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Mirrors > Home > MPE Home > Th. List > Mathboxes > cycpmfvlem | Structured version Visualization version GIF version |
Description: Lemma for cycpmfv1 31099 and cycpmfv2 31100. (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 31095 | . . 3 ⊢ (𝜑 → (𝐶‘𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))) |
6 | 5 | fveq1d 6719 | . 2 ⊢ (𝜑 → ((𝐶‘𝑊)‘(𝑊‘𝑁)) = ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))‘(𝑊‘𝑁))) |
7 | f1oi 6698 | . . . 4 ⊢ ( I ↾ (𝐷 ∖ ran 𝑊)):(𝐷 ∖ ran 𝑊)–1-1-onto→(𝐷 ∖ ran 𝑊) | |
8 | f1ofn 6662 | . . . 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 12208 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ) | |
11 | cshwf 14365 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 1 ∈ ℤ) → (𝑊 cyclShift 1):(0..^(♯‘𝑊))⟶𝐷) | |
12 | 3, 10, 11 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → (𝑊 cyclShift 1):(0..^(♯‘𝑊))⟶𝐷) |
13 | 12 | ffnd 6546 | . . . 4 ⊢ (𝜑 → (𝑊 cyclShift 1) Fn (0..^(♯‘𝑊))) |
14 | df-f1 6385 | . . . . . . . 8 ⊢ (𝑊:dom 𝑊–1-1→𝐷 ↔ (𝑊:dom 𝑊⟶𝐷 ∧ Fun ◡𝑊)) | |
15 | 4, 14 | sylib 221 | . . . . . . 7 ⊢ (𝜑 → (𝑊:dom 𝑊⟶𝐷 ∧ Fun ◡𝑊)) |
16 | 15 | simprd 499 | . . . . . 6 ⊢ (𝜑 → Fun ◡𝑊) |
17 | 16 | funfnd 6411 | . . . . 5 ⊢ (𝜑 → ◡𝑊 Fn dom ◡𝑊) |
18 | df-rn 5562 | . . . . . 6 ⊢ ran 𝑊 = dom ◡𝑊 | |
19 | 18 | fneq2i 6477 | . . . . 5 ⊢ (◡𝑊 Fn ran 𝑊 ↔ ◡𝑊 Fn dom ◡𝑊) |
20 | 17, 19 | sylibr 237 | . . . 4 ⊢ (𝜑 → ◡𝑊 Fn ran 𝑊) |
21 | dfdm4 5764 | . . . . . 6 ⊢ dom 𝑊 = ran ◡𝑊 | |
22 | 21 | eqimss2i 3960 | . . . . 5 ⊢ ran ◡𝑊 ⊆ dom 𝑊 |
23 | wrdfn 14083 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝐷 → 𝑊 Fn (0..^(♯‘𝑊))) | |
24 | 3, 23 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 Fn (0..^(♯‘𝑊))) |
25 | 24 | fndmd 6483 | . . . . 5 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊))) |
26 | 22, 25 | sseqtrid 3953 | . . . 4 ⊢ (𝜑 → ran ◡𝑊 ⊆ (0..^(♯‘𝑊))) |
27 | fnco 6494 | . . . 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 1373 | . . 3 ⊢ (𝜑 → ((𝑊 cyclShift 1) ∘ ◡𝑊) Fn ran 𝑊) |
29 | disjdifr 4387 | . . . 4 ⊢ ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) = ∅ | |
30 | 29 | a1i 11 | . . 3 ⊢ (𝜑 → ((𝐷 ∖ ran 𝑊) ∩ ran 𝑊) = ∅) |
31 | cycpmfvlem.1 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝑊))) | |
32 | fnfvelrn 6901 | . . . 4 ⊢ ((𝑊 Fn (0..^(♯‘𝑊)) ∧ 𝑁 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝑁) ∈ ran 𝑊) | |
33 | 24, 31, 32 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑊‘𝑁) ∈ ran 𝑊) |
34 | fvun2 6803 | . . 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))) | |
35 | 9, 28, 30, 33, 34 | syl112anc 1376 | . 2 ⊢ (𝜑 → ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))‘(𝑊‘𝑁)) = (((𝑊 cyclShift 1) ∘ ◡𝑊)‘(𝑊‘𝑁))) |
36 | 6, 35 | eqtrd 2777 | 1 ⊢ (𝜑 → ((𝐶‘𝑊)‘(𝑊‘𝑁)) = (((𝑊 cyclShift 1) ∘ ◡𝑊)‘(𝑊‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∖ cdif 3863 ∪ cun 3864 ∩ cin 3865 ⊆ wss 3866 ∅c0 4237 I cid 5454 ◡ccnv 5550 dom cdm 5551 ran crn 5552 ↾ cres 5553 ∘ ccom 5555 Fun wfun 6374 Fn wfn 6375 ⟶wf 6376 –1-1→wf1 6377 –1-1-onto→wf1o 6379 ‘cfv 6380 (class class class)co 7213 0cc0 10729 1c1 10730 ℤcz 12176 ..^cfzo 13238 ♯chash 13896 Word cword 14069 cyclShift ccsh 14353 toCycctocyc 31092 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-inf 9059 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-fz 13096 df-fzo 13239 df-fl 13367 df-mod 13443 df-hash 13897 df-word 14070 df-concat 14126 df-substr 14206 df-pfx 14236 df-csh 14354 df-tocyc 31093 |
This theorem is referenced by: cycpmfv1 31099 cycpmfv2 31100 |
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