Theorem 1wlkdlem4 27000

Description: Lemma 4 for 1wlkd 27001. (Contributed by AV, 22-Jan-2021.)

Hypotheses

Ref	Expression
1wlkd.p	|- P = <" X Y ">
1wlkd.f	|- F = <" J ">
1wlkd.x	|- ( ph -> X e. V )
1wlkd.y	|- ( ph -> Y e. V )
1wlkd.l	|- ( ( ph /\ X = Y ) -> ( I ` J ) = { X } )
1wlkd.j	|- ( ( ph /\ X =/= Y ) -> { X , Y } C_ ( I ` J ) )

Assertion

Ref	Expression
1wlkdlem4	|- ( 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 1wlkdlem4

Step	Hyp	Ref	Expression
1		1wlkd.f	. . . . . . . . . 10 |- F = <" J ">
2	1	fveq1i 6192	. . . . . . . . 9 |- ( F ` 0 ) = ( <" J "> ` 0 )
3		1wlkd.p	. . . . . . . . . . . 12 |- P = <" X Y ">
4		1wlkd.x	. . . . . . . . . . . 12 |- ( ph -> X e. V )
5		1wlkd.y	. . . . . . . . . . . 12 |- ( ph -> Y e. V )
6		1wlkd.l	. . . . . . . . . . . 12 |- ( ( ph /\ X = Y ) -> ( I ` J ) = { X } )
7		1wlkd.j	. . . . . . . . . . . 12 |- ( ( ph /\ X =/= Y ) -> { X , Y } C_ ( I ` J ) )
8	3, 1, 4, 5, 6, 7	1wlkdlem2 26998	. . . . . . . . . . 11 |- ( ph -> X e. ( I ` J ) )
9	8	elfvexd 6222	. . . . . . . . . 10 |- ( ph -> J e. _V )
10		s1fv 13390	. . . . . . . . . 10 |- ( J e. _V -> ( <" J "> ` 0 ) = J )
11	9, 10	syl 17	. . . . . . . . 9 |- ( ph -> ( <" J "> ` 0 ) = J )
12	2, 11	syl5eq 2668	. . . . . . . 8 |- ( ph -> ( F ` 0 ) = J )
13	12	fveq2d 6195	. . . . . . 7 |- ( ph -> ( I ` ( F ` 0 ) ) = ( I ` J ) )
14	13	adantr 481	. . . . . 6 |- ( ( ph /\ X = Y ) -> ( I ` ( F ` 0 ) ) = ( I ` J ) )
15	14, 6	eqtrd 2656	. . . . 5 |- ( ( ph /\ X = Y ) -> ( I ` ( F ` 0 ) ) = { X } )
16		df-ne 2795	. . . . . . 7 |- ( X =/= Y <-> -. X = Y )
17	16, 7	sylan2br 493	. . . . . 6 |- ( ( ph /\ -. X = Y ) -> { X , Y } C_ ( I ` J ) )
18	13	adantr 481	. . . . . 6 |- ( ( ph /\ -. X = Y ) -> ( I ` ( F ` 0 ) ) = ( I ` J ) )
19	17, 18	sseqtr4d 3642	. . . . 5 |- ( ( ph /\ -. X = Y ) -> { X , Y } C_ ( I ` ( F ` 0 ) ) )
20	15, 19	ifpimpda 1028	. . . 4 |- ( ph -> if- ( X = Y , ( I ` ( F ` 0 ) ) = { X } , { X , Y } C_ ( I ` ( F ` 0 ) ) ) )
21	3	fveq1i 6192	. . . . . 6 |- ( P ` 0 ) = ( <" X Y "> ` 0 )
22		s2fv0 13632	. . . . . . 7 |- ( X e. V -> ( <" X Y "> ` 0 ) = X )
23	4, 22	syl 17	. . . . . 6 |- ( ph -> ( <" X Y "> ` 0 ) = X )
24	21, 23	syl5eq 2668	. . . . 5 |- ( ph -> ( P ` 0 ) = X )
25	3	fveq1i 6192	. . . . . 6 |- ( P ` 1 ) = ( <" X Y "> ` 1 )
26		s2fv1 13633	. . . . . . 7 |- ( Y e. V -> ( <" X Y "> ` 1 ) = Y )
27	5, 26	syl 17	. . . . . 6 |- ( ph -> ( <" X Y "> ` 1 ) = Y )
28	25, 27	syl5eq 2668	. . . . 5 |- ( ph -> ( P ` 1 ) = Y )
29		eqeq12 2635	. . . . . 6 |- ( ( ( P ` 0 ) = X /\ ( P ` 1 ) = Y ) -> ( ( P ` 0 ) = ( P ` 1 ) <-> X = Y ) )
30		sneq 4187	. . . . . . . 8 |- ( ( P ` 0 ) = X -> { ( P ` 0 ) } = { X } )
31	30	adantr 481	. . . . . . 7 |- ( ( ( P ` 0 ) = X /\ ( P ` 1 ) = Y ) -> { ( P ` 0 ) } = { X } )
32	31	eqeq2d 2632	. . . . . 6 |- ( ( ( P ` 0 ) = X /\ ( P ` 1 ) = Y ) -> ( ( I ` ( F ` 0 ) ) = { ( P ` 0 ) } <-> ( I ` ( F ` 0 ) ) = { X } ) )
33		preq12 4270	. . . . . . 7 |- ( ( ( P ` 0 ) = X /\ ( P ` 1 ) = Y ) -> { ( P ` 0 ) , ( P ` 1 ) } = { X , Y } )
34	33	sseq1d 3632	. . . . . 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 1027	. . . . 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 693	. . . 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 247	. . 3 |- ( ph -> if- ( ( P ` 0 ) = ( P ` 1 ) , ( I ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( I ` ( F ` 0 ) ) ) )
38		c0ex 10034	. . . 4 |- 0 e. _V
39		oveq1 6657	. . . . . 6 |- ( k = 0 -> ( k + 1 ) = ( 0 + 1 ) )
40		0p1e1 11132	. . . . . 6 |- ( 0 + 1 ) = 1
41	39, 40	syl6eq 2672	. . . . 5 |- ( k = 0 -> ( k + 1 ) = 1 )
42		wkslem2 26504	. . . . 5 |- ( ( k = 0 /\ ( k + 1 ) = 1 ) -> (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 ) ) ) ) )
43	41, 42	mpdan 702	. . . 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 ) ) ) ) )
44	38, 43	ralsn 4222	. . 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 ) ) ) )
45	37, 44	sylibr 224	. 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 ) ) ) )
46	1	fveq2i 6194	. . . . . . 7 |- ( # ` F ) = ( # ` <" J "> )
47		s1len 13385	. . . . . . 7 |- ( # ` <" J "> ) = 1
48	46, 47	eqtri 2644	. . . . . 6 |- ( # ` F ) = 1
49	48	oveq2i 6661	. . . . 5 |- ( 0..^ ( # ` F ) ) = ( 0..^ 1 )
50		fzo01 12550	. . . . 5 |- ( 0..^ 1 ) = { 0 }
51	49, 50	eqtri 2644	. . . 4 |- ( 0..^ ( # ` F ) ) = { 0 }
52	51	a1i 11	. . 3 |- ( ph -> ( 0..^ ( # ` F ) ) = { 0 } )
53	52	raleqdv 3144	. 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 ) ) ) ) )
54	45, 53	mpbird 247	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 196   /\ wa 384  if-wif 1012   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   C_ wss 3574   {csn 4177   {cpr 4179   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939  ..^cfzo 12465   #chash 13117   <"cs1 13294   <"cs2 13586

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-ifp 1013  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-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

This theorem is referenced by:  1wlkd  27001