Theorem 3wlkdlem10 27029

Description: Lemma 10 for 3wlkd 27030. (Contributed by Alexander van der Vekens, 12-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 ) )
3wlkd.e	|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )

Assertion

Ref	Expression
3wlkdlem10	|- ( ph -> A. k e. ( 0..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )

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

Allowed substitution hint:   ph( k)

Proof of Theorem 3wlkdlem10

Step	Hyp	Ref	Expression
1		3wlkd.p	. . . 4 |- P = <" A B C D ">
2		3wlkd.f	. . . 4 |- F = <" J K L ">
3		3wlkd.s	. . . 4 |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )
4		3wlkd.n	. . . 4 |- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )
5		3wlkd.e	. . . 4 |- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )
6	1, 2, 3, 4, 5	3wlkdlem9 27028	. . 3 |- ( ph -> ( { A , B } C_ ( I ` ( F ` 0 ) ) /\ { B , C } C_ ( I ` ( F ` 1 ) ) /\ { C , D } C_ ( I ` ( F ` 2 ) ) ) )
7	1, 2, 3	3wlkdlem3 27021	. . . 4 |- ( ph -> ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) )
8		preq12 4270	. . . . . . 7 |- ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) -> { ( P ` 0 ) , ( P ` 1 ) } = { A , B } )
9	8	adantr 481	. . . . . 6 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> { ( P ` 0 ) , ( P ` 1 ) } = { A , B } )
10	9	sseq1d 3632	. . . . 5 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) <-> { A , B } C_ ( I ` ( F ` 0 ) ) ) )
11		simplr 792	. . . . . . 7 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( P ` 1 ) = B )
12		simprl 794	. . . . . . 7 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( P ` 2 ) = C )
13	11, 12	preq12d 4276	. . . . . 6 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> { ( P ` 1 ) , ( P ` 2 ) } = { B , C } )
14	13	sseq1d 3632	. . . . 5 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( { ( P ` 1 ) , ( P ` 2 ) } C_ ( I ` ( F ` 1 ) ) <-> { B , C } C_ ( I ` ( F ` 1 ) ) ) )
15		preq12 4270	. . . . . . 7 |- ( ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) -> { ( P ` 2 ) , ( P ` 3 ) } = { C , D } )
16	15	adantl 482	. . . . . 6 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> { ( P ` 2 ) , ( P ` 3 ) } = { C , D } )
17	16	sseq1d 3632	. . . . 5 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( { ( P ` 2 ) , ( P ` 3 ) } C_ ( I ` ( F ` 2 ) ) <-> { C , D } C_ ( I ` ( F ` 2 ) ) ) )
18	10, 14, 17	3anbi123d 1399	. . . 4 |- ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( ( { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) /\ { ( P ` 1 ) , ( P ` 2 ) } C_ ( I ` ( F ` 1 ) ) /\ { ( P ` 2 ) , ( P ` 3 ) } C_ ( I ` ( F ` 2 ) ) ) <-> ( { A , B } C_ ( I ` ( F ` 0 ) ) /\ { B , C } C_ ( I ` ( F ` 1 ) ) /\ { C , D } C_ ( I ` ( F ` 2 ) ) ) ) )
19	7, 18	syl 17	. . 3 |- ( ph -> ( ( { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) /\ { ( P ` 1 ) , ( P ` 2 ) } C_ ( I ` ( F ` 1 ) ) /\ { ( P ` 2 ) , ( P ` 3 ) } C_ ( I ` ( F ` 2 ) ) ) <-> ( { A , B } C_ ( I ` ( F ` 0 ) ) /\ { B , C } C_ ( I ` ( F ` 1 ) ) /\ { C , D } C_ ( I ` ( F ` 2 ) ) ) ) )
20	6, 19	mpbird 247	. 2 |- ( ph -> ( { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) /\ { ( P ` 1 ) , ( P ` 2 ) } C_ ( I ` ( F ` 1 ) ) /\ { ( P ` 2 ) , ( P ` 3 ) } C_ ( I ` ( F ` 2 ) ) ) )
21	1, 2	3wlkdlem2 27020	. . . 4 |- ( 0..^ ( # ` F ) ) = { 0 , 1 , 2 }
22	21	raleqi 3142	. . 3 |- ( A. k e. ( 0..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) <-> A. k e. { 0 , 1 , 2 } { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )
23		c0ex 10034	. . . 4 |- 0 e. _V
24		1ex 10035	. . . 4 |- 1 e. _V
25		2ex 11092	. . . 4 |- 2 e. _V
26		fveq2 6191	. . . . . 6 |- ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
27		oveq1 6657	. . . . . . . 8 |- ( k = 0 -> ( k + 1 ) = ( 0 + 1 ) )
28		0p1e1 11132	. . . . . . . 8 |- ( 0 + 1 ) = 1
29	27, 28	syl6eq 2672	. . . . . . 7 |- ( k = 0 -> ( k + 1 ) = 1 )
30	29	fveq2d 6195	. . . . . 6 |- ( k = 0 -> ( P ` ( k + 1 ) ) = ( P ` 1 ) )
31	26, 30	preq12d 4276	. . . . 5 |- ( k = 0 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 0 ) , ( P ` 1 ) } )
32		fveq2 6191	. . . . . 6 |- ( k = 0 -> ( F ` k ) = ( F ` 0 ) )
33	32	fveq2d 6195	. . . . 5 |- ( k = 0 -> ( I ` ( F ` k ) ) = ( I ` ( F ` 0 ) ) )
34	31, 33	sseq12d 3634	. . . 4 |- ( k = 0 -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) <-> { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) ) )
35		fveq2 6191	. . . . . 6 |- ( k = 1 -> ( P ` k ) = ( P ` 1 ) )
36		oveq1 6657	. . . . . . . 8 |- ( k = 1 -> ( k + 1 ) = ( 1 + 1 ) )
37		1p1e2 11134	. . . . . . . 8 |- ( 1 + 1 ) = 2
38	36, 37	syl6eq 2672	. . . . . . 7 |- ( k = 1 -> ( k + 1 ) = 2 )
39	38	fveq2d 6195	. . . . . 6 |- ( k = 1 -> ( P ` ( k + 1 ) ) = ( P ` 2 ) )
40	35, 39	preq12d 4276	. . . . 5 |- ( k = 1 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 1 ) , ( P ` 2 ) } )
41		fveq2 6191	. . . . . 6 |- ( k = 1 -> ( F ` k ) = ( F ` 1 ) )
42	41	fveq2d 6195	. . . . 5 |- ( k = 1 -> ( I ` ( F ` k ) ) = ( I ` ( F ` 1 ) ) )
43	40, 42	sseq12d 3634	. . . 4 |- ( k = 1 -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) <-> { ( P ` 1 ) , ( P ` 2 ) } C_ ( I ` ( F ` 1 ) ) ) )
44		fveq2 6191	. . . . . 6 |- ( k = 2 -> ( P ` k ) = ( P ` 2 ) )
45		oveq1 6657	. . . . . . . 8 |- ( k = 2 -> ( k + 1 ) = ( 2 + 1 ) )
46		2p1e3 11151	. . . . . . . 8 |- ( 2 + 1 ) = 3
47	45, 46	syl6eq 2672	. . . . . . 7 |- ( k = 2 -> ( k + 1 ) = 3 )
48	47	fveq2d 6195	. . . . . 6 |- ( k = 2 -> ( P ` ( k + 1 ) ) = ( P ` 3 ) )
49	44, 48	preq12d 4276	. . . . 5 |- ( k = 2 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 2 ) , ( P ` 3 ) } )
50		fveq2 6191	. . . . . 6 |- ( k = 2 -> ( F ` k ) = ( F ` 2 ) )
51	50	fveq2d 6195	. . . . 5 |- ( k = 2 -> ( I ` ( F ` k ) ) = ( I ` ( F ` 2 ) ) )
52	49, 51	sseq12d 3634	. . . 4 |- ( k = 2 -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) <-> { ( P ` 2 ) , ( P ` 3 ) } C_ ( I ` ( F ` 2 ) ) ) )
53	23, 24, 25, 34, 43, 52	raltp 4240	. . 3 |- ( A. k e. { 0 , 1 , 2 } { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) <-> ( { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) /\ { ( P ` 1 ) , ( P ` 2 ) } C_ ( I ` ( F ` 1 ) ) /\ { ( P ` 2 ) , ( P ` 3 ) } C_ ( I ` ( F ` 2 ) ) ) )
54	22, 53	bitri 264	. 2 |- ( A. k e. ( 0..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) <-> ( { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) /\ { ( P ` 1 ) , ( P ` 2 ) } C_ ( I ` ( F ` 1 ) ) /\ { ( P ` 2 ) , ( P ` 3 ) } C_ ( I ` ( F ` 2 ) ) ) )
55	20, 54	sylibr 224	1 |- ( ph -> A. k e. ( 0..^ ( # ` F ) ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   C_ wss 3574   {cpr 4179   {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