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Theorem oesuclem 6540
Description: Lemma for oesuc 6542. (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 5881 . . . 4  |-  ( A  =  (/)  ->  ( A  ^o  suc  B )  =  ( (/)  ^o  suc  B ) )
2 oesuclem.1 . . . . . . . 8  |-  Lim  X
3 limord 4467 . . . . . . . 8  |-  ( Lim 
X  ->  Ord  X )
42, 3ax-mp 8 . . . . . . 7  |-  Ord  X
5 ordelord 4430 . . . . . . 7  |-  ( ( Ord  X  /\  B  e.  X )  ->  Ord  B )
64, 5mpan 651 . . . . . 6  |-  ( B  e.  X  ->  Ord  B )
7 0elsuc 4642 . . . . . 6  |-  ( Ord 
B  ->  (/)  e.  suc  B )
86, 7syl 15 . . . . 5  |-  ( B  e.  X  ->  (/)  e.  suc  B )
9 limsuc 4656 . . . . . . 7  |-  ( Lim 
X  ->  ( B  e.  X  <->  suc  B  e.  X
) )
102, 9ax-mp 8 . . . . . 6  |-  ( B  e.  X  <->  suc  B  e.  X )
11 ordelon 4432 . . . . . . . 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 6536 . . . . . . 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 2350 . . 3  |-  ( ( B  e.  X  /\  A  =  (/) )  -> 
( A  ^o  suc  B )  =  (/) )
18 oveq1 5881 . . . . 5  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
19 id 19 . . . . 5  |-  ( A  =  (/)  ->  A  =  (/) )
2018, 19oveq12d 5892 . . . 4  |-  ( A  =  (/)  ->  ( ( A  ^o  B )  .o  A )  =  ( ( (/)  ^o  B
)  .o  (/) ) )
21 ordelon 4432 . . . . . . 7  |-  ( ( Ord  X  /\  B  e.  X )  ->  B  e.  On )
224, 21mpan 651 . . . . . 6  |-  ( B  e.  X  ->  B  e.  On )
23 oveq2 5882 . . . . . . . . 9  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  =  (
(/)  ^o  (/) ) )
24 oe0m0 6535 . . . . . . . . . 10  |-  ( (/)  ^o  (/) )  =  1o
25 1on 6502 . . . . . . . . . 10  |-  1o  e.  On
2624, 25eqeltri 2366 . . . . . . . . 9  |-  ( (/)  ^o  (/) )  e.  On
2723, 26syl6eqel 2384 . . . . . . . 8  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  e.  On )
2827adantl 452 . . . . . . 7  |-  ( ( B  e.  X  /\  B  =  (/) )  -> 
( (/)  ^o  B )  e.  On )
29 oe0m1 6536 . . . . . . . . . . 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 4461 . . . . . . . . 9  |-  (/)  e.  On
3331, 32syl6eqel 2384 . . . . . . . 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 6528 . . . . . 6  |-  ( ( B  e.  On  /\  B  e.  X )  ->  ( (/)  ^o  B )  e.  On )
3622, 35mpancom 650 . . . . 5  |-  ( B  e.  X  ->  ( (/) 
^o  B )  e.  On )
37 om0 6532 . . . . 5  |-  ( (
(/)  ^o  B )  e.  On  ->  ( ( (/) 
^o  B )  .o  (/) )  =  (/) )
3836, 37syl 15 . . . 4  |-  ( B  e.  X  ->  (
( (/)  ^o  B )  .o  (/) )  =  (/) )
3920, 38sylan9eqr 2350 . . 3  |-  ( ( B  e.  X  /\  A  =  (/) )  -> 
( ( A  ^o  B )  .o  A
)  =  (/) )
4017, 39eqtr4d 2331 . 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 6530 . . . 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 5899 . . . . 5  |-  ( A  ^o  B )  e. 
_V
47 oveq1 5881 . . . . . 6  |-  ( x  =  ( A  ^o  B )  ->  (
x  .o  A )  =  ( ( A  ^o  B )  .o  A ) )
48 eqid 2296 . . . . . 6  |-  ( x  e.  _V  |->  ( x  .o  A ) )  =  ( x  e. 
_V  |->  ( x  .o  A ) )
49 ovex 5899 . . . . . 6  |-  ( ( A  ^o  B )  .o  A )  e. 
_V
5047, 48, 49fvmpt 5618 . . . . 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 6530 . . . . . 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 5545 . . . 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 2342 . . 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 2338 . 2  |-  ( ( ( A  e.  On  /\  B  e.  X )  /\  (/)  e.  A )  ->  ( A  ^o  suc  B )  =  ( ( A  ^o  B
)  .o  A ) )
5740, 56oe0lem 6528 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 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468    e. cmpt 4093   Ord word 4407   Oncon0 4408   Lim wlim 4409   suc csuc 4410   ` cfv 5271  (class class class)co 5874   reccrdg 6438   1oc1o 6488    .o comu 6493    ^o coe 6494
This theorem is referenced by:  oesuc  6542  onesuc  6545
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-1o 6495  df-omul 6500  df-oexp 6501
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