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Theorem oesuclem 7167
Description: Lemma for oesuc 7169. (Contributed by NM, 31-Dec-2004.) (Revised by Mario Carneiro, 15-Nov-2014.)
Hypotheses
Ref Expression
oesuclem.1  |-  Lim  X
oesuclem.2  |-  ( B  e.  X  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  suc  B )  =  ( ( x  e.  _V  |->  ( x  .o  A ) ) `  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
) ) )
Assertion
Ref Expression
oesuclem  |-  ( ( A  e.  On  /\  B  e.  X )  ->  ( A  ^o  suc  B )  =  ( ( A  ^o  B )  .o  A ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    X( x)

Proof of Theorem oesuclem
StepHypRef Expression
1 oveq1 6277 . . . 4  |-  ( A  =  (/)  ->  ( A  ^o  suc  B )  =  ( (/)  ^o  suc  B ) )
2 oesuclem.1 . . . . . . . 8  |-  Lim  X
3 limord 4926 . . . . . . . 8  |-  ( Lim 
X  ->  Ord  X )
42, 3ax-mp 5 . . . . . . 7  |-  Ord  X
5 ordelord 4889 . . . . . . 7  |-  ( ( Ord  X  /\  B  e.  X )  ->  Ord  B )
64, 5mpan 668 . . . . . 6  |-  ( B  e.  X  ->  Ord  B )
7 0elsuc 6643 . . . . . 6  |-  ( Ord 
B  ->  (/)  e.  suc  B )
86, 7syl 16 . . . . 5  |-  ( B  e.  X  ->  (/)  e.  suc  B )
9 limsuc 6657 . . . . . . 7  |-  ( Lim 
X  ->  ( B  e.  X  <->  suc  B  e.  X
) )
102, 9ax-mp 5 . . . . . 6  |-  ( B  e.  X  <->  suc  B  e.  X )
11 ordelon 4891 . . . . . . . 8  |-  ( ( Ord  X  /\  suc  B  e.  X )  ->  suc  B  e.  On )
124, 11mpan 668 . . . . . . 7  |-  ( suc 
B  e.  X  ->  suc  B  e.  On )
13 oe0m1 7163 . . . . . . 7  |-  ( suc 
B  e.  On  ->  (
(/)  e.  suc  B  <->  ( (/)  ^o  suc  B )  =  (/) ) )
1412, 13syl 16 . . . . . 6  |-  ( suc 
B  e.  X  -> 
( (/)  e.  suc  B  <->  (
(/)  ^o  suc  B )  =  (/) ) )
1510, 14sylbi 195 . . . . 5  |-  ( B  e.  X  ->  ( (/) 
e.  suc  B  <->  ( (/)  ^o  suc  B )  =  (/) ) )
168, 15mpbid 210 . . . 4  |-  ( B  e.  X  ->  ( (/) 
^o  suc  B )  =  (/) )
171, 16sylan9eqr 2517 . . 3  |-  ( ( B  e.  X  /\  A  =  (/) )  -> 
( A  ^o  suc  B )  =  (/) )
18 oveq1 6277 . . . . 5  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
19 id 22 . . . . 5  |-  ( A  =  (/)  ->  A  =  (/) )
2018, 19oveq12d 6288 . . . 4  |-  ( A  =  (/)  ->  ( ( A  ^o  B )  .o  A )  =  ( ( (/)  ^o  B
)  .o  (/) ) )
21 ordelon 4891 . . . . . . 7  |-  ( ( Ord  X  /\  B  e.  X )  ->  B  e.  On )
224, 21mpan 668 . . . . . 6  |-  ( B  e.  X  ->  B  e.  On )
23 oveq2 6278 . . . . . . . . 9  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  =  (
(/)  ^o  (/) ) )
24 oe0m0 7162 . . . . . . . . . 10  |-  ( (/)  ^o  (/) )  =  1o
25 1on 7129 . . . . . . . . . 10  |-  1o  e.  On
2624, 25eqeltri 2538 . . . . . . . . 9  |-  ( (/)  ^o  (/) )  e.  On
2723, 26syl6eqel 2550 . . . . . . . 8  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  e.  On )
2827adantl 464 . . . . . . 7  |-  ( ( B  e.  X  /\  B  =  (/) )  -> 
( (/)  ^o  B )  e.  On )
29 oe0m1 7163 . . . . . . . . . . 11  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  ( (/)  ^o  B
)  =  (/) ) )
3022, 29syl 16 . . . . . . . . . 10  |-  ( B  e.  X  ->  ( (/) 
e.  B  <->  ( (/)  ^o  B
)  =  (/) ) )
3130biimpa 482 . . . . . . . . 9  |-  ( ( B  e.  X  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  =  (/) )
32 0elon 4920 . . . . . . . . 9  |-  (/)  e.  On
3331, 32syl6eqel 2550 . . . . . . . 8  |-  ( ( B  e.  X  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  e.  On )
3433adantll 711 . . . . . . 7  |-  ( ( ( B  e.  On  /\  B  e.  X )  /\  (/)  e.  B )  ->  ( (/)  ^o  B
)  e.  On )
3528, 34oe0lem 7155 . . . . . 6  |-  ( ( B  e.  On  /\  B  e.  X )  ->  ( (/)  ^o  B )  e.  On )
3622, 35mpancom 667 . . . . 5  |-  ( B  e.  X  ->  ( (/) 
^o  B )  e.  On )
37 om0 7159 . . . . 5  |-  ( (
(/)  ^o  B )  e.  On  ->  ( ( (/) 
^o  B )  .o  (/) )  =  (/) )
3836, 37syl 16 . . . 4  |-  ( B  e.  X  ->  (
( (/)  ^o  B )  .o  (/) )  =  (/) )
3920, 38sylan9eqr 2517 . . 3  |-  ( ( B  e.  X  /\  A  =  (/) )  -> 
( ( A  ^o  B )  .o  A
)  =  (/) )
4017, 39eqtr4d 2498 . 2  |-  ( ( B  e.  X  /\  A  =  (/) )  -> 
( A  ^o  suc  B )  =  ( ( A  ^o  B )  .o  A ) )
41 oesuclem.2 . . . 4  |-  ( B  e.  X  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  suc  B )  =  ( ( x  e.  _V  |->  ( x  .o  A ) ) `  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
) ) )
4241ad2antlr 724 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  X )  /\  (/)  e.  A )  ->  ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  suc  B )  =  ( ( x  e.  _V  |->  ( x  .o  A ) ) `
 ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )
) )
4310, 12sylbi 195 . . . 4  |-  ( B  e.  X  ->  suc  B  e.  On )
44 oevn0 7157 . . . 4  |-  ( ( ( A  e.  On  /\ 
suc  B  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  suc  B )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  suc  B ) )
4543, 44sylanl2 649 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  X )  /\  (/)  e.  A )  ->  ( A  ^o  suc  B )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  suc  B ) )
46 ovex 6298 . . . . 5  |-  ( A  ^o  B )  e. 
_V
47 oveq1 6277 . . . . . 6  |-  ( x  =  ( A  ^o  B )  ->  (
x  .o  A )  =  ( ( A  ^o  B )  .o  A ) )
48 eqid 2454 . . . . . 6  |-  ( x  e.  _V  |->  ( x  .o  A ) )  =  ( x  e. 
_V  |->  ( x  .o  A ) )
49 ovex 6298 . . . . . 6  |-  ( ( A  ^o  B )  .o  A )  e. 
_V
5047, 48, 49fvmpt 5931 . . . . 5  |-  ( ( A  ^o  B )  e.  _V  ->  (
( x  e.  _V  |->  ( x  .o  A
) ) `  ( A  ^o  B ) )  =  ( ( A  ^o  B )  .o  A ) )
5146, 50ax-mp 5 . . . 4  |-  ( ( x  e.  _V  |->  ( x  .o  A ) ) `  ( A  ^o  B ) )  =  ( ( A  ^o  B )  .o  A )
52 oevn0 7157 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B ) )
5322, 52sylanl2 649 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  X )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B ) )
5453fveq2d 5852 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  X )  /\  (/)  e.  A )  ->  ( ( x  e.  _V  |->  ( x  .o  A ) ) `
 ( A  ^o  B ) )  =  ( ( x  e. 
_V  |->  ( x  .o  A ) ) `  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B ) ) )
5551, 54syl5eqr 2509 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  X )  /\  (/)  e.  A )  ->  ( ( A  ^o  B )  .o  A )  =  ( ( x  e.  _V  |->  ( x  .o  A
) ) `  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
) ) )
5642, 45, 553eqtr4d 2505 . 2  |-  ( ( ( A  e.  On  /\  B  e.  X )  /\  (/)  e.  A )  ->  ( A  ^o  suc  B )  =  ( ( A  ^o  B
)  .o  A ) )
5740, 56oe0lem 7155 1  |-  ( ( A  e.  On  /\  B  e.  X )  ->  ( A  ^o  suc  B )  =  ( ( A  ^o  B )  .o  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   (/)c0 3783    |-> cmpt 4497   Ord word 4866   Oncon0 4867   Lim wlim 4868   suc csuc 4869   ` cfv 5570  (class class class)co 6270   reccrdg 7067   1oc1o 7115    .o comu 7120    ^o coe 7121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-1o 7122  df-omul 7127  df-oexp 7128
This theorem is referenced by:  oesuc  7169  onesuc  7172
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