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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.
StepHypRef Expression
1 oveq2 5571 . . . . . 6  |-  ( n  =  (/)  ->  ( A  +o  n )  =  ( A  +o  (/) ) )
21fveq2d 5233 . . . . 5  |-  ( n  =  (/)  ->  ( G `
 ( A  +o  n ) )  =  ( G `  ( A  +o  (/) ) ) )
3 fveq2 5229 . . . . . 6  |-  ( n  =  (/)  ->  ( G `
 n )  =  ( G `  (/) ) )
43oveq2d 5579 . . . . 5  |-  ( n  =  (/)  ->  ( ( G `  A )  +  ( G `  n ) )  =  ( ( G `  A )  +  ( G `  (/) ) ) )
52, 4eqeq12d 2097 . . . 4  |-  ( n  =  (/)  ->  ( ( G `  ( A  +o  n ) )  =  ( ( G `
 A )  +  ( G `  n
) )  <->  ( G `  ( A  +o  (/) ) )  =  ( ( G `
 A )  +  ( G `  (/) ) ) ) )
65imbi2d 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
) )
87fveq2d 5233 . . . . 5  |-  ( n  =  z  ->  ( G `  ( A  +o  n ) )  =  ( G `  ( A  +o  z ) ) )
9 fveq2 5229 . . . . . 6  |-  ( n  =  z  ->  ( G `  n )  =  ( G `  z ) )
109oveq2d 5579 . . . . 5  |-  ( n  =  z  ->  (
( G `  A
)  +  ( G `
 n ) )  =  ( ( G `
 A )  +  ( G `  z
) ) )
118, 10eqeq12d 2097 . . . 4  |-  ( n  =  z  ->  (
( G `  ( A  +o  n ) )  =  ( ( G `
 A )  +  ( G `  n
) )  <->  ( G `  ( A  +o  z
) )  =  ( ( G `  A
)  +  ( G `
 z ) ) ) )
1211imbi2d 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 ) )
1413fveq2d 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 )
)
1615oveq2d 5579 . . . . 5  |-  ( n  =  suc  z  -> 
( ( G `  A )  +  ( G `  n ) )  =  ( ( G `  A )  +  ( G `  suc  z ) ) )
1714, 16eqeq12d 2097 . . . 4  |-  ( n  =  suc  z  -> 
( ( G `  ( A  +o  n
) )  =  ( ( G `  A
)  +  ( G `
 n ) )  <-> 
( G `  ( A  +o  suc  z ) )  =  ( ( G `  A )  +  ( G `  suc  z ) ) ) )
1817imbi2d 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
) )
2019fveq2d 5233 . . . . 5  |-  ( n  =  B  ->  ( G `  ( A  +o  n ) )  =  ( G `  ( A  +o  B ) ) )
21 fveq2 5229 . . . . . 6  |-  ( n  =  B  ->  ( G `  n )  =  ( G `  B ) )
2221oveq2d 5579 . . . . 5  |-  ( n  =  B  ->  (
( G `  A
)  +  ( G `
 n ) )  =  ( ( G `
 A )  +  ( G `  B
) ) )
2320, 22eqeq12d 2097 . . . 4  |-  ( n  =  B  ->  (
( G `  ( A  +o  n ) )  =  ( ( G `
 A )  +  ( G `  n
) )  <->  ( G `  ( A  +o  B
) )  =  ( ( G `  A
)  +  ( G `
 B ) ) ) )
2423imbi2d 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 )
2625frechashgf1o 9562 . . . . . . . 8  |-  G : om
-1-1-onto-> NN0
27 f1of 5177 . . . . . . . 8  |-  ( G : om -1-1-onto-> NN0  ->  G : om
--> NN0 )
2826, 27ax-mp 7 . . . . . . 7  |-  G : om
--> NN0
2928ffvelrni 5353 . . . . . 6  |-  ( A  e.  om  ->  ( G `  A )  e.  NN0 )
3029nn0cnd 8462 . . . . 5  |-  ( A  e.  om  ->  ( G `  A )  e.  CC )
3130addid1d 7376 . . . 4  |-  ( A  e.  om  ->  (
( G `  A
)  +  0 )  =  ( G `  A ) )
32 0zd 8496 . . . . . 6  |-  ( A  e.  om  ->  0  e.  ZZ )
3332, 25frec2uz0d 9533 . . . . 5  |-  ( A  e.  om  ->  ( G `  (/) )  =  0 )
3433oveq2d 5579 . . . 4  |-  ( A  e.  om  ->  (
( G `  A
)  +  ( G `
 (/) ) )  =  ( ( G `  A )  +  0 ) )
35 nna0 6138 . . . . 5  |-  ( A  e.  om  ->  ( A  +o  (/) )  =  A )
3635fveq2d 5233 . . . 4  |-  ( A  e.  om  ->  ( G `  ( A  +o  (/) ) )  =  ( G `  A
) )
3731, 34, 363eqtr4rd 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
) )
3938fveq2d 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 )
4240, 25, 41frec2uzsucd 9535 . . . . . . . . 9  |-  ( ( A  e.  om  /\  z  e.  om )  ->  ( G `  suc  ( A  +o  z
) )  =  ( ( G `  ( A  +o  z ) )  +  1 ) )
4339, 42eqtrd 2115 . . . . . . . 8  |-  ( ( A  e.  om  /\  z  e.  om )  ->  ( G `  ( A  +o  suc  z ) )  =  ( ( G `  ( A  +o  z ) )  +  1 ) )
44433adant3 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 ) )
45303ad2ant1 960 . . . . . . . . 9  |-  ( ( A  e.  om  /\  z  e.  om  /\  ( G `  ( A  +o  z ) )  =  ( ( G `  A )  +  ( G `  z ) ) )  ->  ( G `  A )  e.  CC )
4628ffvelrni 5353 . . . . . . . . . . 11  |-  ( z  e.  om  ->  ( G `  z )  e.  NN0 )
4746nn0cnd 8462 . . . . . . . . . 10  |-  ( z  e.  om  ->  ( G `  z )  e.  CC )
48473ad2ant2 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 )
5045, 48, 49addassd 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 ) )
52513ad2ant3 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 )
5553, 25, 54frec2uzsucd 9535 . . . . . . . . . 10  |-  ( z  e.  om  ->  ( G `  suc  z )  =  ( ( G `
 z )  +  1 ) )
5655oveq2d 5579 . . . . . . . . 9  |-  ( z  e.  om  ->  (
( G `  A
)  +  ( G `
 suc  z )
)  =  ( ( G `  A )  +  ( ( G `
 z )  +  1 ) ) )
57563ad2ant2 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 ) ) )
5850, 52, 573eqtr4d 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
) ) )
5944, 58eqtrd 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 ) ) )
60593expia 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 )
) ) )
6160expcom 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 ) ) ) ) )
6261a2d 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 ) ) ) ) )
636, 12, 18, 24, 37, 62finds 4369 . 2  |-  ( B  e.  om  ->  ( A  e.  om  ->  ( G `  ( A  +o  B ) )  =  ( ( G `
 A )  +  ( G `  B
) ) ) )
6463impcom 123 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( G `  ( A  +o  B ) )  =  ( ( G `
 A )  +  ( G `  B
) ) )
Colors of variables: wff set class
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
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