Theorem pthdlem2lem 26663

Description: Lemma for pthdlem2 26664. (Contributed by AV, 10-Feb-2021.)

Hypotheses

Ref	Expression
pthd.p	|- ( ph -> P e. Word _V )
pthd.r	|- R = ( ( # ` P ) - 1 )
pthd.s	|- ( ph -> A. i e. ( 0..^ ( # ` P ) ) A. j e. ( 1..^ R ) ( i =/= j -> ( P ` i ) =/= ( P ` j ) ) )

Assertion

Ref	Expression
pthdlem2lem	|- ( ( ph /\ ( # ` P ) e. NN /\ ( I = 0 \/ I = R ) ) -> ( P ` I ) e/ ( P " ( 1..^ R ) ) )

Distinct variable groups:   P, i, j   R, i, j   ph, i, j   i, I, j

Proof of Theorem pthdlem2lem

Step	Hyp	Ref	Expression
1		pthd.s	. . . . . 6 |- ( ph -> A. i e. ( 0..^ ( # ` P ) ) A. j e. ( 1..^ R ) ( i =/= j -> ( P ` i ) =/= ( P ` j ) ) )
2	1	3ad2ant1 1082	. . . . 5 |- ( ( ph /\ ( # ` P ) e. NN /\ ( I = 0 \/ I = R ) ) -> A. i e. ( 0..^ ( # ` P ) ) A. j e. ( 1..^ R ) ( i =/= j -> ( P ` i ) =/= ( P ` j ) ) )
3		ralcom 3098	. . . . . 6 |- ( A. i e. ( 0..^ ( # ` P ) ) A. j e. ( 1..^ R ) ( i =/= j -> ( P ` i ) =/= ( P ` j ) ) <-> A. j e. ( 1..^ R ) A. i e. ( 0..^ ( # ` P ) ) ( i =/= j -> ( P ` i ) =/= ( P ` j ) ) )
4		elfzo1 12517	. . . . . . . . . . . . . . . . 17 |- ( j e. ( 1..^ R ) <-> ( j e. NN /\ R e. NN /\ j < R ) )
5		nnne0 11053	. . . . . . . . . . . . . . . . . . 19 |- ( j e. NN -> j =/= 0 )
6	5	necomd 2849	. . . . . . . . . . . . . . . . . 18 |- ( j e. NN -> 0 =/= j )
7	6	3ad2ant1 1082	. . . . . . . . . . . . . . . . 17 |- ( ( j e. NN /\ R e. NN /\ j < R ) -> 0 =/= j )
8	4, 7	sylbi 207	. . . . . . . . . . . . . . . 16 |- ( j e. ( 1..^ R ) -> 0 =/= j )
9	8	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ( # ` P ) e. NN /\ j e. ( 1..^ R ) ) -> 0 =/= j )
10		neeq1 2856	. . . . . . . . . . . . . . 15 |- ( I = 0 -> ( I =/= j <-> 0 =/= j ) )
11	9, 10	syl5ibr 236	. . . . . . . . . . . . . 14 |- ( I = 0 -> ( ( ( # ` P ) e. NN /\ j e. ( 1..^ R ) ) -> I =/= j ) )
12	11	expd 452	. . . . . . . . . . . . 13 |- ( I = 0 -> ( ( # ` P ) e. NN -> ( j e. ( 1..^ R ) -> I =/= j ) ) )
13		nnre 11027	. . . . . . . . . . . . . . . . . . . . 21 |- ( j e. NN -> j e. RR )
14	13	adantr 481	. . . . . . . . . . . . . . . . . . . 20 |- ( ( j e. NN /\ R e. NN ) -> j e. RR )
15		nnre 11027	. . . . . . . . . . . . . . . . . . . . 21 |- ( R e. NN -> R e. RR )
16	15	adantl 482	. . . . . . . . . . . . . . . . . . . 20 |- ( ( j e. NN /\ R e. NN ) -> R e. RR )
17	14, 16	ltlend 10182	. . . . . . . . . . . . . . . . . . 19 |- ( ( j e. NN /\ R e. NN ) -> ( j < R <-> ( j <_ R /\ R =/= j ) ) )
18		simpr 477	. . . . . . . . . . . . . . . . . . 19 |- ( ( j <_ R /\ R =/= j ) -> R =/= j )
19	17, 18	syl6bi 243	. . . . . . . . . . . . . . . . . 18 |- ( ( j e. NN /\ R e. NN ) -> ( j < R -> R =/= j ) )
20	19	3impia 1261	. . . . . . . . . . . . . . . . 17 |- ( ( j e. NN /\ R e. NN /\ j < R ) -> R =/= j )
21	4, 20	sylbi 207	. . . . . . . . . . . . . . . 16 |- ( j e. ( 1..^ R ) -> R =/= j )
22	21	adantl 482	. . . . . . . . . . . . . . 15 |- ( ( ( # ` P ) e. NN /\ j e. ( 1..^ R ) ) -> R =/= j )
23		neeq1 2856	. . . . . . . . . . . . . . 15 |- ( I = R -> ( I =/= j <-> R =/= j ) )
24	22, 23	syl5ibr 236	. . . . . . . . . . . . . 14 |- ( I = R -> ( ( ( # ` P ) e. NN /\ j e. ( 1..^ R ) ) -> I =/= j ) )
25	24	expd 452	. . . . . . . . . . . . 13 |- ( I = R -> ( ( # ` P ) e. NN -> ( j e. ( 1..^ R ) -> I =/= j ) ) )
26	12, 25	jaoi 394	. . . . . . . . . . . 12 |- ( ( I = 0 \/ I = R ) -> ( ( # ` P ) e. NN -> ( j e. ( 1..^ R ) -> I =/= j ) ) )
27	26	impcom 446	. . . . . . . . . . 11 |- ( ( ( # ` P ) e. NN /\ ( I = 0 \/ I = R ) ) -> ( j e. ( 1..^ R ) -> I =/= j ) )
28	27	3adant1 1079	. . . . . . . . . 10 |- ( ( ph /\ ( # ` P ) e. NN /\ ( I = 0 \/ I = R ) ) -> ( j e. ( 1..^ R ) -> I =/= j ) )
29	28	imp 445	. . . . . . . . 9 |- ( ( ( ph /\ ( # ` P ) e. NN /\ ( I = 0 \/ I = R ) ) /\ j e. ( 1..^ R ) ) -> I =/= j )
30		lbfzo0 12507	. . . . . . . . . . . . . . . 16 |- ( 0 e. ( 0..^ ( # ` P ) ) <-> ( # ` P ) e. NN )
31	30	biimpri 218	. . . . . . . . . . . . . . 15 |- ( ( # ` P ) e. NN -> 0 e. ( 0..^ ( # ` P ) ) )
32		eleq1 2689	. . . . . . . . . . . . . . 15 |- ( I = 0 -> ( I e. ( 0..^ ( # ` P ) ) <-> 0 e. ( 0..^ ( # ` P ) ) ) )
33	31, 32	syl5ibr 236	. . . . . . . . . . . . . 14 |- ( I = 0 -> ( ( # ` P ) e. NN -> I e. ( 0..^ ( # ` P ) ) ) )
34		pthd.r	. . . . . . . . . . . . . . . 16 |- R = ( ( # ` P ) - 1 )
35		fzo0end 12560	. . . . . . . . . . . . . . . 16 |- ( ( # ` P ) e. NN -> ( ( # ` P ) - 1 ) e. ( 0..^ ( # ` P ) ) )
36	34, 35	syl5eqel 2705	. . . . . . . . . . . . . . 15 |- ( ( # ` P ) e. NN -> R e. ( 0..^ ( # ` P ) ) )
37		eleq1 2689	. . . . . . . . . . . . . . 15 |- ( I = R -> ( I e. ( 0..^ ( # ` P ) ) <-> R e. ( 0..^ ( # ` P ) ) ) )
38	36, 37	syl5ibr 236	. . . . . . . . . . . . . 14 |- ( I = R -> ( ( # ` P ) e. NN -> I e. ( 0..^ ( # ` P ) ) ) )
39	33, 38	jaoi 394	. . . . . . . . . . . . 13 |- ( ( I = 0 \/ I = R ) -> ( ( # ` P ) e. NN -> I e. ( 0..^ ( # ` P ) ) ) )
40	39	impcom 446	. . . . . . . . . . . 12 |- ( ( ( # ` P ) e. NN /\ ( I = 0 \/ I = R ) ) -> I e. ( 0..^ ( # ` P ) ) )
41	40	3adant1 1079	. . . . . . . . . . 11 |- ( ( ph /\ ( # ` P ) e. NN /\ ( I = 0 \/ I = R ) ) -> I e. ( 0..^ ( # ` P ) ) )
42	41	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ ( # ` P ) e. NN /\ ( I = 0 \/ I = R ) ) /\ j e. ( 1..^ R ) ) -> I e. ( 0..^ ( # ` P ) ) )
43		neeq1 2856	. . . . . . . . . . . 12 |- ( i = I -> ( i =/= j <-> I =/= j ) )
44		fveq2 6191	. . . . . . . . . . . . 13 |- ( i = I -> ( P ` i ) = ( P ` I ) )
45	44	neeq1d 2853	. . . . . . . . . . . 12 |- ( i = I -> ( ( P ` i ) =/= ( P ` j ) <-> ( P ` I ) =/= ( P ` j ) ) )
46	43, 45	imbi12d 334	. . . . . . . . . . 11 |- ( i = I -> ( ( i =/= j -> ( P ` i ) =/= ( P ` j ) ) <-> ( I =/= j -> ( P ` I ) =/= ( P ` j ) ) ) )
47	46	rspcv 3305	. . . . . . . . . 10 |- ( I e. ( 0..^ ( # ` P ) ) -> ( A. i e. ( 0..^ ( # ` P ) ) ( i =/= j -> ( P ` i ) =/= ( P ` j ) ) -> ( I =/= j -> ( P ` I ) =/= ( P ` j ) ) ) )
48	42, 47	syl 17	. . . . . . . . 9 |- ( ( ( ph /\ ( # ` P ) e. NN /\ ( I = 0 \/ I = R ) ) /\ j e. ( 1..^ R ) ) -> ( A. i e. ( 0..^ ( # ` P ) ) ( i =/= j -> ( P ` i ) =/= ( P ` j ) ) -> ( I =/= j -> ( P ` I ) =/= ( P ` j ) ) ) )
49	29, 48	mpid 44	. . . . . . . 8 |- ( ( ( ph /\ ( # ` P ) e. NN /\ ( I = 0 \/ I = R ) ) /\ j e. ( 1..^ R ) ) -> ( A. i e. ( 0..^ ( # ` P ) ) ( i =/= j -> ( P ` i ) =/= ( P ` j ) ) -> ( P ` I ) =/= ( P ` j ) ) )
50		nesym 2850	. . . . . . . 8 |- ( ( P ` I ) =/= ( P ` j ) <-> -. ( P ` j ) = ( P ` I ) )
51	49, 50	syl6ib 241	. . . . . . 7 |- ( ( ( ph /\ ( # ` P ) e. NN /\ ( I = 0 \/ I = R ) ) /\ j e. ( 1..^ R ) ) -> ( A. i e. ( 0..^ ( # ` P ) ) ( i =/= j -> ( P ` i ) =/= ( P ` j ) ) -> -. ( P ` j ) = ( P ` I ) ) )
52	51	ralimdva 2962	. . . . . 6 |- ( ( ph /\ ( # ` P ) e. NN /\ ( I = 0 \/ I = R ) ) -> ( A. j e. ( 1..^ R ) A. i e. ( 0..^ ( # ` P ) ) ( i =/= j -> ( P ` i ) =/= ( P ` j ) ) -> A. j e. ( 1..^ R ) -. ( P ` j ) = ( P ` I ) ) )
53	3, 52	syl5bi 232	. . . . 5 |- ( ( ph /\ ( # ` P ) e. NN /\ ( I = 0 \/ I = R ) ) -> ( A. i e. ( 0..^ ( # ` P ) ) A. j e. ( 1..^ R ) ( i =/= j -> ( P ` i ) =/= ( P ` j ) ) -> A. j e. ( 1..^ R ) -. ( P ` j ) = ( P ` I ) ) )
54	2, 53	mpd 15	. . . 4 |- ( ( ph /\ ( # ` P ) e. NN /\ ( I = 0 \/ I = R ) ) -> A. j e. ( 1..^ R ) -. ( P ` j ) = ( P ` I ) )
55		ralnex 2992	. . . 4 |- ( A. j e. ( 1..^ R ) -. ( P ` j ) = ( P ` I ) <-> -. E. j e. ( 1..^ R ) ( P ` j ) = ( P ` I ) )
56	54, 55	sylib 208	. . 3 |- ( ( ph /\ ( # ` P ) e. NN /\ ( I = 0 \/ I = R ) ) -> -. E. j e. ( 1..^ R ) ( P ` j ) = ( P ` I ) )
57		pthd.p	. . . . . 6 |- ( ph -> P e. Word _V )
58		wrdf 13310	. . . . . 6 |- ( P e. Word _V -> P : ( 0..^ ( # ` P ) ) --> _V )
59		ffun 6048	. . . . . 6 |- ( P : ( 0..^ ( # ` P ) ) --> _V -> Fun P )
60	57, 58, 59	3syl 18	. . . . 5 |- ( ph -> Fun P )
61	60	3ad2ant1 1082	. . . 4 |- ( ( ph /\ ( # ` P ) e. NN /\ ( I = 0 \/ I = R ) ) -> Fun P )
62		fvelima 6248	. . . . 5 |- ( ( Fun P /\ ( P ` I ) e. ( P " ( 1..^ R ) ) ) -> E. j e. ( 1..^ R ) ( P ` j ) = ( P ` I ) )
63	62	ex 450	. . . 4 |- ( Fun P -> ( ( P ` I ) e. ( P " ( 1..^ R ) ) -> E. j e. ( 1..^ R ) ( P ` j ) = ( P ` I ) ) )
64	61, 63	syl 17	. . 3 |- ( ( ph /\ ( # ` P ) e. NN /\ ( I = 0 \/ I = R ) ) -> ( ( P ` I ) e. ( P " ( 1..^ R ) ) -> E. j e. ( 1..^ R ) ( P ` j ) = ( P ` I ) ) )
65	56, 64	mtod 189	. 2 |- ( ( ph /\ ( # ` P ) e. NN /\ ( I = 0 \/ I = R ) ) -> -. ( P ` I ) e. ( P " ( 1..^ R ) ) )
66		df-nel 2898	. 2 |- ( ( P ` I ) e/ ( P " ( 1..^ R ) ) <-> -. ( P ` I ) e. ( P " ( 1..^ R ) ) )
67	65, 66	sylibr 224	1 |- ( ( ph /\ ( # ` P ) e. NN /\ ( I = 0 \/ I = R ) ) -> ( P ` I ) e/ ( P " ( 1..^ R ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   A.wral 2912   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   "cima 5117   Fun wfun 5882   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020  ..^cfzo 12465   #chash 13117  Word cword 13291

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-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

This theorem is referenced by:  pthdlem2  26664