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Theorem oesuclem 6524
Description: Lemma for oesuc 6526. (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 5865 . . . 4  |-  ( A  =  (/)  ->  ( A  ^o  suc  B )  =  ( (/)  ^o  suc  B ) )
2 oesuclem.1 . . . . . . . 8  |-  Lim  X
3 limord 4451 . . . . . . . 8  |-  ( Lim 
X  ->  Ord  X )
42, 3ax-mp 8 . . . . . . 7  |-  Ord  X
5 ordelord 4414 . . . . . . 7  |-  ( ( Ord  X  /\  B  e.  X )  ->  Ord  B )
64, 5mpan 651 . . . . . 6  |-  ( B  e.  X  ->  Ord  B )
7 0elsuc 4626 . . . . . 6  |-  ( Ord 
B  ->  (/)  e.  suc  B )
86, 7syl 15 . . . . 5  |-  ( B  e.  X  ->  (/)  e.  suc  B )
9 limsuc 4640 . . . . . . 7  |-  ( Lim 
X  ->  ( B  e.  X  <->  suc  B  e.  X
) )
102, 9ax-mp 8 . . . . . 6  |-  ( B  e.  X  <->  suc  B  e.  X )
11 ordelon 4416 . . . . . . . 8  |-  ( ( Ord  X  /\  suc  B  e.  X )  ->  suc  B  e.  On )
124, 11mpan 651 . . . . . . 7  |-  ( suc 
B  e.  X  ->  suc  B  e.  On )
13 oe0m1 6520 . . . . . . 7  |-  ( suc 
B  e.  On  ->  (
(/)  e.  suc  B  <->  ( (/)  ^o  suc  B )  =  (/) ) )
1412, 13syl 15 . . . . . 6  |-  ( suc 
B  e.  X  -> 
( (/)  e.  suc  B  <->  (
(/)  ^o  suc  B )  =  (/) ) )
1510, 14sylbi 187 . . . . 5  |-  ( B  e.  X  ->  ( (/) 
e.  suc  B  <->  ( (/)  ^o  suc  B )  =  (/) ) )
168, 15mpbid 201 . . . 4  |-  ( B  e.  X  ->  ( (/) 
^o  suc  B )  =  (/) )
171, 16sylan9eqr 2337 . . 3  |-  ( ( B  e.  X  /\  A  =  (/) )  -> 
( A  ^o  suc  B )  =  (/) )
18 oveq1 5865 . . . . 5  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
19 id 19 . . . . 5  |-  ( A  =  (/)  ->  A  =  (/) )
2018, 19oveq12d 5876 . . . 4  |-  ( A  =  (/)  ->  ( ( A  ^o  B )  .o  A )  =  ( ( (/)  ^o  B
)  .o  (/) ) )
21 ordelon 4416 . . . . . . 7  |-  ( ( Ord  X  /\  B  e.  X )  ->  B  e.  On )
224, 21mpan 651 . . . . . 6  |-  ( B  e.  X  ->  B  e.  On )
23 oveq2 5866 . . . . . . . . 9  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  =  (
(/)  ^o  (/) ) )
24 oe0m0 6519 . . . . . . . . . 10  |-  ( (/)  ^o  (/) )  =  1o
25 1on 6486 . . . . . . . . . 10  |-  1o  e.  On
2624, 25eqeltri 2353 . . . . . . . . 9  |-  ( (/)  ^o  (/) )  e.  On
2723, 26syl6eqel 2371 . . . . . . . 8  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  e.  On )
2827adantl 452 . . . . . . 7  |-  ( ( B  e.  X  /\  B  =  (/) )  -> 
( (/)  ^o  B )  e.  On )
29 oe0m1 6520 . . . . . . . . . . 11  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  ( (/)  ^o  B
)  =  (/) ) )
3022, 29syl 15 . . . . . . . . . 10  |-  ( B  e.  X  ->  ( (/) 
e.  B  <->  ( (/)  ^o  B
)  =  (/) ) )
3130biimpa 470 . . . . . . . . 9  |-  ( ( B  e.  X  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  =  (/) )
32 0elon 4445 . . . . . . . . 9  |-  (/)  e.  On
3331, 32syl6eqel 2371 . . . . . . . 8  |-  ( ( B  e.  X  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  e.  On )
3433adantll 694 . . . . . . 7  |-  ( ( ( B  e.  On  /\  B  e.  X )  /\  (/)  e.  B )  ->  ( (/)  ^o  B
)  e.  On )
3528, 34oe0lem 6512 . . . . . 6  |-  ( ( B  e.  On  /\  B  e.  X )  ->  ( (/)  ^o  B )  e.  On )
3622, 35mpancom 650 . . . . 5  |-  ( B  e.  X  ->  ( (/) 
^o  B )  e.  On )
37 om0 6516 . . . . 5  |-  ( (
(/)  ^o  B )  e.  On  ->  ( ( (/) 
^o  B )  .o  (/) )  =  (/) )
3836, 37syl 15 . . . 4  |-  ( B  e.  X  ->  (
( (/)  ^o  B )  .o  (/) )  =  (/) )
3920, 38sylan9eqr 2337 . . 3  |-  ( ( B  e.  X  /\  A  =  (/) )  -> 
( ( A  ^o  B )  .o  A
)  =  (/) )
4017, 39eqtr4d 2318 . 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 707 . . 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 187 . . . 4  |-  ( B  e.  X  ->  suc  B  e.  On )
44 oevn0 6514 . . . 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 632 . . 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 5883 . . . . 5  |-  ( A  ^o  B )  e. 
_V
47 oveq1 5865 . . . . . 6  |-  ( x  =  ( A  ^o  B )  ->  (
x  .o  A )  =  ( ( A  ^o  B )  .o  A ) )
48 eqid 2283 . . . . . 6  |-  ( x  e.  _V  |->  ( x  .o  A ) )  =  ( x  e. 
_V  |->  ( x  .o  A ) )
49 ovex 5883 . . . . . 6  |-  ( ( A  ^o  B )  .o  A )  e. 
_V
5047, 48, 49fvmpt 5602 . . . . 5  |-  ( ( A  ^o  B )  e.  _V  ->  (
( x  e.  _V  |->  ( x  .o  A
) ) `  ( A  ^o  B ) )  =  ( ( A  ^o  B )  .o  A ) )
5146, 50ax-mp 8 . . . 4  |-  ( ( x  e.  _V  |->  ( x  .o  A ) ) `  ( A  ^o  B ) )  =  ( ( A  ^o  B )  .o  A )
52 oevn0 6514 . . . . . 6  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B ) )
5322, 52sylanl2 632 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  X )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B ) )
5453fveq2d 5529 . . . 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 2329 . . 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 2325 . 2  |-  ( ( ( A  e.  On  /\  B  e.  X )  /\  (/)  e.  A )  ->  ( A  ^o  suc  B )  =  ( ( A  ^o  B
)  .o  A ) )
5740, 56oe0lem 6512 1  |-  ( ( A  e.  On  /\  B  e.  X )  ->  ( A  ^o  suc  B )  =  ( ( A  ^o  B )  .o  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455    e. cmpt 4077   Ord word 4391   Oncon0 4392   Lim wlim 4393   suc csuc 4394   ` cfv 5255  (class class class)co 5858   reccrdg 6422   1oc1o 6472    .o comu 6477    ^o coe 6478
This theorem is referenced by:  oesuc  6526  onesuc  6529
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-1o 6479  df-omul 6484  df-oexp 6485
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