Theorem 3wlkdlem5 27023

Description: Lemma 5 for 3wlkd 27030. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 7-Feb-2021.)

Hypotheses

Ref	Expression
3wlkd.p	|- P = <" A B C D ">
3wlkd.f	|- F = <" J K L ">
3wlkd.s	|- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )
3wlkd.n	|- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )

Assertion

Ref	Expression
3wlkdlem5	|- ( ph -> A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) )

Distinct variable groups:   A, k   B, k   C, k   D, k   k, J   k, K   k, L   k, V   k, F   P, k

Allowed substitution hint:   ph( k)

Proof of Theorem 3wlkdlem5

Step	Hyp	Ref	Expression
1		3wlkd.n	. . . 4 |- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )
2		simpl 473	. . . . 5 |- ( ( A =/= B /\ A =/= C ) -> A =/= B )
3		simpl 473	. . . . 5 |- ( ( B =/= C /\ B =/= D ) -> B =/= C )
4		id 22	. . . . 5 |- ( C =/= D -> C =/= D )
5	2, 3, 4	3anim123i 1247	. . . 4 |- ( ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) -> ( A =/= B /\ B =/= C /\ C =/= D ) )
6	1, 5	syl 17	. . 3 |- ( ph -> ( A =/= B /\ B =/= C /\ C =/= D ) )
7		3wlkd.p	. . . . 5 |- P = <" A B C D ">
8		3wlkd.f	. . . . 5 |- F = <" J K L ">
9		3wlkd.s	. . . . 5 |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )
10	7, 8, 9	3wlkdlem3 27021	. . . 4 |- ( ph -> ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) )
11		simpl 473	. . . . . . 7 |- ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) -> ( P ` 0 ) = A )
12		simpr 477	. . . . . . 7 |- ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) -> ( P ` 1 ) = B )
13	11, 12	neeq12d 2855	. . . . . 6 |- ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) -> ( ( P ` 0 ) =/= ( P ` 1 ) <-> A =/= B ) )
14	13	adantr 481	. . . . 5 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( ( P ` 0 ) =/= ( P ` 1 ) <-> A =/= B ) )
15	12	adantr 481	. . . . . 6 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( P ` 1 ) = B )
16		simpl 473	. . . . . . 7 |- ( ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) -> ( P ` 2 ) = C )
17	16	adantl 482	. . . . . 6 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( P ` 2 ) = C )
18	15, 17	neeq12d 2855	. . . . 5 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( ( P ` 1 ) =/= ( P ` 2 ) <-> B =/= C ) )
19		simpr 477	. . . . . . 7 |- ( ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) -> ( P ` 3 ) = D )
20	16, 19	neeq12d 2855	. . . . . 6 |- ( ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) -> ( ( P ` 2 ) =/= ( P ` 3 ) <-> C =/= D ) )
21	20	adantl 482	. . . . 5 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( ( P ` 2 ) =/= ( P ` 3 ) <-> C =/= D ) )
22	14, 18, 21	3anbi123d 1399	. . . 4 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 3 ) ) <-> ( A =/= B /\ B =/= C /\ C =/= D ) ) )
23	10, 22	syl 17	. . 3 |- ( ph -> ( ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 3 ) ) <-> ( A =/= B /\ B =/= C /\ C =/= D ) ) )
24	6, 23	mpbird 247	. 2 |- ( ph -> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 3 ) ) )
25	7, 8	3wlkdlem2 27020	. . . 4 |- ( 0..^ ( # ` F ) ) = { 0 , 1 , 2 }
26	25	raleqi 3142	. . 3 |- ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> A. k e. { 0 , 1 , 2 } ( P ` k ) =/= ( P ` ( k + 1 ) ) )
27		c0ex 10034	. . . 4 |- 0 e. _V
28		1ex 10035	. . . 4 |- 1 e. _V
29		2ex 11092	. . . 4 |- 2 e. _V
30		fveq2 6191	. . . . 5 |- ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
31		oveq1 6657	. . . . . . 7 |- ( k = 0 -> ( k + 1 ) = ( 0 + 1 ) )
32		0p1e1 11132	. . . . . . 7 |- ( 0 + 1 ) = 1
33	31, 32	syl6eq 2672	. . . . . 6 |- ( k = 0 -> ( k + 1 ) = 1 )
34	33	fveq2d 6195	. . . . 5 |- ( k = 0 -> ( P ` ( k + 1 ) ) = ( P ` 1 ) )
35	30, 34	neeq12d 2855	. . . 4 |- ( k = 0 -> ( ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> ( P ` 0 ) =/= ( P ` 1 ) ) )
36		fveq2 6191	. . . . 5 |- ( k = 1 -> ( P ` k ) = ( P ` 1 ) )
37		oveq1 6657	. . . . . . 7 |- ( k = 1 -> ( k + 1 ) = ( 1 + 1 ) )
38		1p1e2 11134	. . . . . . 7 |- ( 1 + 1 ) = 2
39	37, 38	syl6eq 2672	. . . . . 6 |- ( k = 1 -> ( k + 1 ) = 2 )
40	39	fveq2d 6195	. . . . 5 |- ( k = 1 -> ( P ` ( k + 1 ) ) = ( P ` 2 ) )
41	36, 40	neeq12d 2855	. . . 4 |- ( k = 1 -> ( ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> ( P ` 1 ) =/= ( P ` 2 ) ) )
42		fveq2 6191	. . . . 5 |- ( k = 2 -> ( P ` k ) = ( P ` 2 ) )
43		oveq1 6657	. . . . . . 7 |- ( k = 2 -> ( k + 1 ) = ( 2 + 1 ) )
44		2p1e3 11151	. . . . . . 7 |- ( 2 + 1 ) = 3
45	43, 44	syl6eq 2672	. . . . . 6 |- ( k = 2 -> ( k + 1 ) = 3 )
46	45	fveq2d 6195	. . . . 5 |- ( k = 2 -> ( P ` ( k + 1 ) ) = ( P ` 3 ) )
47	42, 46	neeq12d 2855	. . . 4 |- ( k = 2 -> ( ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> ( P ` 2 ) =/= ( P ` 3 ) ) )
48	27, 28, 29, 35, 41, 47	raltp 4240	. . 3 |- ( A. k e. { 0 , 1 , 2 } ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 3 ) ) )
49	26, 48	bitri 264	. 2 |- ( A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) <-> ( ( P ` 0 ) =/= ( P ` 1 ) /\ ( P ` 1 ) =/= ( P ` 2 ) /\ ( P ` 2 ) =/= ( P ` 3 ) ) )
50	24, 49	sylibr 224	1 |- ( ph -> A. k e. ( 0..^ ( # ` F ) ) ( P ` k ) =/= ( P ` ( k + 1 ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   {ctp 4181   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   2c2 11070   3c3 11071  ..^cfzo 12465   #chash 13117   <"cs3 13587   <"cs4 13588

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-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-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-s2 13593  df-s3 13594  df-s4 13595

This theorem is referenced by:  3wlkd  27030