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Theorem wwlksnextinj 26794
Description: Lemma for wwlksnextbij 26797. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.)
Hypotheses
Ref Expression
wwlksnextbij0.v |- V = (Vtx ` G )
wwlksnextbij0.e |- E = (Edg ` G )
wwlksnextbij0.d |- D = { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) }
wwlksnextbij.r |- R = { n e. V | { ( lastS ` W ) , n } e. E }
wwlksnextbij.f |- F = ( t e. D |-> ( lastS ` t ) )
Assertion
Ref Expression
wwlksnextinj |- ( N e. NN0 -> F : D -1-1-> R )
Distinct variable groups:   w, G   w, N   w, W   t, D   n, E   w, E   t, N, w   t, R   n, V   w, V   n, W   t, n
Allowed substitution hints:   D( w, n)   R( w, n)   E( t)   F( w, t, n)   G( t, n)   N( n)   V( t)   W( t)
Proof of Theorem wwlksnextinj
Dummy variables d x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wwlksnextbij0.v . . 3 |- V = (Vtx ` G )
2 wwlksnextbij0.e . . 3 |- E = (Edg ` G )
3 wwlksnextbij0.d . . 3 |- D = { w e. Word V | ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) }
4 wwlksnextbij.r . . 3 |- R = { n e. V | { ( lastS ` W ) , n } e. E }
5 wwlksnextbij.f . . 3 |- F = ( t e. D |-> ( lastS ` t ) )
61, 2, 3, 4, 5wwlksnextfun 26793 . 2 |- ( N e. NN0 -> F : D --> R )
7 fveq2 6191 . . . . . . 7 |- ( t = d -> ( lastS ` t ) = ( lastS ` d ) )
8 fvex 6201 . . . . . . 7 |- ( lastS ` d ) e. _V
97, 5, 8fvmpt 6282 . . . . . 6 |- ( d e. D -> ( F ` d ) = ( lastS ` d ) )
10 fveq2 6191 . . . . . . 7 |- ( t = x -> ( lastS ` t ) = ( lastS ` x ) )
11 fvex 6201 . . . . . . 7 |- ( lastS ` x ) e. _V
1210, 5, 11fvmpt 6282 . . . . . 6 |- ( x e. D -> ( F ` x ) = ( lastS ` x ) )
139, 12eqeqan12d 2638 . . . . 5 |- ( ( d e. D /\ x e. D ) -> ( ( F ` d ) = ( F ` x ) <-> ( lastS ` d ) = ( lastS ` x ) ) )
1413adantl 482 . . . 4 |- ( ( N e. NN0 /\ ( d e. D /\ x e. D ) ) -> ( ( F ` d ) = ( F ` x ) <-> ( lastS ` d ) = ( lastS ` x ) ) )
15 fveq2 6191 . . . . . . . . 9 |- ( w = d -> ( # ` w ) = ( # ` d ) )
1615eqeq1d 2624 . . . . . . . 8 |- ( w = d -> ( ( # ` w ) = ( N + 2 ) <-> ( # ` d ) = ( N + 2 ) ) )
17 oveq1 6657 . . . . . . . . 9 |- ( w = d -> ( w substr <. 0 , ( N + 1 ) >. ) = ( d substr <. 0 , ( N + 1 ) >. ) )
1817eqeq1d 2624 . . . . . . . 8 |- ( w = d -> ( ( w substr <. 0 , ( N + 1 ) >. ) = W <-> ( d substr <. 0 , ( N + 1 ) >. ) = W ) )
19 fveq2 6191 . . . . . . . . . 10 |- ( w = d -> ( lastS ` w ) = ( lastS ` d ) )
2019preq2d 4275 . . . . . . . . 9 |- ( w = d -> { ( lastS ` W ) , ( lastS ` w ) } = { ( lastS ` W ) , ( lastS ` d ) } )
2120eleq1d 2686 . . . . . . . 8 |- ( w = d -> ( { ( lastS ` W ) , ( lastS ` w ) } e. E <-> { ( lastS ` W ) , ( lastS ` d ) } e. E ) )
2216, 18, 213anbi123d 1399 . . . . . . 7 |- ( w = d -> ( ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) <-> ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) )
2322, 3elrab2 3366 . . . . . 6 |- ( d e. D <-> ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) )
24 fveq2 6191 . . . . . . . . 9 |- ( w = x -> ( # ` w ) = ( # ` x ) )
2524eqeq1d 2624 . . . . . . . 8 |- ( w = x -> ( ( # ` w ) = ( N + 2 ) <-> ( # ` x ) = ( N + 2 ) ) )
26 oveq1 6657 . . . . . . . . 9 |- ( w = x -> ( w substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( N + 1 ) >. ) )
2726eqeq1d 2624 . . . . . . . 8 |- ( w = x -> ( ( w substr <. 0 , ( N + 1 ) >. ) = W <-> ( x substr <. 0 , ( N + 1 ) >. ) = W ) )
28 fveq2 6191 . . . . . . . . . 10 |- ( w = x -> ( lastS ` w ) = ( lastS ` x ) )
2928preq2d 4275 . . . . . . . . 9 |- ( w = x -> { ( lastS ` W ) , ( lastS ` w ) } = { ( lastS ` W ) , ( lastS ` x ) } )
3029eleq1d 2686 . . . . . . . 8 |- ( w = x -> ( { ( lastS ` W ) , ( lastS ` w ) } e. E <-> { ( lastS ` W ) , ( lastS ` x ) } e. E ) )
3125, 27, 303anbi123d 1399 . . . . . . 7 |- ( w = x -> ( ( ( # ` w ) = ( N + 2 ) /\ ( w substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` w ) } e. E ) <-> ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) )
3231, 3elrab2 3366 . . . . . 6 |- ( x e. D <-> ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) )
33 eqtr3 2643 . . . . . . . . . . . . . . . . 17 |- ( ( ( # ` d ) = ( N + 2 ) /\ ( # ` x ) = ( N + 2 ) ) -> ( # ` d ) = ( # ` x ) )
3433expcom 451 . . . . . . . . . . . . . . . 16 |- ( ( # ` x ) = ( N + 2 ) -> ( ( # ` d ) = ( N + 2 ) -> ( # ` d ) = ( # ` x ) ) )
35343ad2ant1 1082 . . . . . . . . . . . . . . 15 |- ( ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) -> ( ( # ` d ) = ( N + 2 ) -> ( # ` d ) = ( # ` x ) ) )
3635adantl 482 . . . . . . . . . . . . . 14 |- ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) -> ( ( # ` d ) = ( N + 2 ) -> ( # ` d ) = ( # ` x ) ) )
3736com12 32 . . . . . . . . . . . . 13 |- ( ( # ` d ) = ( N + 2 ) -> ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) -> ( # ` d ) = ( # ` x ) ) )
38373ad2ant1 1082 . . . . . . . . . . . 12 |- ( ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) -> ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) -> ( # ` d ) = ( # ` x ) ) )
3938adantl 482 . . . . . . . . . . 11 |- ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) -> ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) -> ( # ` d ) = ( # ` x ) ) )
4039imp 445 . . . . . . . . . 10 |- ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) ) -> ( # ` d ) = ( # ` x ) )
4140adantr 481 . . . . . . . . 9 |- ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) ) /\ N e. NN0 ) -> ( # ` d ) = ( # ` x ) )
4241adantr 481 . . . . . . . 8 |- ( ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) ) /\ N e. NN0 ) /\ ( lastS ` d ) = ( lastS ` x ) ) -> ( # ` d ) = ( # ` x ) )
43 simpr 477 . . . . . . . 8 |- ( ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) ) /\ N e. NN0 ) /\ ( lastS ` d ) = ( lastS ` x ) ) -> ( lastS ` d ) = ( lastS ` x ) )
44 eqtr3 2643 . . . . . . . . . . . . . . . . . . . 20 |- ( ( ( d substr <. 0 , ( N + 1 ) >. ) = W /\ ( x substr <. 0 , ( N + 1 ) >. ) = W ) -> ( d substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( N + 1 ) >. ) )
45 1e2m1 11136 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- 1 = ( 2 - 1 )
4645a1i 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( N e. NN0 -> 1 = ( 2 - 1 ) )
4746oveq2d 6666 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( N e. NN0 -> ( N + 1 ) = ( N + ( 2 - 1 ) ) )
48 nn0cn 11302 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( N e. NN0 -> N e. CC )
49 2cnd 11093 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( N e. NN0 -> 2 e. CC )
50 1cnd 10056 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ( N e. NN0 -> 1 e. CC )
5148, 49, 50addsubassd 10412 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( N e. NN0 -> ( ( N + 2 ) - 1 ) = ( N + ( 2 - 1 ) ) )
5247, 51eqtr4d 2659 . . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( N e. NN0 -> ( N + 1 ) = ( ( N + 2 ) - 1 ) )
5352adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( N e. NN0 /\ ( # ` d ) = ( N + 2 ) ) -> ( N + 1 ) = ( ( N + 2 ) - 1 ) )
54 oveq1 6657 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ( ( # ` d ) = ( N + 2 ) -> ( ( # ` d ) - 1 ) = ( ( N + 2 ) - 1 ) )
5554eqeq2d 2632 . . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( # ` d ) = ( N + 2 ) -> ( ( N + 1 ) = ( ( # ` d ) - 1 ) <-> ( N + 1 ) = ( ( N + 2 ) - 1 ) ) )
5655adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( N e. NN0 /\ ( # ` d ) = ( N + 2 ) ) -> ( ( N + 1 ) = ( ( # ` d ) - 1 ) <-> ( N + 1 ) = ( ( N + 2 ) - 1 ) ) )
5753, 56mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( N e. NN0 /\ ( # ` d ) = ( N + 2 ) ) -> ( N + 1 ) = ( ( # ` d ) - 1 ) )
58 opeq2 4403 . . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ( ( N + 1 ) = ( ( # ` d ) - 1 ) -> <. 0 , ( N + 1 ) >. = <. 0 , ( ( # ` d ) - 1 ) >. )
5958oveq2d 6666 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( N + 1 ) = ( ( # ` d ) - 1 ) -> ( d substr <. 0 , ( N + 1 ) >. ) = ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) )
6058oveq2d 6666 . . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( N + 1 ) = ( ( # ` d ) - 1 ) -> ( x substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) )
6159, 60eqeq12d 2637 . . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( N + 1 ) = ( ( # ` d ) - 1 ) -> ( ( d substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( N + 1 ) >. ) <-> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) )
6257, 61syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( N e. NN0 /\ ( # ` d ) = ( N + 2 ) ) -> ( ( d substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( N + 1 ) >. ) <-> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) )
6362biimpd 219 . . . . . . . . . . . . . . . . . . . . . 22 |- ( ( N e. NN0 /\ ( # ` d ) = ( N + 2 ) ) -> ( ( d substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( N + 1 ) >. ) -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) )
6463ex 450 . . . . . . . . . . . . . . . . . . . . 21 |- ( N e. NN0 -> ( ( # ` d ) = ( N + 2 ) -> ( ( d substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( N + 1 ) >. ) -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
6564com13 88 . . . . . . . . . . . . . . . . . . . 20 |- ( ( d substr <. 0 , ( N + 1 ) >. ) = ( x substr <. 0 , ( N + 1 ) >. ) -> ( ( # ` d ) = ( N + 2 ) -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
6644, 65syl 17 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( d substr <. 0 , ( N + 1 ) >. ) = W /\ ( x substr <. 0 , ( N + 1 ) >. ) = W ) -> ( ( # ` d ) = ( N + 2 ) -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
6766ex 450 . . . . . . . . . . . . . . . . . 18 |- ( ( d substr <. 0 , ( N + 1 ) >. ) = W -> ( ( x substr <. 0 , ( N + 1 ) >. ) = W -> ( ( # ` d ) = ( N + 2 ) -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) ) )
6867com23 86 . . . . . . . . . . . . . . . . 17 |- ( ( d substr <. 0 , ( N + 1 ) >. ) = W -> ( ( # ` d ) = ( N + 2 ) -> ( ( x substr <. 0 , ( N + 1 ) >. ) = W -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) ) )
6968impcom 446 . . . . . . . . . . . . . . . 16 |- ( ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W ) -> ( ( x substr <. 0 , ( N + 1 ) >. ) = W -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
7069com12 32 . . . . . . . . . . . . . . 15 |- ( ( x substr <. 0 , ( N + 1 ) >. ) = W -> ( ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W ) -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
71703ad2ant2 1083 . . . . . . . . . . . . . 14 |- ( ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) -> ( ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W ) -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
7271adantl 482 . . . . . . . . . . . . 13 |- ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) -> ( ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W ) -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
7372com12 32 . . . . . . . . . . . 12 |- ( ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W ) -> ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
74733adant3 1081 . . . . . . . . . . 11 |- ( ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) -> ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
7574adantl 482 . . . . . . . . . 10 |- ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) -> ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) -> ( N e. NN0 -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
7675imp31 448 . . . . . . . . 9 |- ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) ) /\ N e. NN0 ) -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) )
7776adantr 481 . . . . . . . 8 |- ( ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) ) /\ N e. NN0 ) /\ ( lastS ` d ) = ( lastS ` x ) ) -> ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) )
78 simpl 473 . . . . . . . . . . . . 13 |- ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) -> d e. Word V )
79 simpl 473 . . . . . . . . . . . . 13 |- ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) -> x e. Word V )
8078, 79anim12i 590 . . . . . . . . . . . 12 |- ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) ) -> ( d e. Word V /\ x e. Word V ) )
8180adantr 481 . . . . . . . . . . 11 |- ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) ) /\ N e. NN0 ) -> ( d e. Word V /\ x e. Word V ) )
82 nn0re 11301 . . . . . . . . . . . . . . . . . . . . 21 |- ( N e. NN0 -> N e. RR )
83 2re 11090 . . . . . . . . . . . . . . . . . . . . . 22 |- 2 e. RR
8483a1i 11 . . . . . . . . . . . . . . . . . . . . 21 |- ( N e. NN0 -> 2 e. RR )
85 nn0ge0 11318 . . . . . . . . . . . . . . . . . . . . 21 |- ( N e. NN0 -> 0 <_ N )
86 2pos 11112 . . . . . . . . . . . . . . . . . . . . . 22 |- 0 < 2
8786a1i 11 . . . . . . . . . . . . . . . . . . . . 21 |- ( N e. NN0 -> 0 < 2 )
8882, 84, 85, 87addgegt0d 10601 . . . . . . . . . . . . . . . . . . . 20 |- ( N e. NN0 -> 0 < ( N + 2 ) )
8988adantl 482 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( # ` d ) = ( N + 2 ) /\ N e. NN0 ) -> 0 < ( N + 2 ) )
90 breq2 4657 . . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` d ) = ( N + 2 ) -> ( 0 < ( # ` d ) <-> 0 < ( N + 2 ) ) )
9190adantr 481 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( # ` d ) = ( N + 2 ) /\ N e. NN0 ) -> ( 0 < ( # ` d ) <-> 0 < ( N + 2 ) ) )
9289, 91mpbird 247 . . . . . . . . . . . . . . . . . 18 |- ( ( ( # ` d ) = ( N + 2 ) /\ N e. NN0 ) -> 0 < ( # ` d ) )
93 hashgt0n0 13156 . . . . . . . . . . . . . . . . . 18 |- ( ( d e. Word V /\ 0 < ( # ` d ) ) -> d =/= (/) )
9492, 93sylan2 491 . . . . . . . . . . . . . . . . 17 |- ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ N e. NN0 ) ) -> d =/= (/) )
9594exp32 631 . . . . . . . . . . . . . . . 16 |- ( d e. Word V -> ( ( # ` d ) = ( N + 2 ) -> ( N e. NN0 -> d =/= (/) ) ) )
9695com12 32 . . . . . . . . . . . . . . 15 |- ( ( # ` d ) = ( N + 2 ) -> ( d e. Word V -> ( N e. NN0 -> d =/= (/) ) ) )
97963ad2ant1 1082 . . . . . . . . . . . . . 14 |- ( ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) -> ( d e. Word V -> ( N e. NN0 -> d =/= (/) ) ) )
9897impcom 446 . . . . . . . . . . . . 13 |- ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) -> ( N e. NN0 -> d =/= (/) ) )
9998adantr 481 . . . . . . . . . . . 12 |- ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) ) -> ( N e. NN0 -> d =/= (/) ) )
10099imp 445 . . . . . . . . . . 11 |- ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) ) /\ N e. NN0 ) -> d =/= (/) )
10188adantl 482 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( # ` x ) = ( N + 2 ) /\ N e. NN0 ) -> 0 < ( N + 2 ) )
102 breq2 4657 . . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` x ) = ( N + 2 ) -> ( 0 < ( # ` x ) <-> 0 < ( N + 2 ) ) )
103102adantr 481 . . . . . . . . . . . . . . . . . . 19 |- ( ( ( # ` x ) = ( N + 2 ) /\ N e. NN0 ) -> ( 0 < ( # ` x ) <-> 0 < ( N + 2 ) ) )
104101, 103mpbird 247 . . . . . . . . . . . . . . . . . 18 |- ( ( ( # ` x ) = ( N + 2 ) /\ N e. NN0 ) -> 0 < ( # ` x ) )
105 hashgt0n0 13156 . . . . . . . . . . . . . . . . . 18 |- ( ( x e. Word V /\ 0 < ( # ` x ) ) -> x =/= (/) )
106104, 105sylan2 491 . . . . . . . . . . . . . . . . 17 |- ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ N e. NN0 ) ) -> x =/= (/) )
107106exp32 631 . . . . . . . . . . . . . . . 16 |- ( x e. Word V -> ( ( # ` x ) = ( N + 2 ) -> ( N e. NN0 -> x =/= (/) ) ) )
108107com12 32 . . . . . . . . . . . . . . 15 |- ( ( # ` x ) = ( N + 2 ) -> ( x e. Word V -> ( N e. NN0 -> x =/= (/) ) ) )
1091083ad2ant1 1082 . . . . . . . . . . . . . 14 |- ( ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) -> ( x e. Word V -> ( N e. NN0 -> x =/= (/) ) ) )
110109impcom 446 . . . . . . . . . . . . 13 |- ( ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) -> ( N e. NN0 -> x =/= (/) ) )
111110adantl 482 . . . . . . . . . . . 12 |- ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) ) -> ( N e. NN0 -> x =/= (/) ) )
112111imp 445 . . . . . . . . . . 11 |- ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) ) /\ N e. NN0 ) -> x =/= (/) )
11381, 100, 112jca32 558 . . . . . . . . . 10 |- ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) ) /\ N e. NN0 ) -> ( ( d e. Word V /\ x e. Word V ) /\ ( d =/= (/) /\ x =/= (/) ) ) )
114113adantr 481 . . . . . . . . 9 |- ( ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) ) /\ N e. NN0 ) /\ ( lastS ` d ) = ( lastS ` x ) ) -> ( ( d e. Word V /\ x e. Word V ) /\ ( d =/= (/) /\ x =/= (/) ) ) )
115 simpl 473 . . . . . . . . . . . 12 |- ( ( d e. Word V /\ x e. Word V ) -> d e. Word V )
116115adantr 481 . . . . . . . . . . 11 |- ( ( ( d e. Word V /\ x e. Word V ) /\ ( d =/= (/) /\ x =/= (/) ) ) -> d e. Word V )
117 simpr 477 . . . . . . . . . . . 12 |- ( ( d e. Word V /\ x e. Word V ) -> x e. Word V )
118117adantr 481 . . . . . . . . . . 11 |- ( ( ( d e. Word V /\ x e. Word V ) /\ ( d =/= (/) /\ x =/= (/) ) ) -> x e. Word V )
119 hashneq0 13155 . . . . . . . . . . . . . . . 16 |- ( d e. Word V -> ( 0 < ( # ` d ) <-> d =/= (/) ) )
120119biimprd 238 . . . . . . . . . . . . . . 15 |- ( d e. Word V -> ( d =/= (/) -> 0 < ( # ` d ) ) )
121120adantr 481 . . . . . . . . . . . . . 14 |- ( ( d e. Word V /\ x e. Word V ) -> ( d =/= (/) -> 0 < ( # ` d ) ) )
122121com12 32 . . . . . . . . . . . . 13 |- ( d =/= (/) -> ( ( d e. Word V /\ x e. Word V ) -> 0 < ( # ` d ) ) )
123122adantr 481 . . . . . . . . . . . 12 |- ( ( d =/= (/) /\ x =/= (/) ) -> ( ( d e. Word V /\ x e. Word V ) -> 0 < ( # ` d ) ) )
124123impcom 446 . . . . . . . . . . 11 |- ( ( ( d e. Word V /\ x e. Word V ) /\ ( d =/= (/) /\ x =/= (/) ) ) -> 0 < ( # ` d ) )
125 2swrd1eqwrdeq 13454 . . . . . . . . . . 11 |- ( ( d e. Word V /\ x e. Word V /\ 0 < ( # ` d ) ) -> ( d = x <-> ( ( # ` d ) = ( # ` x ) /\ ( ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) /\ ( lastS ` d ) = ( lastS ` x ) ) ) ) )
126116, 118, 124, 125syl3anc 1326 . . . . . . . . . 10 |- ( ( ( d e. Word V /\ x e. Word V ) /\ ( d =/= (/) /\ x =/= (/) ) ) -> ( d = x <-> ( ( # ` d ) = ( # ` x ) /\ ( ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) /\ ( lastS ` d ) = ( lastS ` x ) ) ) ) )
127 ancom 466 . . . . . . . . . . . 12 |- ( ( ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) /\ ( lastS ` d ) = ( lastS ` x ) ) <-> ( ( lastS ` d ) = ( lastS ` x ) /\ ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) )
128127anbi2i 730 . . . . . . . . . . 11 |- ( ( ( # ` d ) = ( # ` x ) /\ ( ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) /\ ( lastS ` d ) = ( lastS ` x ) ) ) <-> ( ( # ` d ) = ( # ` x ) /\ ( ( lastS ` d ) = ( lastS ` x ) /\ ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
129 3anass 1042 . . . . . . . . . . 11 |- ( ( ( # ` d ) = ( # ` x ) /\ ( lastS ` d ) = ( lastS ` x ) /\ ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) <-> ( ( # ` d ) = ( # ` x ) /\ ( ( lastS ` d ) = ( lastS ` x ) /\ ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
130128, 129bitr4i 267 . . . . . . . . . 10 |- ( ( ( # ` d ) = ( # ` x ) /\ ( ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) /\ ( lastS ` d ) = ( lastS ` x ) ) ) <-> ( ( # ` d ) = ( # ` x ) /\ ( lastS ` d ) = ( lastS ` x ) /\ ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) )
131126, 130syl6bb 276 . . . . . . . . 9 |- ( ( ( d e. Word V /\ x e. Word V ) /\ ( d =/= (/) /\ x =/= (/) ) ) -> ( d = x <-> ( ( # ` d ) = ( # ` x ) /\ ( lastS ` d ) = ( lastS ` x ) /\ ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
132114, 131syl 17 . . . . . . . 8 |- ( ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) ) /\ N e. NN0 ) /\ ( lastS ` d ) = ( lastS ` x ) ) -> ( d = x <-> ( ( # ` d ) = ( # ` x ) /\ ( lastS ` d ) = ( lastS ` x ) /\ ( d substr <. 0 , ( ( # ` d ) - 1 ) >. ) = ( x substr <. 0 , ( ( # ` d ) - 1 ) >. ) ) ) )
13342, 43, 77, 132mpbir3and 1245 . . . . . . 7 |- ( ( ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) ) /\ N e. NN0 ) /\ ( lastS ` d ) = ( lastS ` x ) ) -> d = x )
134133exp31 630 . . . . . 6 |- ( ( ( d e. Word V /\ ( ( # ` d ) = ( N + 2 ) /\ ( d substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` d ) } e. E ) ) /\ ( x e. Word V /\ ( ( # ` x ) = ( N + 2 ) /\ ( x substr <. 0 , ( N + 1 ) >. ) = W /\ { ( lastS ` W ) , ( lastS ` x ) } e. E ) ) ) -> ( N e. NN0 -> ( ( lastS ` d ) = ( lastS ` x ) -> d = x ) ) )
13523, 32, 134syl2anb 496 . . . . 5 |- ( ( d e. D /\ x e. D ) -> ( N e. NN0 -> ( ( lastS ` d ) = ( lastS ` x ) -> d = x ) ) )
136135impcom 446 . . . 4 |- ( ( N e. NN0 /\ ( d e. D /\ x e. D ) ) -> ( ( lastS ` d ) = ( lastS ` x ) -> d = x ) )
13714, 136sylbid 230 . . 3 |- ( ( N e. NN0 /\ ( d e. D /\ x e. D ) ) -> ( ( F ` d ) = ( F ` x ) -> d = x ) )
138137ralrimivva 2971 . 2 |- ( N e. NN0 -> A. d e. D A. x e. D ( ( F ` d ) = ( F ` x ) -> d = x ) )
139 dff13 6512 . 2 |- ( F : D -1-1-> R <-> ( F : D --> R /\ A. d e. D A. x e. D ( ( F ` d ) = ( F ` x ) -> d = x ) ) )
1406, 138, 139sylanbrc 698 1 |- ( N e. NN0 -> F : D -1-1-> R )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {crab 2916   (/)c0 3915   {cpr 4179   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   -1-1->wf1 5885   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   - cmin 10266   2c2 11070   NN0cn0 11292   #chash 13117  Word cword 13291   lastS clsw 13292   substr csubstr 13295  Vtxcvtx 25874  Edgcedg 25939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-s1 13302  df-substr 13303
This theorem is referenced by:  wwlksnextbij0  26796
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