Theorem cycpmcl 30758

Description: Cyclic permutations are permutations. (Contributed by Thierry Arnoux, 24-Sep-2023.)

Hypotheses

Ref	Expression
tocycval.1	⊢ 𝐶 = (toCyc‘𝐷)
tocycfv.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
tocycfv.w	⊢ (𝜑 → 𝑊 ∈ Word 𝐷)
tocycfv.1	⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)
cycpmcl.s	⊢ 𝑆 = (SymGrp‘𝐷)

Assertion

Ref	Expression
cycpmcl	⊢ (𝜑 → (𝐶‘𝑊) ∈ (Base‘𝑆))

Proof of Theorem cycpmcl

Step	Hyp	Ref	Expression
1		f1oi 6652	. . . . 5 ⊢ ( I ↾ (𝐷 ∖ ran 𝑊)):(𝐷 ∖ ran 𝑊)–1-1-onto→(𝐷 ∖ ran 𝑊)
2	1	a1i 11	. . . 4 ⊢ (𝜑 → ( I ↾ (𝐷 ∖ ran 𝑊)):(𝐷 ∖ ran 𝑊)–1-1-onto→(𝐷 ∖ ran 𝑊))
3		tocycfv.w	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷)
4		1zzd 12014	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ ℤ)
5		cshwf 14162	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 1 ∈ ℤ) → (𝑊 cyclShift 1):(0..^(♯‘𝑊))⟶𝐷)
6	3, 4, 5	syl2anc 586	. . . . . . . 8 ⊢ (𝜑 → (𝑊 cyclShift 1):(0..^(♯‘𝑊))⟶𝐷)
7	6	ffnd 6515	. . . . . . 7 ⊢ (𝜑 → (𝑊 cyclShift 1) Fn (0..^(♯‘𝑊)))
8		tocycfv.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷)
9		df-f1 6360	. . . . . . . . . 10 ⊢ (𝑊:dom 𝑊–1-1→𝐷 ↔ (𝑊:dom 𝑊⟶𝐷 ∧ Fun ◡𝑊))
10	8, 9	sylib 220	. . . . . . . . 9 ⊢ (𝜑 → (𝑊:dom 𝑊⟶𝐷 ∧ Fun ◡𝑊))
11	10	simprd 498	. . . . . . . 8 ⊢ (𝜑 → Fun ◡𝑊)
12		eqid 2821	. . . . . . . . 9 ⊢ (𝑊 cyclShift 1) = (𝑊 cyclShift 1)
13		cshinj 14173	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝐷 ∧ Fun ◡𝑊 ∧ 1 ∈ ℤ) → ((𝑊 cyclShift 1) = (𝑊 cyclShift 1) → Fun ◡(𝑊 cyclShift 1)))
14	12, 13	mpi 20	. . . . . . . 8 ⊢ ((𝑊 ∈ Word 𝐷 ∧ Fun ◡𝑊 ∧ 1 ∈ ℤ) → Fun ◡(𝑊 cyclShift 1))
15	3, 11, 4, 14	syl3anc 1367	. . . . . . 7 ⊢ (𝜑 → Fun ◡(𝑊 cyclShift 1))
16		f1orn 6625	. . . . . . 7 ⊢ ((𝑊 cyclShift 1):(0..^(♯‘𝑊))–1-1-onto→ran (𝑊 cyclShift 1) ↔ ((𝑊 cyclShift 1) Fn (0..^(♯‘𝑊)) ∧ Fun ◡(𝑊 cyclShift 1)))
17	7, 15, 16	sylanbrc 585	. . . . . 6 ⊢ (𝜑 → (𝑊 cyclShift 1):(0..^(♯‘𝑊))–1-1-onto→ran (𝑊 cyclShift 1))
18		eqidd 2822	. . . . . . 7 ⊢ (𝜑 → (𝑊 cyclShift 1) = (𝑊 cyclShift 1))
19		wrdf 13867	. . . . . . . . 9 ⊢ (𝑊 ∈ Word 𝐷 → 𝑊:(0..^(♯‘𝑊))⟶𝐷)
20	3, 19	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑊:(0..^(♯‘𝑊))⟶𝐷)
21	20	fdmd 6523	. . . . . . 7 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊)))
22		cshwrnid 30635	. . . . . . . . 9 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 1 ∈ ℤ) → ran (𝑊 cyclShift 1) = ran 𝑊)
23	3, 4, 22	syl2anc 586	. . . . . . . 8 ⊢ (𝜑 → ran (𝑊 cyclShift 1) = ran 𝑊)
24	23	eqcomd 2827	. . . . . . 7 ⊢ (𝜑 → ran 𝑊 = ran (𝑊 cyclShift 1))
25	18, 21, 24	f1oeq123d 6610	. . . . . 6 ⊢ (𝜑 → ((𝑊 cyclShift 1):dom 𝑊–1-1-onto→ran 𝑊 ↔ (𝑊 cyclShift 1):(0..^(♯‘𝑊))–1-1-onto→ran (𝑊 cyclShift 1)))
26	17, 25	mpbird 259	. . . . 5 ⊢ (𝜑 → (𝑊 cyclShift 1):dom 𝑊–1-1-onto→ran 𝑊)
27		f1f1orn 6626	. . . . . 6 ⊢ (𝑊:dom 𝑊–1-1→𝐷 → 𝑊:dom 𝑊–1-1-onto→ran 𝑊)
28		f1ocnv 6627	. . . . . 6 ⊢ (𝑊:dom 𝑊–1-1-onto→ran 𝑊 → ◡𝑊:ran 𝑊–1-1-onto→dom 𝑊)
29	8, 27, 28	3syl 18	. . . . 5 ⊢ (𝜑 → ◡𝑊:ran 𝑊–1-1-onto→dom 𝑊)
30		f1oco 6637	. . . . 5 ⊢ (((𝑊 cyclShift 1):dom 𝑊–1-1-onto→ran 𝑊 ∧ ◡𝑊:ran 𝑊–1-1-onto→dom 𝑊) → ((𝑊 cyclShift 1) ∘ ◡𝑊):ran 𝑊–1-1-onto→ran 𝑊)
31	26, 29, 30	syl2anc 586	. . . 4 ⊢ (𝜑 → ((𝑊 cyclShift 1) ∘ ◡𝑊):ran 𝑊–1-1-onto→ran 𝑊)
32		incom 4178	. . . . . 6 ⊢ (ran 𝑊 ∩ (𝐷 ∖ ran 𝑊)) = ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊)
33		disjdif 4421	. . . . . 6 ⊢ (ran 𝑊 ∩ (𝐷 ∖ ran 𝑊)) = ∅
34	32, 33	eqtr3i 2846	. . . . 5 ⊢ ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) = ∅
35	34	a1i 11	. . . 4 ⊢ (𝜑 → ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) = ∅)
36		f1oun 6634	. . . 4 ⊢ (((( I ↾ (𝐷 ∖ ran 𝑊)):(𝐷 ∖ ran 𝑊)–1-1-onto→(𝐷 ∖ ran 𝑊) ∧ ((𝑊 cyclShift 1) ∘ ◡𝑊):ran 𝑊–1-1-onto→ran 𝑊) ∧ (((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) = ∅ ∧ ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) = ∅)) → (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊)):((𝐷 ∖ ran 𝑊) ∪ ran 𝑊)–1-1-onto→((𝐷 ∖ ran 𝑊) ∪ ran 𝑊))
37	2, 31, 35, 35, 36	syl22anc 836	. . 3 ⊢ (𝜑 → (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊)):((𝐷 ∖ ran 𝑊) ∪ ran 𝑊)–1-1-onto→((𝐷 ∖ ran 𝑊) ∪ ran 𝑊))
38		tocycval.1	. . . . 5 ⊢ 𝐶 = (toCyc‘𝐷)
39		tocycfv.d	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
40	38, 39, 3, 8	tocycfv 30751	. . . 4 ⊢ (𝜑 → (𝐶‘𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊)))
41		uncom 4129	. . . . 5 ⊢ (ran 𝑊 ∪ (𝐷 ∖ ran 𝑊)) = ((𝐷 ∖ ran 𝑊) ∪ ran 𝑊)
42	20	frnd 6521	. . . . . 6 ⊢ (𝜑 → ran 𝑊 ⊆ 𝐷)
43		undif 4430	. . . . . 6 ⊢ (ran 𝑊 ⊆ 𝐷 ↔ (ran 𝑊 ∪ (𝐷 ∖ ran 𝑊)) = 𝐷)
44	42, 43	sylib 220	. . . . 5 ⊢ (𝜑 → (ran 𝑊 ∪ (𝐷 ∖ ran 𝑊)) = 𝐷)
45	41, 44	syl5reqr 2871	. . . 4 ⊢ (𝜑 → 𝐷 = ((𝐷 ∖ ran 𝑊) ∪ ran 𝑊))
46	40, 45, 45	f1oeq123d 6610	. . 3 ⊢ (𝜑 → ((𝐶‘𝑊):𝐷–1-1-onto→𝐷 ↔ (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊)):((𝐷 ∖ ran 𝑊) ∪ ran 𝑊)–1-1-onto→((𝐷 ∖ ran 𝑊) ∪ ran 𝑊)))
47	37, 46	mpbird 259	. 2 ⊢ (𝜑 → (𝐶‘𝑊):𝐷–1-1-onto→𝐷)
48		fvex 6683	. . 3 ⊢ (𝐶‘𝑊) ∈ V
49		cycpmcl.s	. . . 4 ⊢ 𝑆 = (SymGrp‘𝐷)
50		eqid 2821	. . . 4 ⊢ (Base‘𝑆) = (Base‘𝑆)
51	49, 50	elsymgbas2 18501	. . 3 ⊢ ((𝐶‘𝑊) ∈ V → ((𝐶‘𝑊) ∈ (Base‘𝑆) ↔ (𝐶‘𝑊):𝐷–1-1-onto→𝐷))
52	48, 51	ax-mp 5	. 2 ⊢ ((𝐶‘𝑊) ∈ (Base‘𝑆) ↔ (𝐶‘𝑊):𝐷–1-1-onto→𝐷)
53	47, 52	sylibr 236	1 ⊢ (𝜑 → (𝐶‘𝑊) ∈ (Base‘𝑆))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  Vcvv 3494   ∖ cdif 3933   ∪ cun 3934   ∩ cin 3935   ⊆ wss 3936  ∅c0 4291   I cid 5459  ^◡ccnv 5554  dom cdm 5555  ran crn 5556   ↾ cres 5557   ∘ ccom 5559  Fun wfun 6349   Fn wfn 6350  ⟶wf 6351  –1-1→wf1 6352  –1-1-onto→wf1o 6354  ‘cfv 6355  (class class class)co 7156  0cc0 10537  1c1 10538  ℤcz 11982  ..^cfzo 13034  ♯chash 13691  Word cword 13862   cyclShift ccsh 14150  Basecbs 16483  SymGrpcsymg 18495  toCycctocyc 30748

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614  ax-pre-sup 10615

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-sup 8906  df-inf 8907  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-div 11298  df-nn 11639  df-2 11701  df-3 11702  df-4 11703  df-5 11704  df-6 11705  df-7 11706  df-8 11707  df-9 11708  df-n0 11899  df-z 11983  df-uz 12245  df-rp 12391  df-fz 12894  df-fzo 13035  df-fl 13163  df-mod 13239  df-hash 13692  df-word 13863  df-concat 13923  df-substr 14003  df-pfx 14033  df-csh 14151  df-struct 16485  df-ndx 16486  df-slot 16487  df-base 16489  df-sets 16490  df-ress 16491  df-plusg 16578  df-tset 16584  df-efmnd 18034  df-symg 18496  df-tocyc 30749

This theorem is referenced by:  tocycf  30759  cycpm2cl  30762  cycpmco2  30775  cycpm3cl  30777  cycpmrn  30785  cyc3evpm  30792  cycpmgcl  30795