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Theorem hashgadd 11606
Description:  G maps ordinal addition to integer addition. (Contributed by Paul Chapman, 30-Nov-2012.) (Revised by Mario Carneiro, 15-Sep-2013.)
Hypothesis
Ref Expression
hashgadd.1  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )
Assertion
Ref Expression
hashgadd  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( G `  ( A  +o  B ) )  =  ( ( G `
 A )  +  ( G `  B
) ) )

Proof of Theorem hashgadd
Dummy variables  n  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6048 . . . . . 6  |-  ( n  =  (/)  ->  ( A  +o  n )  =  ( A  +o  (/) ) )
21fveq2d 5691 . . . . 5  |-  ( n  =  (/)  ->  ( G `
 ( A  +o  n ) )  =  ( G `  ( A  +o  (/) ) ) )
3 fveq2 5687 . . . . . 6  |-  ( n  =  (/)  ->  ( G `
 n )  =  ( G `  (/) ) )
43oveq2d 6056 . . . . 5  |-  ( n  =  (/)  ->  ( ( G `  A )  +  ( G `  n ) )  =  ( ( G `  A )  +  ( G `  (/) ) ) )
52, 4eqeq12d 2418 . . . 4  |-  ( n  =  (/)  ->  ( ( G `  ( A  +o  n ) )  =  ( ( G `
 A )  +  ( G `  n
) )  <->  ( G `  ( A  +o  (/) ) )  =  ( ( G `
 A )  +  ( G `  (/) ) ) ) )
65imbi2d 308 . . 3  |-  ( n  =  (/)  ->  ( ( A  e.  om  ->  ( G `  ( A  +o  n ) )  =  ( ( G `
 A )  +  ( G `  n
) ) )  <->  ( A  e.  om  ->  ( G `  ( A  +o  (/) ) )  =  ( ( G `
 A )  +  ( G `  (/) ) ) ) ) )
7 oveq2 6048 . . . . . 6  |-  ( n  =  z  ->  ( A  +o  n )  =  ( A  +o  z
) )
87fveq2d 5691 . . . . 5  |-  ( n  =  z  ->  ( G `  ( A  +o  n ) )  =  ( G `  ( A  +o  z ) ) )
9 fveq2 5687 . . . . . 6  |-  ( n  =  z  ->  ( G `  n )  =  ( G `  z ) )
109oveq2d 6056 . . . . 5  |-  ( n  =  z  ->  (
( G `  A
)  +  ( G `
 n ) )  =  ( ( G `
 A )  +  ( G `  z
) ) )
118, 10eqeq12d 2418 . . . 4  |-  ( n  =  z  ->  (
( G `  ( A  +o  n ) )  =  ( ( G `
 A )  +  ( G `  n
) )  <->  ( G `  ( A  +o  z
) )  =  ( ( G `  A
)  +  ( G `
 z ) ) ) )
1211imbi2d 308 . . 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 6048 . . . . . 6  |-  ( n  =  suc  z  -> 
( A  +o  n
)  =  ( A  +o  suc  z ) )
1413fveq2d 5691 . . . . 5  |-  ( n  =  suc  z  -> 
( G `  ( A  +o  n ) )  =  ( G `  ( A  +o  suc  z
) ) )
15 fveq2 5687 . . . . . 6  |-  ( n  =  suc  z  -> 
( G `  n
)  =  ( G `
 suc  z )
)
1615oveq2d 6056 . . . . 5  |-  ( n  =  suc  z  -> 
( ( G `  A )  +  ( G `  n ) )  =  ( ( G `  A )  +  ( G `  suc  z ) ) )
1714, 16eqeq12d 2418 . . . 4  |-  ( n  =  suc  z  -> 
( ( G `  ( A  +o  n
) )  =  ( ( G `  A
)  +  ( G `
 n ) )  <-> 
( G `  ( A  +o  suc  z ) )  =  ( ( G `  A )  +  ( G `  suc  z ) ) ) )
1817imbi2d 308 . . 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 6048 . . . . . 6  |-  ( n  =  B  ->  ( A  +o  n )  =  ( A  +o  B
) )
2019fveq2d 5691 . . . . 5  |-  ( n  =  B  ->  ( G `  ( A  +o  n ) )  =  ( G `  ( A  +o  B ) ) )
21 fveq2 5687 . . . . . 6  |-  ( n  =  B  ->  ( G `  n )  =  ( G `  B ) )
2221oveq2d 6056 . . . . 5  |-  ( n  =  B  ->  (
( G `  A
)  +  ( G `
 n ) )  =  ( ( G `
 A )  +  ( G `  B
) ) )
2320, 22eqeq12d 2418 . . . 4  |-  ( n  =  B  ->  (
( G `  ( A  +o  n ) )  =  ( ( G `
 A )  +  ( G `  n
) )  <->  ( G `  ( A  +o  B
) )  =  ( ( G `  A
)  +  ( G `
 B ) ) ) )
2423imbi2d 308 . . 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 hashgadd.1 . . . . . . . . 9  |-  G  =  ( rec ( ( x  e.  _V  |->  ( x  +  1 ) ) ,  0 )  |`  om )
2625hashgf1o 11265 . . . . . . . 8  |-  G : om
-1-1-onto-> NN0
27 f1of 5633 . . . . . . . 8  |-  ( G : om -1-1-onto-> NN0  ->  G : om
--> NN0 )
2826, 27ax-mp 8 . . . . . . 7  |-  G : om
--> NN0
2928ffvelrni 5828 . . . . . 6  |-  ( A  e.  om  ->  ( G `  A )  e.  NN0 )
3029nn0cnd 10232 . . . . 5  |-  ( A  e.  om  ->  ( G `  A )  e.  CC )
3130addid1d 9222 . . . 4  |-  ( A  e.  om  ->  (
( G `  A
)  +  0 )  =  ( G `  A ) )
32 0z 10249 . . . . . . 7  |-  0  e.  ZZ
3332, 25om2uz0i 11242 . . . . . 6  |-  ( G `
 (/) )  =  0
3433oveq2i 6051 . . . . 5  |-  ( ( G `  A )  +  ( G `  (/) ) )  =  ( ( G `  A
)  +  0 )
3534a1i 11 . . . 4  |-  ( A  e.  om  ->  (
( G `  A
)  +  ( G `
 (/) ) )  =  ( ( G `  A )  +  0 ) )
36 nna0 6806 . . . . 5  |-  ( A  e.  om  ->  ( A  +o  (/) )  =  A )
3736fveq2d 5691 . . . 4  |-  ( A  e.  om  ->  ( G `  ( A  +o  (/) ) )  =  ( G `  A
) )
3831, 35, 373eqtr4rd 2447 . . 3  |-  ( A  e.  om  ->  ( G `  ( A  +o  (/) ) )  =  ( ( G `  A )  +  ( G `  (/) ) ) )
39 nnasuc 6808 . . . . . . . . . 10  |-  ( ( A  e.  om  /\  z  e.  om )  ->  ( A  +o  suc  z )  =  suc  ( A  +o  z
) )
4039fveq2d 5691 . . . . . . . . 9  |-  ( ( A  e.  om  /\  z  e.  om )  ->  ( G `  ( A  +o  suc  z ) )  =  ( G `
 suc  ( A  +o  z ) ) )
41 nnacl 6813 . . . . . . . . . 10  |-  ( ( A  e.  om  /\  z  e.  om )  ->  ( A  +o  z
)  e.  om )
4232, 25om2uzsuci 11243 . . . . . . . . . 10  |-  ( ( A  +o  z )  e.  om  ->  ( G `  suc  ( A  +o  z ) )  =  ( ( G `
 ( A  +o  z ) )  +  1 ) )
4341, 42syl 16 . . . . . . . . 9  |-  ( ( A  e.  om  /\  z  e.  om )  ->  ( G `  suc  ( A  +o  z
) )  =  ( ( G `  ( A  +o  z ) )  +  1 ) )
4440, 43eqtrd 2436 . . . . . . . 8  |-  ( ( A  e.  om  /\  z  e.  om )  ->  ( G `  ( A  +o  suc  z ) )  =  ( ( G `  ( A  +o  z ) )  +  1 ) )
45443adant3 977 . . . . . . 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 ) )
4628ffvelrni 5828 . . . . . . . . . . 11  |-  ( z  e.  om  ->  ( G `  z )  e.  NN0 )
4746nn0cnd 10232 . . . . . . . . . 10  |-  ( z  e.  om  ->  ( G `  z )  e.  CC )
48 ax-1cn 9004 . . . . . . . . . . 11  |-  1  e.  CC
49 addass 9033 . . . . . . . . . . 11  |-  ( ( ( G `  A
)  e.  CC  /\  ( G `  z )  e.  CC  /\  1  e.  CC )  ->  (
( ( G `  A )  +  ( G `  z ) )  +  1 )  =  ( ( G `
 A )  +  ( ( G `  z )  +  1 ) ) )
5048, 49mp3an3 1268 . . . . . . . . . 10  |-  ( ( ( G `  A
)  e.  CC  /\  ( G `  z )  e.  CC )  -> 
( ( ( G `
 A )  +  ( G `  z
) )  +  1 )  =  ( ( G `  A )  +  ( ( G `
 z )  +  1 ) ) )
5130, 47, 50syl2an 464 . . . . . . . . 9  |-  ( ( A  e.  om  /\  z  e.  om )  ->  ( ( ( G `
 A )  +  ( G `  z
) )  +  1 )  =  ( ( G `  A )  +  ( ( G `
 z )  +  1 ) ) )
52513adant3 977 . . . . . . . 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 ) ) )
53 oveq1 6047 . . . . . . . . 9  |-  ( ( G `  ( A  +o  z ) )  =  ( ( G `
 A )  +  ( G `  z
) )  ->  (
( G `  ( A  +o  z ) )  +  1 )  =  ( ( ( G `
 A )  +  ( G `  z
) )  +  1 ) )
54533ad2ant3 980 . . . . . . . 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 ) )
5532, 25om2uzsuci 11243 . . . . . . . . . 10  |-  ( z  e.  om  ->  ( G `  suc  z )  =  ( ( G `
 z )  +  1 ) )
5655oveq2d 6056 . . . . . . . . 9  |-  ( z  e.  om  ->  (
( G `  A
)  +  ( G `
 suc  z )
)  =  ( ( G `  A )  +  ( ( G `
 z )  +  1 ) ) )
57563ad2ant2 979 . . . . . . . 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 ) ) )
5852, 54, 573eqtr4d 2446 . . . . . . 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
) ) )
5945, 58eqtrd 2436 . . . . . 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 1155 . . . . 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 425 . . . 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 24 . . 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, 38, 62finds 4830 . 2  |-  ( B  e.  om  ->  ( A  e.  om  ->  ( G `  ( A  +o  B ) )  =  ( ( G `
 A )  +  ( G `  B
) ) ) )
6463impcom 420 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721   _Vcvv 2916   (/)c0 3588    e. cmpt 4226   suc csuc 4543   omcom 4804    |` cres 4839   -->wf 5409   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040   reccrdg 6626    +o coa 6680   CCcc 8944   0cc0 8946   1c1 8947    + caddc 8949   NN0cn0 10177
This theorem is referenced by:  hashdom  11608  hashun  11611
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-recs 6592  df-rdg 6627  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-n0 10178  df-z 10239  df-uz 10445
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