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Mirrors > Home > MPE Home > Th. List > Mathboxes > cycpmfvlem | Structured version Visualization version GIF version |
Description: Lemma for cycpmfv1 30366 and cycpmfv2 30367. (Contributed by Thierry Arnoux, 22-Sep-2023.) |
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) |
cycpmfvlem.1 | ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝑊))) |
Ref | Expression |
---|---|
cycpmfvlem | ⊢ (𝜑 → ((𝐶‘𝑊)‘(𝑊‘𝑁)) = (((𝑊 cyclShift 1) ∘ ◡𝑊)‘(𝑊‘𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tocycval.1 | . . . 4 ⊢ 𝐶 = (toCyc‘𝐷) | |
2 | tocycfv.d | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉) | |
3 | tocycfv.w | . . . 4 ⊢ (𝜑 → 𝑊 ∈ Word 𝐷) | |
4 | tocycfv.1 | . . . 4 ⊢ (𝜑 → 𝑊:dom 𝑊–1-1→𝐷) | |
5 | 1, 2, 3, 4 | tocycfv 30362 | . . 3 ⊢ (𝜑 → (𝐶‘𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))) |
6 | 5 | fveq1d 6532 | . 2 ⊢ (𝜑 → ((𝐶‘𝑊)‘(𝑊‘𝑁)) = ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))‘(𝑊‘𝑁))) |
7 | f1oi 6512 | . . . 4 ⊢ ( I ↾ (𝐷 ∖ ran 𝑊)):(𝐷 ∖ ran 𝑊)–1-1-onto→(𝐷 ∖ ran 𝑊) | |
8 | f1ofn 6476 | . . . 4 ⊢ (( I ↾ (𝐷 ∖ ran 𝑊)):(𝐷 ∖ ran 𝑊)–1-1-onto→(𝐷 ∖ ran 𝑊) → ( I ↾ (𝐷 ∖ ran 𝑊)) Fn (𝐷 ∖ ran 𝑊)) | |
9 | 7, 8 | mp1i 13 | . . 3 ⊢ (𝜑 → ( I ↾ (𝐷 ∖ ran 𝑊)) Fn (𝐷 ∖ ran 𝑊)) |
10 | 1zzd 11851 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ) | |
11 | cshwf 13986 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 1 ∈ ℤ) → (𝑊 cyclShift 1):(0..^(♯‘𝑊))⟶𝐷) | |
12 | 3, 10, 11 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑊 cyclShift 1):(0..^(♯‘𝑊))⟶𝐷) |
13 | 12 | ffnd 6375 | . . . 4 ⊢ (𝜑 → (𝑊 cyclShift 1) Fn (0..^(♯‘𝑊))) |
14 | df-f1 6222 | . . . . . . . 8 ⊢ (𝑊:dom 𝑊–1-1→𝐷 ↔ (𝑊:dom 𝑊⟶𝐷 ∧ Fun ◡𝑊)) | |
15 | 4, 14 | sylib 219 | . . . . . . 7 ⊢ (𝜑 → (𝑊:dom 𝑊⟶𝐷 ∧ Fun ◡𝑊)) |
16 | 15 | simprd 496 | . . . . . 6 ⊢ (𝜑 → Fun ◡𝑊) |
17 | 16 | funfnd 6248 | . . . . 5 ⊢ (𝜑 → ◡𝑊 Fn dom ◡𝑊) |
18 | df-rn 5446 | . . . . . 6 ⊢ ran 𝑊 = dom ◡𝑊 | |
19 | 18 | fneq2i 6313 | . . . . 5 ⊢ (◡𝑊 Fn ran 𝑊 ↔ ◡𝑊 Fn dom ◡𝑊) |
20 | 17, 19 | sylibr 235 | . . . 4 ⊢ (𝜑 → ◡𝑊 Fn ran 𝑊) |
21 | dfdm4 5642 | . . . . . 6 ⊢ dom 𝑊 = ran ◡𝑊 | |
22 | 21 | eqimss2i 3942 | . . . . 5 ⊢ ran ◡𝑊 ⊆ dom 𝑊 |
23 | wrdfn 13710 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝐷 → 𝑊 Fn (0..^(♯‘𝑊))) | |
24 | 3, 23 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 Fn (0..^(♯‘𝑊))) |
25 | 24 | fndmd 6318 | . . . . 5 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊))) |
26 | 22, 25 | sseqtrid 3935 | . . . 4 ⊢ (𝜑 → ran ◡𝑊 ⊆ (0..^(♯‘𝑊))) |
27 | fnco 6327 | . . . 4 ⊢ (((𝑊 cyclShift 1) Fn (0..^(♯‘𝑊)) ∧ ◡𝑊 Fn ran 𝑊 ∧ ran ◡𝑊 ⊆ (0..^(♯‘𝑊))) → ((𝑊 cyclShift 1) ∘ ◡𝑊) Fn ran 𝑊) | |
28 | 13, 20, 26, 27 | syl3anc 1362 | . . 3 ⊢ (𝜑 → ((𝑊 cyclShift 1) ∘ ◡𝑊) Fn ran 𝑊) |
29 | incom 4094 | . . . . 5 ⊢ (ran 𝑊 ∩ (𝐷 ∖ ran 𝑊)) = ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) | |
30 | disjdif 4329 | . . . . 5 ⊢ (ran 𝑊 ∩ (𝐷 ∖ ran 𝑊)) = ∅ | |
31 | 29, 30 | eqtr3i 2819 | . . . 4 ⊢ ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) = ∅ |
32 | 31 | a1i 11 | . . 3 ⊢ (𝜑 → ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) = ∅) |
33 | cycpmfvlem.1 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝑊))) | |
34 | fnfvelrn 6704 | . . . 4 ⊢ ((𝑊 Fn (0..^(♯‘𝑊)) ∧ 𝑁 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝑁) ∈ ran 𝑊) | |
35 | 24, 33, 34 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑊‘𝑁) ∈ ran 𝑊) |
36 | fvun2 6614 | . . 3 ⊢ ((( I ↾ (𝐷 ∖ ran 𝑊)) Fn (𝐷 ∖ ran 𝑊) ∧ ((𝑊 cyclShift 1) ∘ ◡𝑊) Fn ran 𝑊 ∧ (((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) = ∅ ∧ (𝑊‘𝑁) ∈ ran 𝑊)) → ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))‘(𝑊‘𝑁)) = (((𝑊 cyclShift 1) ∘ ◡𝑊)‘(𝑊‘𝑁))) | |
37 | 9, 28, 32, 35, 36 | syl112anc 1365 | . 2 ⊢ (𝜑 → ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))‘(𝑊‘𝑁)) = (((𝑊 cyclShift 1) ∘ ◡𝑊)‘(𝑊‘𝑁))) |
38 | 6, 37 | eqtrd 2829 | 1 ⊢ (𝜑 → ((𝐶‘𝑊)‘(𝑊‘𝑁)) = (((𝑊 cyclShift 1) ∘ ◡𝑊)‘(𝑊‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1520 ∈ wcel 2079 ∖ cdif 3851 ∪ cun 3852 ∩ cin 3853 ⊆ wss 3854 ∅c0 4206 I cid 5339 ◡ccnv 5434 dom cdm 5435 ran crn 5436 ↾ cres 5437 ∘ ccom 5439 Fun wfun 6211 Fn wfn 6212 ⟶wf 6213 –1-1→wf1 6214 –1-1-onto→wf1o 6216 ‘cfv 6217 (class class class)co 7007 0cc0 10372 1c1 10373 ℤcz 11818 ..^cfzo 12872 ♯chash 13528 Word cword 13695 cyclShift ccsh 13974 toCycctocyc 30359 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-rep 5075 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 ax-cnex 10428 ax-resscn 10429 ax-1cn 10430 ax-icn 10431 ax-addcl 10432 ax-addrcl 10433 ax-mulcl 10434 ax-mulrcl 10435 ax-mulcom 10436 ax-addass 10437 ax-mulass 10438 ax-distr 10439 ax-i2m1 10440 ax-1ne0 10441 ax-1rid 10442 ax-rnegex 10443 ax-rrecex 10444 ax-cnre 10445 ax-pre-lttri 10446 ax-pre-lttrn 10447 ax-pre-ltadd 10448 ax-pre-mulgt0 10449 ax-pre-sup 10450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1079 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-nel 3089 df-ral 3108 df-rex 3109 df-reu 3110 df-rmo 3111 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-pss 3871 df-nul 4207 df-if 4376 df-pw 4449 df-sn 4467 df-pr 4469 df-tp 4471 df-op 4473 df-uni 4740 df-int 4777 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-tr 5058 df-id 5340 df-eprel 5345 df-po 5354 df-so 5355 df-fr 5394 df-we 5396 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-res 5447 df-ima 5448 df-pred 6015 df-ord 6061 df-on 6062 df-lim 6063 df-suc 6064 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-f1 6222 df-fo 6223 df-f1o 6224 df-fv 6225 df-riota 6968 df-ov 7010 df-oprab 7011 df-mpo 7012 df-om 7428 df-1st 7536 df-2nd 7537 df-wrecs 7789 df-recs 7851 df-rdg 7889 df-1o 7944 df-oadd 7948 df-er 8130 df-map 8249 df-en 8348 df-dom 8349 df-sdom 8350 df-fin 8351 df-sup 8742 df-inf 8743 df-card 9203 df-pnf 10512 df-mnf 10513 df-xr 10514 df-ltxr 10515 df-le 10516 df-sub 10708 df-neg 10709 df-div 11135 df-nn 11476 df-n0 11735 df-z 11819 df-uz 12083 df-rp 12229 df-fz 12732 df-fzo 12873 df-fl 13000 df-mod 13076 df-hash 13529 df-word 13696 df-concat 13757 df-substr 13827 df-pfx 13857 df-csh 13975 df-tocyc 30360 |
This theorem is referenced by: cycpmfv1 30366 cycpmfv2 30367 |
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