A e. om /\ z e. om ) -> 0 e. ZZ )
41		nnacl 6439	. . . . . . . . . 10 |- ( ( A e. om /\ z e. om ) -> ( A +o z ) e. om )
42	40, 25, 41	frec2uzsucd 10326	. . . . . . . . 9 |- ( ( A e. om /\ z e. om ) -> ( G ` suc ( A +o z ) ) = ( ( G ` ( A +o z ) ) + 1 ) )
43	39, 42	eqtrd 2197	. . . . . . . 8 |- ( ( A e. om /\ z e. om ) -> ( G ` ( A +o suc z ) ) = ( ( G ` ( A +o z ) ) + 1 ) )
44	43	3adant3 1006	. . . . . . 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 1007	. . . . . . . . 9 |- ( ( A e. om /\ z e. om /\ ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) -> ( G ` A ) e. CC )
46	28	ffvelrni 5613	. . . . . . . . . . 11 |- ( z e. om -> ( G ` z ) e. NN0 )
47	46	nn0cnd 9160	. . . . . . . . . 10 |- ( z e. om -> ( G ` z ) e. CC )
48	47	3ad2ant2 1008	. . . . . . . . 9 |- ( ( A e. om /\ z e. om /\ ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) -> ( G ` z ) e. CC )
49		1cnd 7906	. . . . . . . . 9 |- ( ( A e. om /\ z e. om /\ ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) ) -> 1 e. CC )
50	45, 48, 49	addassd 7912	. . . . . . . 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 5843	. . . . . . . . 9 |- ( ( G ` ( A +o z ) ) = ( ( G ` A ) + ( G ` z ) ) -> ( ( G ` ( A +o z ) ) + 1 ) = ( ( ( G ` A ) + ( G ` z ) ) + 1 ) )
52	51	3ad2ant3 1009	. . . . . . . 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 9194	. . . . . . . . . . 11 |- ( z e. om -> 0 e. ZZ )
54		id 19	. . . . . . . . . . 11 |- ( z e. om -> z e. om )
55	53, 25, 54	frec2uzsucd 10326	. . . . . . . . . 10 |- ( z e. om -> ( G ` suc z ) = ( ( G ` z ) + 1 ) )
56	55	oveq2d 5852	. . . . . . . . 9 |- ( z e. om -> ( ( G ` A ) + ( G ` suc z ) ) = ( ( G ` A ) + ( ( G ` z ) + 1 ) ) )
57	56	3ad2ant2 1008	. . . . . . . 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 2207	. . . . . . 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 2197	. . . . . 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 1194	. . . . 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 115	. . . 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 4571	. 2 |- ( B e. om -> ( A e. om -> ( G ` ( A +o B ) ) = ( ( G ` A ) + ( G ` B ) ) ) )
64	63	impcom 124	1 |- ( ( A e. om /\ B e. om ) -> ( G ` ( A +o B ) ) = ( ( G ` A ) + ( G ` B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 967   = wceq 1342   e. wcel 2135   (/)c0 3404   |-> cmpt 4037   suc csuc 4337   omcom 4561   -->wf 5178   -1-1-onto->wf1o 5181   ` cfv 5182  (class class class)co 5836  freccfrec 6349   +o coa 6372   CCcc 7742   0cc0 7744   1c1 7745   + caddc 7747   NN0cn0 9105   ZZcz 9182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559  ax-cnex 7835  ax-resscn 7836  ax-1cn 7837  ax-1re 7838  ax-icn 7839  ax-addcl 7840  ax-addrcl 7841  ax-mulcl 7842  ax-addcom 7844  ax-addass 7846  ax-distr 7848  ax-i2m1 7849  ax-0lt1 7850  ax-0id 7852  ax-rnegex 7853  ax-cnre 7855  ax-pre-ltirr 7856  ax-pre-ltwlin 7857  ax-pre-lttrn 7858  ax-pre-ltadd 7860

This theorem depends on definitions:  df-bi 116  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-iord 4338  df-on 4340  df-ilim 4341  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-riota 5792  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-irdg 6329  df-frec 6350  df-oadd 6379  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-sub 8062  df-neg 8063  df-inn 8849  df-n0 9106  df-z 9183  df-uz 9458

This theorem is referenced by:  hashun  10707