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Theorem oesuclem 6957
Description: Lemma for oesuc 6959. (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 6093 . . . 4  |-  ( A  =  (/)  ->  ( A  ^o  suc  B )  =  ( (/)  ^o  suc  B ) )
2 oesuclem.1 . . . . . . . 8  |-  Lim  X
3 limord 4773 . . . . . . . 8  |-  ( Lim 
X  ->  Ord  X )
42, 3ax-mp 5 . . . . . . 7  |-  Ord  X
5 ordelord 4736 . . . . . . 7  |-  ( ( Ord  X  /\  B  e.  X )  ->  Ord  B )
64, 5mpan 670 . . . . . 6  |-  ( B  e.  X  ->  Ord  B )
7 0elsuc 6441 . . . . . 6  |-  ( Ord 
B  ->  (/)  e.  suc  B )
86, 7syl 16 . . . . 5  |-  ( B  e.  X  ->  (/)  e.  suc  B )
9 limsuc 6455 . . . . . . 7  |-  ( Lim 
X  ->  ( B  e.  X  <->  suc  B  e.  X
) )
102, 9ax-mp 5 . . . . . 6  |-  ( B  e.  X  <->  suc  B  e.  X )
11 ordelon 4738 . . . . . . . 8  |-  ( ( Ord  X  /\  suc  B  e.  X )  ->  suc  B  e.  On )
124, 11mpan 670 . . . . . . 7  |-  ( suc 
B  e.  X  ->  suc  B  e.  On )
13 oe0m1 6953 . . . . . . 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 2492 . . 3  |-  ( ( B  e.  X  /\  A  =  (/) )  -> 
( A  ^o  suc  B )  =  (/) )
18 oveq1 6093 . . . . 5  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
19 id 22 . . . . 5  |-  ( A  =  (/)  ->  A  =  (/) )
2018, 19oveq12d 6104 . . . 4  |-  ( A  =  (/)  ->  ( ( A  ^o  B )  .o  A )  =  ( ( (/)  ^o  B
)  .o  (/) ) )
21 ordelon 4738 . . . . . . 7  |-  ( ( Ord  X  /\  B  e.  X )  ->  B  e.  On )
224, 21mpan 670 . . . . . 6  |-  ( B  e.  X  ->  B  e.  On )
23 oveq2 6094 . . . . . . . . 9  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  =  (
(/)  ^o  (/) ) )
24 oe0m0 6952 . . . . . . . . . 10  |-  ( (/)  ^o  (/) )  =  1o
25 1on 6919 . . . . . . . . . 10  |-  1o  e.  On
2624, 25eqeltri 2508 . . . . . . . . 9  |-  ( (/)  ^o  (/) )  e.  On
2723, 26syl6eqel 2526 . . . . . . . 8  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  e.  On )
2827adantl 466 . . . . . . 7  |-  ( ( B  e.  X  /\  B  =  (/) )  -> 
( (/)  ^o  B )  e.  On )
29 oe0m1 6953 . . . . . . . . . . 11  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  ( (/)  ^o  B
)  =  (/) ) )
3022, 29syl 16 . . . . . . . . . 10  |-  ( B  e.  X  ->  ( (/) 
e.  B  <->  ( (/)  ^o  B
)  =  (/) ) )
3130biimpa 484 . . . . . . . . 9  |-  ( ( B  e.  X  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  =  (/) )
32 0elon 4767 . . . . . . . . 9  |-  (/)  e.  On
3331, 32syl6eqel 2526 . . . . . . . 8  |-  ( ( B  e.  X  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  e.  On )
3433adantll 713 . . . . . . 7  |-  ( ( ( B  e.  On  /\  B  e.  X )  /\  (/)  e.  B )  ->  ( (/)  ^o  B
)  e.  On )
3528, 34oe0lem 6945 . . . . . 6  |-  ( ( B  e.  On  /\  B  e.  X )  ->  ( (/)  ^o  B )  e.  On )
3622, 35mpancom 669 . . . . 5  |-  ( B  e.  X  ->  ( (/) 
^o  B )  e.  On )
37 om0 6949 . . . . 5  |-  ( (
(/)  ^o  B )  e.  On  ->  ( ( (/) 
^o  B )  .o  (/) )  =  (/) )
3836, 37syl 16 . . . 4  |-  ( B  e.  X  ->  (
( (/)  ^o  B )  .o  (/) )  =  (/) )
3920, 38sylan9eqr 2492 . . 3  |-  ( ( B  e.  X  /\  A  =  (/) )  -> 
( ( A  ^o  B )  .o  A
)  =  (/) )
4017, 39eqtr4d 2473 . 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 726 . . 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 6947 . . . 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 651 . . 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 6111 . . . . 5  |-  ( A  ^o  B )  e. 
_V
47 oveq1 6093 . . . . . 6  |-  ( x  =  ( A  ^o  B )  ->  (
x  .o  A )  =  ( ( A  ^o  B )  .o  A ) )
48 eqid 2438 . . . . . 6  |-  ( x  e.  _V  |->  ( x  .o  A ) )  =  ( x  e. 
_V  |->  ( x  .o  A ) )
49 ovex 6111 . . . . . 6  |-  ( ( A  ^o  B )  .o  A )  e. 
_V
5047, 48, 49fvmpt 5769 . . . . 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 6947 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B ) )
5322, 52sylanl2 651 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  X )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B ) )
5453fveq2d 5690 . . . 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 2484 . . 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 2480 . 2  |-  ( ( ( A  e.  On  /\  B  e.  X )  /\  (/)  e.  A )  ->  ( A  ^o  suc  B )  =  ( ( A  ^o  B
)  .o  A ) )
5740, 56oe0lem 6945 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 369    = wceq 1369    e. wcel 1756   _Vcvv 2967   (/)c0 3632    e. cmpt 4345   Ord word 4713   Oncon0 4714   Lim wlim 4715   suc csuc 4716   ` cfv 5413  (class class class)co 6086   reccrdg 6857   1oc1o 6905    .o comu 6910    ^o coe 6911
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-recs 6824  df-rdg 6858  df-1o 6912  df-omul 6917  df-oexp 6918
This theorem is referenced by:  oesuc  6959  onesuc  6962
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