Theorem 11wlkdlem4 40028

Description: Lemma 4 for 11wlkd 40029. (Contributed by AV, 22-Jan-2021.)

Hypotheses

Ref	Expression
11wlkd.p	|- P = <" X Y ">
11wlkd.f	|- F = <" J ">
11wlkd.x	|- ( ph -> X e. V )
11wlkd.y	|- ( ph -> Y e. V )
11wlkd.l	|- ( ( ph /\ X = Y ) -> ( I ` J ) = { X } )
11wlkd.j	|- ( ( ph /\ X =/= Y ) -> { X , Y } C_ ( I ` J ) )

Assertion

Ref	Expression
11wlkdlem4	|- ( ph -> A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) )

Distinct variable groups:   k, F   k, I   P, k

Allowed substitution hints:   ph( k)   J( k)   V( k)   X( k)   Y( k)

Proof of Theorem 11wlkdlem4

Step	Hyp	Ref	Expression
1		11wlkd.f	. . . . . . . . . 10 |- F = <" J ">
2	1	fveq1i 5880	. . . . . . . . 9 |- ( F ` 0 ) = ( <" J "> ` 0 )
3		11wlkd.p	. . . . . . . . . . . 12 |- P = <" X Y ">
4		11wlkd.x	. . . . . . . . . . . 12 |- ( ph -> X e. V )
5		11wlkd.y	. . . . . . . . . . . 12 |- ( ph -> Y e. V )
6		11wlkd.l	. . . . . . . . . . . 12 |- ( ( ph /\ X = Y ) -> ( I ` J ) = { X } )
7		11wlkd.j	. . . . . . . . . . . 12 |- ( ( ph /\ X =/= Y ) -> { X , Y } C_ ( I ` J ) )
8	3, 1, 4, 5, 6, 7	11wlkdlem2 40026	. . . . . . . . . . 11 |- ( ph -> X e. ( I ` J ) )
9	8	elfvexd 5907	. . . . . . . . . 10 |- ( ph -> J e. _V )
10		s1fv 12801	. . . . . . . . . 10 |- ( J e. _V -> ( <" J "> ` 0 ) = J )
11	9, 10	syl 17	. . . . . . . . 9 |- ( ph -> ( <" J "> ` 0 ) = J )
12	2, 11	syl5eq 2517	. . . . . . . 8 |- ( ph -> ( F ` 0 ) = J )
13	12	fveq2d 5883	. . . . . . 7 |- ( ph -> ( I ` ( F ` 0 ) ) = ( I ` J ) )
14	13	adantr 472	. . . . . 6 |- ( ( ph /\ X = Y ) -> ( I ` ( F ` 0 ) ) = ( I ` J ) )
15	14, 6	eqtrd 2505	. . . . 5 |- ( ( ph /\ X = Y ) -> ( I ` ( F ` 0 ) ) = { X } )
16		df-ne 2643	. . . . . . 7 |- ( X =/= Y <-> -. X = Y )
17	16, 7	sylan2br 484	. . . . . 6 |- ( ( ph /\ -. X = Y ) -> { X , Y } C_ ( I ` J ) )
18	13	adantr 472	. . . . . 6 |- ( ( ph /\ -. X = Y ) -> ( I ` ( F ` 0 ) ) = ( I ` J ) )
19	17, 18	sseqtr4d 3455	. . . . 5 |- ( ( ph /\ -. X = Y ) -> { X , Y } C_ ( I ` ( F ` 0 ) ) )
20	15, 19	ifpimpda 999	. . . 4 |- ( ph -> if- ( X = Y , ( I ` ( F ` 0 ) ) = { X } , { X , Y } C_ ( I ` ( F ` 0 ) ) ) )
21	3	fveq1i 5880	. . . . . 6 |- ( P ` 0 ) = ( <" X Y "> ` 0 )
22		s2fv0 13041	. . . . . . 7 |- ( X e. V -> ( <" X Y "> ` 0 ) = X )
23	4, 22	syl 17	. . . . . 6 |- ( ph -> ( <" X Y "> ` 0 ) = X )
24	21, 23	syl5eq 2517	. . . . 5 |- ( ph -> ( P ` 0 ) = X )
25	3	fveq1i 5880	. . . . . 6 |- ( P ` 1 ) = ( <" X Y "> ` 1 )
26		s2fv1 13042	. . . . . . 7 |- ( Y e. V -> ( <" X Y "> ` 1 ) = Y )
27	5, 26	syl 17	. . . . . 6 |- ( ph -> ( <" X Y "> ` 1 ) = Y )
28	25, 27	syl5eq 2517	. . . . 5 |- ( ph -> ( P ` 1 ) = Y )
29		eqeq12 2484	. . . . . 6 |- ( ( ( P ` 0 ) = X /\ ( P ` 1 ) = Y ) -> ( ( P ` 0 ) = ( P ` 1 ) <-> X = Y ) )
30		sneq 3969	. . . . . . . 8 |- ( ( P ` 0 ) = X -> { ( P ` 0 ) } = { X } )
31	30	adantr 472	. . . . . . 7 |- ( ( ( P ` 0 ) = X /\ ( P ` 1 ) = Y ) -> { ( P ` 0 ) } = { X } )
32	31	eqeq2d 2481	. . . . . 6 |- ( ( ( P ` 0 ) = X /\ ( P ` 1 ) = Y ) -> ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) } <-> ( I ` ( F ` 0 ) ) = { X } ) )
33		preq12 4044	. . . . . . 7 |- ( ( ( P ` 0 ) = X /\ ( P ` 1 ) = Y ) -> { ( P ` 0 ) , ( P ` 1 ) } = { X , Y } )
34	33	sseq1d 3445	. . . . . 6 |- ( ( ( P ` 0 ) = X /\ ( P ` 1 ) = Y ) -> ( { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) <-> { X , Y } C_ ( I ` ( F ` 0 ) ) ) )
35	29, 32, 34	ifpbi123d 998	. . . . 5 |- ( ( ( P ` 0 ) = X /\ ( P ` 1 ) = Y ) -> (if- ( ( P ` 0 ) = ( P ` 1 ) , ( I ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) ) <-> if- ( X = Y , ( I ` ( F ` 0 ) ) = { X } , { X , Y } C_ ( I ` ( F ` 0 ) ) ) ) )
36	24, 28, 35	syl2anc 673	. . . 4 |- ( ph -> (if- ( ( P ` 0 ) = ( P ` 1 ) , ( I ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) ) <-> if- ( X = Y , ( I ` ( F ` 0 ) ) = { X } , { X , Y } C_ ( I ` ( F ` 0 ) ) ) ) )
37	20, 36	mpbird 240	. . 3 |- ( ph -> if- ( ( P ` 0 ) = ( P ` 1 ) , ( I ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) ) )
38		c0ex 9655	. . . 4 |- 0 e. _V
39		fveq2 5879	. . . . . 6 |- ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
40		oveq1 6315	. . . . . . . 8 |- ( k = 0 -> ( k + 1 ) = ( 0 + 1 ) )
41		0p1e1 10743	. . . . . . . 8 |- ( 0 + 1 ) = 1
42	40, 41	syl6eq 2521	. . . . . . 7 |- ( k = 0 -> ( k + 1 ) = 1 )
43	42	fveq2d 5883	. . . . . 6 |- ( k = 0 -> ( P ` ( k + 1 ) ) = ( P ` 1 ) )
44	39, 43	eqeq12d 2486	. . . . 5 |- ( k = 0 -> ( ( P ` k ) = ( P ` ( k + 1 ) ) <-> ( P ` 0 ) = ( P ` 1 ) ) )
45		fveq2 5879	. . . . . . 7 |- ( k = 0 -> ( F ` k ) = ( F ` 0 ) )
46	45	fveq2d 5883	. . . . . 6 |- ( k = 0 -> ( I ` ( F ` k ) ) = ( I ` ( F ` 0 ) ) )
47	39	sneqd 3971	. . . . . 6 |- ( k = 0 -> { ( P ` k ) } = { ( P ` 0 ) } )
48	46, 47	eqeq12d 2486	. . . . 5 |- ( k = 0 -> ( ( I ` ( F ` k ) ) = { ( P ` k ) } <-> ( I ` ( F ` 0 ) ) = { ( P ` 0 ) } ) )
49	39, 43	preq12d 4050	. . . . . 6 |- ( k = 0 -> { ( P ` k ) , ( P ` ( k + 1 ) ) } = { ( P ` 0 ) , ( P ` 1 ) } )
50	49, 46	sseq12d 3447	. . . . 5 |- ( k = 0 -> ( { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) <-> { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) ) )
51	44, 48, 50	ifpbi123d 998	. . . 4 |- ( k = 0 -> (if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) <-> if- ( ( P ` 0 ) = ( P ` 1 ) , ( I ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) ) ) )
52	38, 51	ralsn 4001	. . 3 |- ( A. k e. { 0 }if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) <-> if- ( ( P ` 0 ) = ( P ` 1 ) , ( I ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) ) )
53	37, 52	sylibr 217	. 2 |- ( ph -> A. k e. { 0 }if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) )
54	1	fveq2i 5882	. . . . . . 7 |- ( # ` F ) = ( # ` <" J "> )
55		s1len 12797	. . . . . . 7 |- ( # ` <" J "> ) = 1
56	54, 55	eqtri 2493	. . . . . 6 |- ( # ` F ) = 1
57	56	oveq2i 6319	. . . . 5 |- ( 0..^ ( # ` F ) ) = ( 0..^ 1 )
58		fzo01 12024	. . . . 5 |- ( 0..^ 1 ) = { 0 }
59	57, 58	eqtri 2493	. . . 4 |- ( 0..^ ( # ` F ) ) = { 0 }
60	59	a1i 11	. . 3 |- ( ph -> ( 0..^ ( # ` F ) ) = { 0 } )
61	60	raleqdv 2979	. 2 |- ( ph -> ( A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) <-> A. k e. { 0 }if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) ) )
62	53, 61	mpbird 240	1 |- ( ph -> A. k e. ( 0..^ ( # ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( I ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( I ` ( F ` k ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 376  if-wif 983   = wceq 1452   e. wcel 1904   =/= wne 2641   A.wral 2756   _Vcvv 3031   C_ wss 3390   {csn 3959   {cpr 3961   ` cfv 5589  (class class class)co 6308   0cc0 9557   1c1 9558   + caddc 9560  ..^cfzo 11942   #chash 12553   <"cs1 12706   <"cs2 12996

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  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-ifp 984  df-3or 1008  df-3an 1009  df-tru 1455  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-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  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-int 4227  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-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943  df-hash 12554  df-word 12711  df-concat 12713  df-s1 12714  df-s2 13003

This theorem is referenced by:  11wlkd  40029