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Mirrors > Home > MPE Home > Th. List > Mathboxes > cycpmfvlem | Structured version Visualization version GIF version |
Description: Lemma for cycpmfv1 32992 and cycpmfv2 32993. (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 32988 | . . 3 ⊢ (𝜑 → (𝐶‘𝑊) = (( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))) |
6 | 5 | fveq1d 6892 | . 2 ⊢ (𝜑 → ((𝐶‘𝑊)‘(𝑊‘𝑁)) = ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))‘(𝑊‘𝑁))) |
7 | f1oi 6870 | . . . 4 ⊢ ( I ↾ (𝐷 ∖ ran 𝑊)):(𝐷 ∖ ran 𝑊)–1-1-onto→(𝐷 ∖ ran 𝑊) | |
8 | f1ofn 6833 | . . . 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 12636 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ) | |
11 | cshwf 14800 | . . . . . 6 ⊢ ((𝑊 ∈ Word 𝐷 ∧ 1 ∈ ℤ) → (𝑊 cyclShift 1):(0..^(♯‘𝑊))⟶𝐷) | |
12 | 3, 10, 11 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → (𝑊 cyclShift 1):(0..^(♯‘𝑊))⟶𝐷) |
13 | 12 | ffnd 6718 | . . . 4 ⊢ (𝜑 → (𝑊 cyclShift 1) Fn (0..^(♯‘𝑊))) |
14 | df-f1 6548 | . . . . . . . 8 ⊢ (𝑊:dom 𝑊–1-1→𝐷 ↔ (𝑊:dom 𝑊⟶𝐷 ∧ Fun ◡𝑊)) | |
15 | 4, 14 | sylib 217 | . . . . . . 7 ⊢ (𝜑 → (𝑊:dom 𝑊⟶𝐷 ∧ Fun ◡𝑊)) |
16 | 15 | simprd 494 | . . . . . 6 ⊢ (𝜑 → Fun ◡𝑊) |
17 | 16 | funfnd 6579 | . . . . 5 ⊢ (𝜑 → ◡𝑊 Fn dom ◡𝑊) |
18 | df-rn 5683 | . . . . . 6 ⊢ ran 𝑊 = dom ◡𝑊 | |
19 | 18 | fneq2i 6647 | . . . . 5 ⊢ (◡𝑊 Fn ran 𝑊 ↔ ◡𝑊 Fn dom ◡𝑊) |
20 | 17, 19 | sylibr 233 | . . . 4 ⊢ (𝜑 → ◡𝑊 Fn ran 𝑊) |
21 | dfdm4 5892 | . . . . . 6 ⊢ dom 𝑊 = ran ◡𝑊 | |
22 | 21 | eqimss2i 4040 | . . . . 5 ⊢ ran ◡𝑊 ⊆ dom 𝑊 |
23 | wrdfn 14528 | . . . . . . 7 ⊢ (𝑊 ∈ Word 𝐷 → 𝑊 Fn (0..^(♯‘𝑊))) | |
24 | 3, 23 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 Fn (0..^(♯‘𝑊))) |
25 | 24 | fndmd 6654 | . . . . 5 ⊢ (𝜑 → dom 𝑊 = (0..^(♯‘𝑊))) |
26 | 22, 25 | sseqtrid 4031 | . . . 4 ⊢ (𝜑 → ran ◡𝑊 ⊆ (0..^(♯‘𝑊))) |
27 | fnco 6667 | . . . 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 1368 | . . 3 ⊢ (𝜑 → ((𝑊 cyclShift 1) ∘ ◡𝑊) Fn ran 𝑊) |
29 | disjdifr 4467 | . . . 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 7083 | . . . 4 ⊢ ((𝑊 Fn (0..^(♯‘𝑊)) ∧ 𝑁 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝑁) ∈ ran 𝑊) | |
33 | 24, 31, 32 | syl2anc 582 | . . 3 ⊢ (𝜑 → (𝑊‘𝑁) ∈ ran 𝑊) |
34 | fvun2 6983 | . . 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 1371 | . 2 ⊢ (𝜑 → ((( I ↾ (𝐷 ∖ ran 𝑊)) ∪ ((𝑊 cyclShift 1) ∘ ◡𝑊))‘(𝑊‘𝑁)) = (((𝑊 cyclShift 1) ∘ ◡𝑊)‘(𝑊‘𝑁))) |
36 | 6, 35 | eqtrd 2766 | 1 ⊢ (𝜑 → ((𝐶‘𝑊)‘(𝑊‘𝑁)) = (((𝑊 cyclShift 1) ∘ ◡𝑊)‘(𝑊‘𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∖ cdif 3943 ∪ cun 3944 ∩ cin 3945 ⊆ wss 3946 ∅c0 4322 I cid 5569 ◡ccnv 5671 dom cdm 5672 ran crn 5673 ↾ cres 5674 ∘ ccom 5676 Fun wfun 6537 Fn wfn 6538 ⟶wf 6539 –1-1→wf1 6540 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7413 0cc0 11146 1c1 11147 ℤcz 12601 ..^cfzo 13672 ♯chash 14339 Word cword 14514 cyclShift ccsh 14788 toCycctocyc 32985 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6302 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7866 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-sup 9475 df-inf 9476 df-card 9972 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12256 df-n0 12516 df-z 12602 df-uz 12866 df-rp 13020 df-fz 13530 df-fzo 13673 df-fl 13803 df-mod 13881 df-hash 14340 df-word 14515 df-concat 14571 df-substr 14641 df-pfx 14671 df-csh 14789 df-tocyc 32986 |
This theorem is referenced by: cycpmfv1 32992 cycpmfv2 32993 |
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