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Theorem rdgeqoa 31843
Description: If a recursive function with an initial value  A at step  N is equal to itself with an initial value  B at step  M, then every finite number of successor steps will also be equal. (Contributed by ML, 21-Oct-2020.)
Assertion
Ref Expression
rdgeqoa  |-  ( ( N  e.  On  /\  M  e.  On  /\  X  e.  om )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  X ) )  =  ( rec ( F ,  B ) `  ( M  +o  X
) ) ) )

Proof of Theorem rdgeqoa
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp3 1032 . 2  |-  ( ( N  e.  On  /\  M  e.  On  /\  X  e.  om )  ->  X  e.  om )
2 eleq1 2537 . . . . 5  |-  ( x  =  X  ->  (
x  e.  om  <->  X  e.  om ) )
323anbi3d 1371 . . . 4  |-  ( x  =  X  ->  (
( N  e.  On  /\  M  e.  On  /\  x  e.  om )  <->  ( N  e.  On  /\  M  e.  On  /\  X  e.  om ) ) )
4 oveq2 6316 . . . . . . 7  |-  ( x  =  X  ->  ( N  +o  x )  =  ( N  +o  X
) )
54fveq2d 5883 . . . . . 6  |-  ( x  =  X  ->  ( rec ( F ,  A
) `  ( N  +o  x ) )  =  ( rec ( F ,  A ) `  ( N  +o  X
) ) )
6 oveq2 6316 . . . . . . 7  |-  ( x  =  X  ->  ( M  +o  x )  =  ( M  +o  X
) )
76fveq2d 5883 . . . . . 6  |-  ( x  =  X  ->  ( rec ( F ,  B
) `  ( M  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  X
) ) )
85, 7eqeq12d 2486 . . . . 5  |-  ( x  =  X  ->  (
( rec ( F ,  A ) `  ( N  +o  x
) )  =  ( rec ( F ,  B ) `  ( M  +o  x ) )  <-> 
( rec ( F ,  A ) `  ( N  +o  X
) )  =  ( rec ( F ,  B ) `  ( M  +o  X ) ) ) )
98imbi2d 323 . . . 4  |-  ( x  =  X  ->  (
( ( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) )  <->  ( ( rec ( F ,  A
) `  N )  =  ( rec ( F ,  B ) `  M )  ->  ( rec ( F ,  A
) `  ( N  +o  X ) )  =  ( rec ( F ,  B ) `  ( M  +o  X
) ) ) ) )
103, 9imbi12d 327 . . 3  |-  ( x  =  X  ->  (
( ( N  e.  On  /\  M  e.  On  /\  x  e. 
om )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) ) )  <-> 
( ( N  e.  On  /\  M  e.  On  /\  X  e. 
om )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  X ) )  =  ( rec ( F ,  B ) `  ( M  +o  X
) ) ) ) ) )
11 peano1 6731 . . . . 5  |-  (/)  e.  om
12 oa0 7236 . . . . . . . . . . . 12  |-  ( N  e.  On  ->  ( N  +o  (/) )  =  N )
1312fveq2d 5883 . . . . . . . . . . 11  |-  ( N  e.  On  ->  ( rec ( F ,  A
) `  ( N  +o  (/) ) )  =  ( rec ( F ,  A ) `  N ) )
1413eqcomd 2477 . . . . . . . . . 10  |-  ( N  e.  On  ->  ( rec ( F ,  A
) `  N )  =  ( rec ( F ,  A ) `  ( N  +o  (/) ) ) )
15 oa0 7236 . . . . . . . . . . . 12  |-  ( M  e.  On  ->  ( M  +o  (/) )  =  M )
1615fveq2d 5883 . . . . . . . . . . 11  |-  ( M  e.  On  ->  ( rec ( F ,  B
) `  ( M  +o  (/) ) )  =  ( rec ( F ,  B ) `  M ) )
1716eqcomd 2477 . . . . . . . . . 10  |-  ( M  e.  On  ->  ( rec ( F ,  B
) `  M )  =  ( rec ( F ,  B ) `  ( M  +o  (/) ) ) )
1814, 17eqeqan12d 2487 . . . . . . . . 9  |-  ( ( N  e.  On  /\  M  e.  On )  ->  ( ( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  <->  ( rec ( F ,  A ) `  ( N  +o  (/) ) )  =  ( rec ( F ,  B ) `  ( M  +o  (/) ) ) ) )
1918biimpd 212 . . . . . . . 8  |-  ( ( N  e.  On  /\  M  e.  On )  ->  ( ( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  (/) ) )  =  ( rec ( F ,  B ) `  ( M  +o  (/) ) ) ) )
20 eleq1 2537 . . . . . . . . . . 11  |-  ( x  =  (/)  ->  ( x  e.  om  <->  (/)  e.  om ) )
21203anbi3d 1371 . . . . . . . . . 10  |-  ( x  =  (/)  ->  ( ( N  e.  On  /\  M  e.  On  /\  x  e.  om )  <->  ( N  e.  On  /\  M  e.  On  /\  (/)  e.  om ) ) )
2211biantru 513 . . . . . . . . . . . 12  |-  ( M  e.  On  <->  ( M  e.  On  /\  (/)  e.  om ) )
2322anbi2i 708 . . . . . . . . . . 11  |-  ( ( N  e.  On  /\  M  e.  On )  <->  ( N  e.  On  /\  ( M  e.  On  /\  (/)  e.  om ) ) )
24 3anass 1011 . . . . . . . . . . 11  |-  ( ( N  e.  On  /\  M  e.  On  /\  (/)  e.  om ) 
<->  ( N  e.  On  /\  ( M  e.  On  /\  (/)  e.  om ) ) )
2523, 24bitr4i 260 . . . . . . . . . 10  |-  ( ( N  e.  On  /\  M  e.  On )  <->  ( N  e.  On  /\  M  e.  On  /\  (/)  e.  om ) )
2621, 25syl6bbr 271 . . . . . . . . 9  |-  ( x  =  (/)  ->  ( ( N  e.  On  /\  M  e.  On  /\  x  e.  om )  <->  ( N  e.  On  /\  M  e.  On ) ) )
27 oveq2 6316 . . . . . . . . . . . 12  |-  ( x  =  (/)  ->  ( N  +o  x )  =  ( N  +o  (/) ) )
2827fveq2d 5883 . . . . . . . . . . 11  |-  ( x  =  (/)  ->  ( rec ( F ,  A
) `  ( N  +o  x ) )  =  ( rec ( F ,  A ) `  ( N  +o  (/) ) ) )
29 oveq2 6316 . . . . . . . . . . . 12  |-  ( x  =  (/)  ->  ( M  +o  x )  =  ( M  +o  (/) ) )
3029fveq2d 5883 . . . . . . . . . . 11  |-  ( x  =  (/)  ->  ( rec ( F ,  B
) `  ( M  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  (/) ) ) )
3128, 30eqeq12d 2486 . . . . . . . . . 10  |-  ( x  =  (/)  ->  ( ( rec ( F ,  A ) `  ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) )  <->  ( rec ( F ,  A ) `
 ( N  +o  (/) ) )  =  ( rec ( F ,  B ) `  ( M  +o  (/) ) ) ) )
3231imbi2d 323 . . . . . . . . 9  |-  ( x  =  (/)  ->  ( ( ( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) )  <->  ( ( rec ( F ,  A
) `  N )  =  ( rec ( F ,  B ) `  M )  ->  ( rec ( F ,  A
) `  ( N  +o  (/) ) )  =  ( rec ( F ,  B ) `  ( M  +o  (/) ) ) ) ) )
3326, 32imbi12d 327 . . . . . . . 8  |-  ( x  =  (/)  ->  ( ( ( N  e.  On  /\  M  e.  On  /\  x  e.  om )  ->  ( ( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) ) )  <-> 
( ( N  e.  On  /\  M  e.  On )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  (/) ) )  =  ( rec ( F ,  B ) `  ( M  +o  (/) ) ) ) ) ) )
3419, 33mpbiri 241 . . . . . . 7  |-  ( x  =  (/)  ->  ( ( N  e.  On  /\  M  e.  On  /\  x  e.  om )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) ) ) )
3534ax-gen 1677 . . . . . 6  |-  A. x
( x  =  (/)  ->  ( ( N  e.  On  /\  M  e.  On  /\  x  e. 
om )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) ) ) )
36 sbc6g 3281 . . . . . 6  |-  ( (/)  e.  om  ->  ( [. (/)  /  x ]. ( ( N  e.  On  /\  M  e.  On  /\  x  e.  om )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) ) )  <->  A. x ( x  =  (/)  ->  ( ( N  e.  On  /\  M  e.  On  /\  x  e. 
om )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) ) ) ) ) )
3735, 36mpbiri 241 . . . . 5  |-  ( (/)  e.  om  ->  [. (/)  /  x ]. ( ( N  e.  On  /\  M  e.  On  /\  x  e. 
om )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) ) ) )
3811, 37ax-mp 5 . . . 4  |-  [. (/)  /  x ]. ( ( N  e.  On  /\  M  e.  On  /\  x  e. 
om )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) ) )
39 peano2b 6727 . . . . 5  |-  ( x  e.  om  <->  suc  x  e. 
om )
40393anbi3i 1223 . . . . . . . 8  |-  ( ( N  e.  On  /\  M  e.  On  /\  x  e.  om )  <->  ( N  e.  On  /\  M  e.  On  /\  suc  x  e.  om ) )
4140imbi1i 332 . . . . . . 7  |-  ( ( ( N  e.  On  /\  M  e.  On  /\  x  e.  om )  ->  ( ( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) ) )  <-> 
( ( N  e.  On  /\  M  e.  On  /\  suc  x  e.  om )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) ) ) )
42 nnon 6717 . . . . . . . . . . . . 13  |-  ( x  e.  om  ->  x  e.  On )
43 oacl 7255 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  On  /\  x  e.  On )  ->  ( N  +o  x
)  e.  On )
44 oacl 7255 . . . . . . . . . . . . . . . . 17  |-  ( ( M  e.  On  /\  x  e.  On )  ->  ( M  +o  x
)  e.  On )
4543, 44anim12i 576 . . . . . . . . . . . . . . . 16  |-  ( ( ( N  e.  On  /\  x  e.  On )  /\  ( M  e.  On  /\  x  e.  On ) )  -> 
( ( N  +o  x )  e.  On  /\  ( M  +o  x
)  e.  On ) )
46453impdir 1348 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  On  /\  M  e.  On  /\  x  e.  On )  ->  (
( N  +o  x
)  e.  On  /\  ( M  +o  x
)  e.  On ) )
47 rdgsuc 7160 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  +o  x )  e.  On  ->  ( rec ( F ,  A
) `  suc  ( N  +o  x ) )  =  ( F `  ( rec ( F ,  A ) `  ( N  +o  x ) ) ) )
48 fveq2 5879 . . . . . . . . . . . . . . . . . 18  |-  ( ( rec ( F ,  A ) `  ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) )  ->  ( F `  ( rec ( F ,  A ) `
 ( N  +o  x ) ) )  =  ( F `  ( rec ( F ,  B ) `  ( M  +o  x ) ) ) )
4947, 48sylan9eqr 2527 . . . . . . . . . . . . . . . . 17  |-  ( ( ( rec ( F ,  A ) `  ( N  +o  x
) )  =  ( rec ( F ,  B ) `  ( M  +o  x ) )  /\  ( N  +o  x )  e.  On )  ->  ( rec ( F ,  A ) `  suc  ( N  +o  x ) )  =  ( F `  ( rec ( F ,  B
) `  ( M  +o  x ) ) ) )
5049adantrr 731 . . . . . . . . . . . . . . . 16  |-  ( ( ( rec ( F ,  A ) `  ( N  +o  x
) )  =  ( rec ( F ,  B ) `  ( M  +o  x ) )  /\  ( ( N  +o  x )  e.  On  /\  ( M  +o  x )  e.  On ) )  -> 
( rec ( F ,  A ) `  suc  ( N  +o  x
) )  =  ( F `  ( rec ( F ,  B
) `  ( M  +o  x ) ) ) )
51 rdgsuc 7160 . . . . . . . . . . . . . . . . 17  |-  ( ( M  +o  x )  e.  On  ->  ( rec ( F ,  B
) `  suc  ( M  +o  x ) )  =  ( F `  ( rec ( F ,  B ) `  ( M  +o  x ) ) ) )
5251ad2antll 743 . . . . . . . . . . . . . . . 16  |-  ( ( ( rec ( F ,  A ) `  ( N  +o  x
) )  =  ( rec ( F ,  B ) `  ( M  +o  x ) )  /\  ( ( N  +o  x )  e.  On  /\  ( M  +o  x )  e.  On ) )  -> 
( rec ( F ,  B ) `  suc  ( M  +o  x
) )  =  ( F `  ( rec ( F ,  B
) `  ( M  +o  x ) ) ) )
5350, 52eqtr4d 2508 . . . . . . . . . . . . . . 15  |-  ( ( ( rec ( F ,  A ) `  ( N  +o  x
) )  =  ( rec ( F ,  B ) `  ( M  +o  x ) )  /\  ( ( N  +o  x )  e.  On  /\  ( M  +o  x )  e.  On ) )  -> 
( rec ( F ,  A ) `  suc  ( N  +o  x
) )  =  ( rec ( F ,  B ) `  suc  ( M  +o  x
) ) )
5446, 53sylan2 482 . . . . . . . . . . . . . 14  |-  ( ( ( rec ( F ,  A ) `  ( N  +o  x
) )  =  ( rec ( F ,  B ) `  ( M  +o  x ) )  /\  ( N  e.  On  /\  M  e.  On  /\  x  e.  On ) )  -> 
( rec ( F ,  A ) `  suc  ( N  +o  x
) )  =  ( rec ( F ,  B ) `  suc  ( M  +o  x
) ) )
5554ancoms 460 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  On  /\  M  e.  On  /\  x  e.  On )  /\  ( rec ( F ,  A ) `  ( N  +o  x
) )  =  ( rec ( F ,  B ) `  ( M  +o  x ) ) )  ->  ( rec ( F ,  A ) `
 suc  ( N  +o  x ) )  =  ( rec ( F ,  B ) `  suc  ( M  +o  x
) ) )
5642, 55syl3anl3 1342 . . . . . . . . . . . 12  |-  ( ( ( N  e.  On  /\  M  e.  On  /\  x  e.  om )  /\  ( rec ( F ,  A ) `  ( N  +o  x
) )  =  ( rec ( F ,  B ) `  ( M  +o  x ) ) )  ->  ( rec ( F ,  A ) `
 suc  ( N  +o  x ) )  =  ( rec ( F ,  B ) `  suc  ( M  +o  x
) ) )
57 onasuc 7248 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  On  /\  x  e.  om )  ->  ( N  +o  suc  x )  =  suc  ( N  +o  x
) )
5857fveq2d 5883 . . . . . . . . . . . . . 14  |-  ( ( N  e.  On  /\  x  e.  om )  ->  ( rec ( F ,  A ) `  ( N  +o  suc  x
) )  =  ( rec ( F ,  A ) `  suc  ( N  +o  x
) ) )
59583adant2 1049 . . . . . . . . . . . . 13  |-  ( ( N  e.  On  /\  M  e.  On  /\  x  e.  om )  ->  ( rec ( F ,  A
) `  ( N  +o  suc  x ) )  =  ( rec ( F ,  A ) `  suc  ( N  +o  x ) ) )
6059adantr 472 . . . . . . . . . . . 12  |-  ( ( ( N  e.  On  /\  M  e.  On  /\  x  e.  om )  /\  ( rec ( F ,  A ) `  ( N  +o  x
) )  =  ( rec ( F ,  B ) `  ( M  +o  x ) ) )  ->  ( rec ( F ,  A ) `
 ( N  +o  suc  x ) )  =  ( rec ( F ,  A ) `  suc  ( N  +o  x
) ) )
61 onasuc 7248 . . . . . . . . . . . . . . 15  |-  ( ( M  e.  On  /\  x  e.  om )  ->  ( M  +o  suc  x )  =  suc  ( M  +o  x
) )
6261fveq2d 5883 . . . . . . . . . . . . . 14  |-  ( ( M  e.  On  /\  x  e.  om )  ->  ( rec ( F ,  B ) `  ( M  +o  suc  x
) )  =  ( rec ( F ,  B ) `  suc  ( M  +o  x
) ) )
63623adant1 1048 . . . . . . . . . . . . 13  |-  ( ( N  e.  On  /\  M  e.  On  /\  x  e.  om )  ->  ( rec ( F ,  B
) `  ( M  +o  suc  x ) )  =  ( rec ( F ,  B ) `  suc  ( M  +o  x ) ) )
6463adantr 472 . . . . . . . . . . . 12  |-  ( ( ( N  e.  On  /\  M  e.  On  /\  x  e.  om )  /\  ( rec ( F ,  A ) `  ( N  +o  x
) )  =  ( rec ( F ,  B ) `  ( M  +o  x ) ) )  ->  ( rec ( F ,  B ) `
 ( M  +o  suc  x ) )  =  ( rec ( F ,  B ) `  suc  ( M  +o  x
) ) )
6556, 60, 643eqtr4d 2515 . . . . . . . . . . 11  |-  ( ( ( N  e.  On  /\  M  e.  On  /\  x  e.  om )  /\  ( rec ( F ,  A ) `  ( N  +o  x
) )  =  ( rec ( F ,  B ) `  ( M  +o  x ) ) )  ->  ( rec ( F ,  A ) `
 ( N  +o  suc  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  suc  x
) ) )
6665ex 441 . . . . . . . . . 10  |-  ( ( N  e.  On  /\  M  e.  On  /\  x  e.  om )  ->  (
( rec ( F ,  A ) `  ( N  +o  x
) )  =  ( rec ( F ,  B ) `  ( M  +o  x ) )  ->  ( rec ( F ,  A ) `  ( N  +o  suc  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  suc  x
) ) ) )
6766imim2d 53 . . . . . . . . 9  |-  ( ( N  e.  On  /\  M  e.  On  /\  x  e.  om )  ->  (
( ( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) )  -> 
( ( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  suc  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  suc  x
) ) ) ) )
6840, 67sylbir 218 . . . . . . . 8  |-  ( ( N  e.  On  /\  M  e.  On  /\  suc  x  e.  om )  ->  ( ( ( rec ( F ,  A
) `  N )  =  ( rec ( F ,  B ) `  M )  ->  ( rec ( F ,  A
) `  ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) )  -> 
( ( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  suc  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  suc  x
) ) ) ) )
6968a2i 14 . . . . . . 7  |-  ( ( ( N  e.  On  /\  M  e.  On  /\  suc  x  e.  om )  ->  ( ( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) ) )  ->  ( ( N  e.  On  /\  M  e.  On  /\  suc  x  e.  om )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  suc  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  suc  x
) ) ) ) )
7041, 69sylbi 200 . . . . . 6  |-  ( ( ( N  e.  On  /\  M  e.  On  /\  x  e.  om )  ->  ( ( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) ) )  ->  ( ( N  e.  On  /\  M  e.  On  /\  suc  x  e.  om )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  suc  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  suc  x
) ) ) ) )
71 sbcimg 3297 . . . . . . 7  |-  ( suc  x  e.  om  ->  (
[. suc  x  /  x ]. ( ( N  e.  On  /\  M  e.  On  /\  x  e. 
om )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) ) )  <-> 
( [. suc  x  /  x ]. ( N  e.  On  /\  M  e.  On  /\  x  e. 
om )  ->  [. suc  x  /  x ]. (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) ) ) ) )
72 sbc3an 3313 . . . . . . . . 9  |-  ( [. suc  x  /  x ]. ( N  e.  On  /\  M  e.  On  /\  x  e.  om )  <->  (
[. suc  x  /  x ]. N  e.  On  /\ 
[. suc  x  /  x ]. M  e.  On  /\ 
[. suc  x  /  x ]. x  e.  om ) )
73 sbcg 3321 . . . . . . . . . 10  |-  ( suc  x  e.  om  ->  (
[. suc  x  /  x ]. N  e.  On  <->  N  e.  On ) )
74 sbcg 3321 . . . . . . . . . 10  |-  ( suc  x  e.  om  ->  (
[. suc  x  /  x ]. M  e.  On  <->  M  e.  On ) )
75 sbcel1v 3314 . . . . . . . . . . 11  |-  ( [. suc  x  /  x ]. x  e.  om  <->  suc  x  e. 
om )
7675a1i 11 . . . . . . . . . 10  |-  ( suc  x  e.  om  ->  (
[. suc  x  /  x ]. x  e.  om  <->  suc  x  e.  om )
)
7773, 74, 763anbi123d 1365 . . . . . . . . 9  |-  ( suc  x  e.  om  ->  ( ( [. suc  x  /  x ]. N  e.  On  /\  [. suc  x  /  x ]. M  e.  On  /\  [. suc  x  /  x ]. x  e.  om )  <->  ( N  e.  On  /\  M  e.  On  /\  suc  x  e.  om ) ) )
7872, 77syl5bb 265 . . . . . . . 8  |-  ( suc  x  e.  om  ->  (
[. suc  x  /  x ]. ( N  e.  On  /\  M  e.  On  /\  x  e. 
om )  <->  ( N  e.  On  /\  M  e.  On  /\  suc  x  e.  om ) ) )
79 sbcimg 3297 . . . . . . . . 9  |-  ( suc  x  e.  om  ->  (
[. suc  x  /  x ]. ( ( rec ( F ,  A
) `  N )  =  ( rec ( F ,  B ) `  M )  ->  ( rec ( F ,  A
) `  ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) )  <->  ( [. suc  x  /  x ]. ( rec ( F ,  A ) `  N
)  =  ( rec ( F ,  B
) `  M )  ->  [. suc  x  /  x ]. ( rec ( F ,  A ) `  ( N  +o  x
) )  =  ( rec ( F ,  B ) `  ( M  +o  x ) ) ) ) )
80 sbcg 3321 . . . . . . . . . 10  |-  ( suc  x  e.  om  ->  (
[. suc  x  /  x ]. ( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  <->  ( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
) ) )
81 sbceqg 3777 . . . . . . . . . . 11  |-  ( suc  x  e.  om  ->  (
[. suc  x  /  x ]. ( rec ( F ,  A ) `  ( N  +o  x
) )  =  ( rec ( F ,  B ) `  ( M  +o  x ) )  <->  [_ suc  x  /  x ]_ ( rec ( F ,  A ) `  ( N  +o  x
) )  =  [_ suc  x  /  x ]_ ( rec ( F ,  B ) `  ( M  +o  x ) ) ) )
82 csbfv12 5914 . . . . . . . . . . . . 13  |-  [_ suc  x  /  x ]_ ( rec ( F ,  A
) `  ( N  +o  x ) )  =  ( [_ suc  x  /  x ]_ rec ( F ,  A ) `  [_ suc  x  /  x ]_ ( N  +o  x ) )
83 csbconstg 3362 . . . . . . . . . . . . . 14  |-  ( suc  x  e.  om  ->  [_
suc  x  /  x ]_ rec ( F ,  A )  =  rec ( F ,  A ) )
84 csbov123 6342 . . . . . . . . . . . . . . 15  |-  [_ suc  x  /  x ]_ ( N  +o  x )  =  ( [_ suc  x  /  x ]_ N [_ suc  x  /  x ]_  +o  [_ suc  x  /  x ]_ x )
85 csbconstg 3362 . . . . . . . . . . . . . . . 16  |-  ( suc  x  e.  om  ->  [_
suc  x  /  x ]_  +o  =  +o  )
86 csbconstg 3362 . . . . . . . . . . . . . . . 16  |-  ( suc  x  e.  om  ->  [_
suc  x  /  x ]_ N  =  N
)
87 csbvarg 3796 . . . . . . . . . . . . . . . 16  |-  ( suc  x  e.  om  ->  [_
suc  x  /  x ]_ x  =  suc  x )
8885, 86, 87oveq123d 6329 . . . . . . . . . . . . . . 15  |-  ( suc  x  e.  om  ->  (
[_ suc  x  /  x ]_ N [_ suc  x  /  x ]_  +o  [_
suc  x  /  x ]_ x )  =  ( N  +o  suc  x
) )
8984, 88syl5eq 2517 . . . . . . . . . . . . . 14  |-  ( suc  x  e.  om  ->  [_
suc  x  /  x ]_ ( N  +o  x
)  =  ( N  +o  suc  x ) )
9083, 89fveq12d 5885 . . . . . . . . . . . . 13  |-  ( suc  x  e.  om  ->  (
[_ suc  x  /  x ]_ rec ( F ,  A ) `  [_
suc  x  /  x ]_ ( N  +o  x
) )  =  ( rec ( F ,  A ) `  ( N  +o  suc  x ) ) )
9182, 90syl5eq 2517 . . . . . . . . . . . 12  |-  ( suc  x  e.  om  ->  [_
suc  x  /  x ]_ ( rec ( F ,  A ) `  ( N  +o  x
) )  =  ( rec ( F ,  A ) `  ( N  +o  suc  x ) ) )
92 csbfv12 5914 . . . . . . . . . . . . 13  |-  [_ suc  x  /  x ]_ ( rec ( F ,  B
) `  ( M  +o  x ) )  =  ( [_ suc  x  /  x ]_ rec ( F ,  B ) `  [_ suc  x  /  x ]_ ( M  +o  x ) )
93 csbconstg 3362 . . . . . . . . . . . . . 14  |-  ( suc  x  e.  om  ->  [_
suc  x  /  x ]_ rec ( F ,  B )  =  rec ( F ,  B ) )
94 csbov123 6342 . . . . . . . . . . . . . . 15  |-  [_ suc  x  /  x ]_ ( M  +o  x )  =  ( [_ suc  x  /  x ]_ M [_ suc  x  /  x ]_  +o  [_ suc  x  /  x ]_ x )
95 csbconstg 3362 . . . . . . . . . . . . . . . 16  |-  ( suc  x  e.  om  ->  [_
suc  x  /  x ]_ M  =  M
)
9685, 95, 87oveq123d 6329 . . . . . . . . . . . . . . 15  |-  ( suc  x  e.  om  ->  (
[_ suc  x  /  x ]_ M [_ suc  x  /  x ]_  +o  [_
suc  x  /  x ]_ x )  =  ( M  +o  suc  x
) )
9794, 96syl5eq 2517 . . . . . . . . . . . . . 14  |-  ( suc  x  e.  om  ->  [_
suc  x  /  x ]_ ( M  +o  x
)  =  ( M  +o  suc  x ) )
9893, 97fveq12d 5885 . . . . . . . . . . . . 13  |-  ( suc  x  e.  om  ->  (
[_ suc  x  /  x ]_ rec ( F ,  B ) `  [_
suc  x  /  x ]_ ( M  +o  x
) )  =  ( rec ( F ,  B ) `  ( M  +o  suc  x ) ) )
9992, 98syl5eq 2517 . . . . . . . . . . . 12  |-  ( suc  x  e.  om  ->  [_
suc  x  /  x ]_ ( rec ( F ,  B ) `  ( M  +o  x
) )  =  ( rec ( F ,  B ) `  ( M  +o  suc  x ) ) )
10091, 99eqeq12d 2486 . . . . . . . . . . 11  |-  ( suc  x  e.  om  ->  (
[_ suc  x  /  x ]_ ( rec ( F ,  A ) `  ( N  +o  x
) )  =  [_ suc  x  /  x ]_ ( rec ( F ,  B ) `  ( M  +o  x ) )  <-> 
( rec ( F ,  A ) `  ( N  +o  suc  x
) )  =  ( rec ( F ,  B ) `  ( M  +o  suc  x ) ) ) )
10181, 100bitrd 261 . . . . . . . . . 10  |-  ( suc  x  e.  om  ->  (
[. suc  x  /  x ]. ( rec ( F ,  A ) `  ( N  +o  x
) )  =  ( rec ( F ,  B ) `  ( M  +o  x ) )  <-> 
( rec ( F ,  A ) `  ( N  +o  suc  x
) )  =  ( rec ( F ,  B ) `  ( M  +o  suc  x ) ) ) )
10280, 101imbi12d 327 . . . . . . . . 9  |-  ( suc  x  e.  om  ->  ( ( [. suc  x  /  x ]. ( rec ( F ,  A
) `  N )  =  ( rec ( F ,  B ) `  M )  ->  [. suc  x  /  x ]. ( rec ( F ,  A
) `  ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) )  <->  ( ( rec ( F ,  A
) `  N )  =  ( rec ( F ,  B ) `  M )  ->  ( rec ( F ,  A
) `  ( N  +o  suc  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  suc  x ) ) ) ) )
10379, 102bitrd 261 . . . . . . . 8  |-  ( suc  x  e.  om  ->  (
[. suc  x  /  x ]. ( ( rec ( F ,  A
) `  N )  =  ( rec ( F ,  B ) `  M )  ->  ( rec ( F ,  A
) `  ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) )  <->  ( ( rec ( F ,  A
) `  N )  =  ( rec ( F ,  B ) `  M )  ->  ( rec ( F ,  A
) `  ( N  +o  suc  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  suc  x ) ) ) ) )
10478, 103imbi12d 327 . . . . . . 7  |-  ( suc  x  e.  om  ->  ( ( [. suc  x  /  x ]. ( N  e.  On  /\  M  e.  On  /\  x  e. 
om )  ->  [. suc  x  /  x ]. (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) ) )  <-> 
( ( N  e.  On  /\  M  e.  On  /\  suc  x  e.  om )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  suc  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  suc  x
) ) ) ) ) )
10571, 104bitrd 261 . . . . . 6  |-  ( suc  x  e.  om  ->  (
[. suc  x  /  x ]. ( ( N  e.  On  /\  M  e.  On  /\  x  e. 
om )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) ) )  <-> 
( ( N  e.  On  /\  M  e.  On  /\  suc  x  e.  om )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  suc  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  suc  x
) ) ) ) ) )
10670, 105syl5ibr 229 . . . . 5  |-  ( suc  x  e.  om  ->  ( ( ( N  e.  On  /\  M  e.  On  /\  x  e. 
om )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) ) )  ->  [. suc  x  /  x ]. ( ( N  e.  On  /\  M  e.  On  /\  x  e. 
om )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) ) ) ) )
10739, 106sylbi 200 . . . 4  |-  ( x  e.  om  ->  (
( ( N  e.  On  /\  M  e.  On  /\  x  e. 
om )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) ) )  ->  [. suc  x  /  x ]. ( ( N  e.  On  /\  M  e.  On  /\  x  e. 
om )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) ) ) ) )
10838, 107findes 6742 . . 3  |-  ( x  e.  om  ->  (
( N  e.  On  /\  M  e.  On  /\  x  e.  om )  ->  ( ( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  x ) )  =  ( rec ( F ,  B ) `  ( M  +o  x
) ) ) ) )
10910, 108vtoclga 3099 . 2  |-  ( X  e.  om  ->  (
( N  e.  On  /\  M  e.  On  /\  X  e.  om )  ->  ( ( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  X ) )  =  ( rec ( F ,  B ) `  ( M  +o  X
) ) ) ) )
1101, 109mpcom 36 1  |-  ( ( N  e.  On  /\  M  e.  On  /\  X  e.  om )  ->  (
( rec ( F ,  A ) `  N )  =  ( rec ( F ,  B ) `  M
)  ->  ( rec ( F ,  A ) `
 ( N  +o  X ) )  =  ( rec ( F ,  B ) `  ( M  +o  X
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007   A.wal 1450    = wceq 1452    e. wcel 1904   [.wsbc 3255   [_csb 3349   (/)c0 3722   Oncon0 5430   suc csuc 5432   ` cfv 5589  (class class class)co 6308   omcom 6711   reccrdg 7145    +o coa 7197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-oadd 7204
This theorem is referenced by:  finxpreclem4  31856
  Copyright terms: Public domain W3C validator