Theorem 3wlkdlem4 27022

Description: Lemma 4 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 ) ) )

Assertion

Ref	Expression
3wlkdlem4	|- ( ph -> A. k e. ( 0 ... ( # ` F ) ) ( P ` k ) e. V )

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 3wlkdlem4

Step	Hyp	Ref	Expression
1		3wlkd.s	. . 3 |- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )
2		3wlkd.p	. . . 4 |- P = <" A B C D ">
3		3wlkd.f	. . . 4 |- F = <" J K L ">
4	2, 3, 1	3wlkdlem3 27021	. . 3 |- ( ph -> ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) )
5		simpl 473	. . . . . . . . 9 |- ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) -> ( P ` 0 ) = A )
6	5	eleq1d 2686	. . . . . . . 8 |- ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) -> ( ( P ` 0 ) e. V <-> A e. V ) )
7		simpr 477	. . . . . . . . 9 |- ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) -> ( P ` 1 ) = B )
8	7	eleq1d 2686	. . . . . . . 8 |- ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) -> ( ( P ` 1 ) e. V <-> B e. V ) )
9	6, 8	anbi12d 747	. . . . . . 7 |- ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) -> ( ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V ) <-> ( A e. V /\ B e. V ) ) )
10	9	biimparc 504	. . . . . 6 |- ( ( ( A e. V /\ B e. V ) /\ ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) ) -> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V ) )
11		c0ex 10034	. . . . . . . 8 |- 0 e. _V
12		1ex 10035	. . . . . . . 8 |- 1 e. _V
13	11, 12	pm3.2i 471	. . . . . . 7 |- ( 0 e. _V /\ 1 e. _V )
14		fveq2 6191	. . . . . . . . 9 |- ( k = 0 -> ( P ` k ) = ( P ` 0 ) )
15	14	eleq1d 2686	. . . . . . . 8 |- ( k = 0 -> ( ( P ` k ) e. V <-> ( P ` 0 ) e. V ) )
16		fveq2 6191	. . . . . . . . 9 |- ( k = 1 -> ( P ` k ) = ( P ` 1 ) )
17	16	eleq1d 2686	. . . . . . . 8 |- ( k = 1 -> ( ( P ` k ) e. V <-> ( P ` 1 ) e. V ) )
18	15, 17	ralprg 4234	. . . . . . 7 |- ( ( 0 e. _V /\ 1 e. _V ) -> ( A. k e. { 0 , 1 } ( P ` k ) e. V <-> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V ) ) )
19	13, 18	mp1i 13	. . . . . 6 |- ( ( ( A e. V /\ B e. V ) /\ ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) ) -> ( A. k e. { 0 , 1 } ( P ` k ) e. V <-> ( ( P ` 0 ) e. V /\ ( P ` 1 ) e. V ) ) )
20	10, 19	mpbird 247	. . . . 5 |- ( ( ( A e. V /\ B e. V ) /\ ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) ) -> A. k e. { 0 , 1 } ( P ` k ) e. V )
21	20	ex 450	. . . 4 |- ( ( A e. V /\ B e. V ) -> ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) -> A. k e. { 0 , 1 } ( P ` k ) e. V ) )
22		simpl 473	. . . . . . . . 9 |- ( ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) -> ( P ` 2 ) = C )
23	22	eleq1d 2686	. . . . . . . 8 |- ( ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) -> ( ( P ` 2 ) e. V <-> C e. V ) )
24		simpr 477	. . . . . . . . 9 |- ( ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) -> ( P ` 3 ) = D )
25	24	eleq1d 2686	. . . . . . . 8 |- ( ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) -> ( ( P ` 3 ) e. V <-> D e. V ) )
26	23, 25	anbi12d 747	. . . . . . 7 |- ( ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) -> ( ( ( P ` 2 ) e. V /\ ( P ` 3 ) e. V ) <-> ( C e. V /\ D e. V ) ) )
27	26	biimparc 504	. . . . . 6 |- ( ( ( C e. V /\ D e. V ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( ( P ` 2 ) e. V /\ ( P ` 3 ) e. V ) )
28		2ex 11092	. . . . . . . 8 |- 2 e. _V
29		3ex 11096	. . . . . . . 8 |- 3 e. _V
30	28, 29	pm3.2i 471	. . . . . . 7 |- ( 2 e. _V /\ 3 e. _V )
31		fveq2 6191	. . . . . . . . 9 |- ( k = 2 -> ( P ` k ) = ( P ` 2 ) )
32	31	eleq1d 2686	. . . . . . . 8 |- ( k = 2 -> ( ( P ` k ) e. V <-> ( P ` 2 ) e. V ) )
33		fveq2 6191	. . . . . . . . 9 |- ( k = 3 -> ( P ` k ) = ( P ` 3 ) )
34	33	eleq1d 2686	. . . . . . . 8 |- ( k = 3 -> ( ( P ` k ) e. V <-> ( P ` 3 ) e. V ) )
35	32, 34	ralprg 4234	. . . . . . 7 |- ( ( 2 e. _V /\ 3 e. _V ) -> ( A. k e. { 2 , 3 } ( P ` k ) e. V <-> ( ( P ` 2 ) e. V /\ ( P ` 3 ) e. V ) ) )
36	30, 35	mp1i 13	. . . . . 6 |- ( ( ( C e. V /\ D e. V ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( A. k e. { 2 , 3 } ( P ` k ) e. V <-> ( ( P ` 2 ) e. V /\ ( P ` 3 ) e. V ) ) )
37	27, 36	mpbird 247	. . . . 5 |- ( ( ( C e. V /\ D e. V ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> A. k e. { 2 , 3 } ( P ` k ) e. V )
38	37	ex 450	. . . 4 |- ( ( C e. V /\ D e. V ) -> ( ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) -> A. k e. { 2 , 3 } ( P ` k ) e. V ) )
39	21, 38	im2anan9 880	. . 3 |- ( ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) -> ( ( ( ( P ` 0 ) = A /\ ( P ` 1 ) = B ) /\ ( ( P ` 2 ) = C /\ ( P ` 3 ) = D ) ) -> ( A. k e. { 0 , 1 } ( P ` k ) e. V /\ A. k e. { 2 , 3 } ( P ` k ) e. V ) ) )
40	1, 4, 39	sylc 65	. 2 |- ( ph -> ( A. k e. { 0 , 1 } ( P ` k ) e. V /\ A. k e. { 2 , 3 } ( P ` k ) e. V ) )
41	3	fveq2i 6194	. . . . . . 7 |- ( # ` F ) = ( # ` <" J K L "> )
42		s3len 13639	. . . . . . 7 |- ( # ` <" J K L "> ) = 3
43	41, 42	eqtri 2644	. . . . . 6 |- ( # ` F ) = 3
44	43	oveq2i 6661	. . . . 5 |- ( 0 ... ( # ` F ) ) = ( 0 ... 3 )
45		fz0to3un2pr 12441	. . . . 5 |- ( 0 ... 3 ) = ( { 0 , 1 } u. { 2 , 3 } )
46	44, 45	eqtri 2644	. . . 4 |- ( 0 ... ( # ` F ) ) = ( { 0 , 1 } u. { 2 , 3 } )
47	46	raleqi 3142	. . 3 |- ( A. k e. ( 0 ... ( # ` F ) ) ( P ` k ) e. V <-> A. k e. ( { 0 , 1 } u. { 2 , 3 } ) ( P ` k ) e. V )
48		ralunb 3794	. . 3 |- ( A. k e. ( { 0 , 1 } u. { 2 , 3 } ) ( P ` k ) e. V <-> ( A. k e. { 0 , 1 } ( P ` k ) e. V /\ A. k e. { 2 , 3 } ( P ` k ) e. V ) )
49	47, 48	bitri 264	. 2 |- ( A. k e. ( 0 ... ( # ` F ) ) ( P ` k ) e. V <-> ( A. k e. { 0 , 1 } ( P ` k ) e. V /\ A. k e. { 2 , 3 } ( P ` k ) e. V ) )
50	40, 49	sylibr 224	1 |- ( ph -> A. k e. ( 0 ... ( # ` F ) ) ( P ` k ) e. V )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   u. cun 3572   {cpr 4179   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   2c2 11070   3c3 11071   ...cfz 12326   #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