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Theorem hashgadd 12155
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 6114 . . . . . 6  |-  ( n  =  (/)  ->  ( A  +o  n )  =  ( A  +o  (/) ) )
21fveq2d 5710 . . . . 5  |-  ( n  =  (/)  ->  ( G `
 ( A  +o  n ) )  =  ( G `  ( A  +o  (/) ) ) )
3 fveq2 5706 . . . . . 6  |-  ( n  =  (/)  ->  ( G `
 n )  =  ( G `  (/) ) )
43oveq2d 6122 . . . . 5  |-  ( n  =  (/)  ->  ( ( G `  A )  +  ( G `  n ) )  =  ( ( G `  A )  +  ( G `  (/) ) ) )
52, 4eqeq12d 2457 . . . 4  |-  ( n  =  (/)  ->  ( ( G `  ( A  +o  n ) )  =  ( ( G `
 A )  +  ( G `  n
) )  <->  ( G `  ( A  +o  (/) ) )  =  ( ( G `
 A )  +  ( G `  (/) ) ) ) )
65imbi2d 316 . . 3  |-  ( n  =  (/)  ->  ( ( A  e.  om  ->  ( G `  ( A  +o  n ) )  =  ( ( G `
 A )  +  ( G `  n
) ) )  <->  ( A  e.  om  ->  ( G `  ( A  +o  (/) ) )  =  ( ( G `
 A )  +  ( G `  (/) ) ) ) ) )
7 oveq2 6114 . . . . . 6  |-  ( n  =  z  ->  ( A  +o  n )  =  ( A  +o  z
) )
87fveq2d 5710 . . . . 5  |-  ( n  =  z  ->  ( G `  ( A  +o  n ) )  =  ( G `  ( A  +o  z ) ) )
9 fveq2 5706 . . . . . 6  |-  ( n  =  z  ->  ( G `  n )  =  ( G `  z ) )
109oveq2d 6122 . . . . 5  |-  ( n  =  z  ->  (
( G `  A
)  +  ( G `
 n ) )  =  ( ( G `
 A )  +  ( G `  z
) ) )
118, 10eqeq12d 2457 . . . 4  |-  ( n  =  z  ->  (
( G `  ( A  +o  n ) )  =  ( ( G `
 A )  +  ( G `  n
) )  <->  ( G `  ( A  +o  z
) )  =  ( ( G `  A
)  +  ( G `
 z ) ) ) )
1211imbi2d 316 . . 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 6114 . . . . . 6  |-  ( n  =  suc  z  -> 
( A  +o  n
)  =  ( A  +o  suc  z ) )
1413fveq2d 5710 . . . . 5  |-  ( n  =  suc  z  -> 
( G `  ( A  +o  n ) )  =  ( G `  ( A  +o  suc  z
) ) )
15 fveq2 5706 . . . . . 6  |-  ( n  =  suc  z  -> 
( G `  n
)  =  ( G `
 suc  z )
)
1615oveq2d 6122 . . . . 5  |-  ( n  =  suc  z  -> 
( ( G `  A )  +  ( G `  n ) )  =  ( ( G `  A )  +  ( G `  suc  z ) ) )
1714, 16eqeq12d 2457 . . . 4  |-  ( n  =  suc  z  -> 
( ( G `  ( A  +o  n
) )  =  ( ( G `  A
)  +  ( G `
 n ) )  <-> 
( G `  ( A  +o  suc  z ) )  =  ( ( G `  A )  +  ( G `  suc  z ) ) ) )
1817imbi2d 316 . . 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 6114 . . . . . 6  |-  ( n  =  B  ->  ( A  +o  n )  =  ( A  +o  B
) )
2019fveq2d 5710 . . . . 5  |-  ( n  =  B  ->  ( G `  ( A  +o  n ) )  =  ( G `  ( A  +o  B ) ) )
21 fveq2 5706 . . . . . 6  |-  ( n  =  B  ->  ( G `  n )  =  ( G `  B ) )
2221oveq2d 6122 . . . . 5  |-  ( n  =  B  ->  (
( G `  A
)  +  ( G `
 n ) )  =  ( ( G `
 A )  +  ( G `  B
) ) )
2320, 22eqeq12d 2457 . . . 4  |-  ( n  =  B  ->  (
( G `  ( A  +o  n ) )  =  ( ( G `
 A )  +  ( G `  n
) )  <->  ( G `  ( A  +o  B
) )  =  ( ( G `  A
)  +  ( G `
 B ) ) ) )
2423imbi2d 316 . . 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 11808 . . . . . . . 8  |-  G : om
-1-1-onto-> NN0
27 f1of 5656 . . . . . . . 8  |-  ( G : om -1-1-onto-> NN0  ->  G : om
--> NN0 )
2826, 27ax-mp 5 . . . . . . 7  |-  G : om
--> NN0
2928ffvelrni 5857 . . . . . 6  |-  ( A  e.  om  ->  ( G `  A )  e.  NN0 )
3029nn0cnd 10653 . . . . 5  |-  ( A  e.  om  ->  ( G `  A )  e.  CC )
3130addid1d 9584 . . . 4  |-  ( A  e.  om  ->  (
( G `  A
)  +  0 )  =  ( G `  A ) )
32 0z 10672 . . . . . . 7  |-  0  e.  ZZ
3332, 25om2uz0i 11785 . . . . . 6  |-  ( G `
 (/) )  =  0
3433oveq2i 6117 . . . . 5  |-  ( ( G `  A )  +  ( G `  (/) ) )  =  ( ( G `  A
)  +  0 )
3534a1i 11 . . . 4  |-  ( A  e.  om  ->  (
( G `  A
)  +  ( G `
 (/) ) )  =  ( ( G `  A )  +  0 ) )
36 nna0 7058 . . . . 5  |-  ( A  e.  om  ->  ( A  +o  (/) )  =  A )
3736fveq2d 5710 . . . 4  |-  ( A  e.  om  ->  ( G `  ( A  +o  (/) ) )  =  ( G `  A
) )
3831, 35, 373eqtr4rd 2486 . . 3  |-  ( A  e.  om  ->  ( G `  ( A  +o  (/) ) )  =  ( ( G `  A )  +  ( G `  (/) ) ) )
39 nnasuc 7060 . . . . . . . . . 10  |-  ( ( A  e.  om  /\  z  e.  om )  ->  ( A  +o  suc  z )  =  suc  ( A  +o  z
) )
4039fveq2d 5710 . . . . . . . . 9  |-  ( ( A  e.  om  /\  z  e.  om )  ->  ( G `  ( A  +o  suc  z ) )  =  ( G `
 suc  ( A  +o  z ) ) )
41 nnacl 7065 . . . . . . . . . 10  |-  ( ( A  e.  om  /\  z  e.  om )  ->  ( A  +o  z
)  e.  om )
4232, 25om2uzsuci 11786 . . . . . . . . . 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 2475 . . . . . . . 8  |-  ( ( A  e.  om  /\  z  e.  om )  ->  ( G `  ( A  +o  suc  z ) )  =  ( ( G `  ( A  +o  z ) )  +  1 ) )
45443adant3 1008 . . . . . . 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 5857 . . . . . . . . . . 11  |-  ( z  e.  om  ->  ( G `  z )  e.  NN0 )
4746nn0cnd 10653 . . . . . . . . . 10  |-  ( z  e.  om  ->  ( G `  z )  e.  CC )
48 ax-1cn 9355 . . . . . . . . . . 11  |-  1  e.  CC
49 addass 9384 . . . . . . . . . . 11  |-  ( ( ( G `  A
)  e.  CC  /\  ( G `  z )  e.  CC  /\  1  e.  CC )  ->  (
( ( G `  A )  +  ( G `  z ) )  +  1 )  =  ( ( G `
 A )  +  ( ( G `  z )  +  1 ) ) )
5048, 49mp3an3 1303 . . . . . . . . . 10  |-  ( ( ( G `  A
)  e.  CC  /\  ( G `  z )  e.  CC )  -> 
( ( ( G `
 A )  +  ( G `  z
) )  +  1 )  =  ( ( G `  A )  +  ( ( G `
 z )  +  1 ) ) )
5130, 47, 50syl2an 477 . . . . . . . . 9  |-  ( ( A  e.  om  /\  z  e.  om )  ->  ( ( ( G `
 A )  +  ( G `  z
) )  +  1 )  =  ( ( G `  A )  +  ( ( G `
 z )  +  1 ) ) )
52513adant3 1008 . . . . . . . 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 6113 . . . . . . . . 9  |-  ( ( G `  ( A  +o  z ) )  =  ( ( G `
 A )  +  ( G `  z
) )  ->  (
( G `  ( A  +o  z ) )  +  1 )  =  ( ( ( G `
 A )  +  ( G `  z
) )  +  1 ) )
54533ad2ant3 1011 . . . . . . . 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 11786 . . . . . . . . . 10  |-  ( z  e.  om  ->  ( G `  suc  z )  =  ( ( G `
 z )  +  1 ) )
5655oveq2d 6122 . . . . . . . . 9  |-  ( z  e.  om  ->  (
( G `  A
)  +  ( G `
 suc  z )
)  =  ( ( G `  A )  +  ( ( G `
 z )  +  1 ) ) )
57563ad2ant2 1010 . . . . . . . 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 2485 . . . . . . 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 2475 . . . . . 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 1189 . . . . 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 435 . . . 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, 38, 62finds 6517 . 2  |-  ( B  e.  om  ->  ( A  e.  om  ->  ( G `  ( A  +o  B ) )  =  ( ( G `
 A )  +  ( G `  B
) ) ) )
6463impcom 430 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( G `  ( A  +o  B ) )  =  ( ( G `
 A )  +  ( G `  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   _Vcvv 2987   (/)c0 3652    e. cmpt 4365   suc csuc 4736    |` cres 4857   -->wf 5429   -1-1-onto->wf1o 5432   ` cfv 5433  (class class class)co 6106   omcom 6491   reccrdg 6880    +o coa 6932   CCcc 9295   0cc0 9297   1c1 9298    + caddc 9300   NN0cn0 10594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-recs 6847  df-rdg 6881  df-oadd 6939  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-nn 10338  df-n0 10595  df-z 10662  df-uz 10877
This theorem is referenced by:  hashdom  12157  hashun  12160
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