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Mirrors > Home > MPE Home > Th. List > efgsf | Structured version Visualization version GIF version |
Description: Value of the auxiliary function 𝑆 defining a sequence of extensions starting at some irreducible word. (Contributed by Mario Carneiro, 1-Oct-2015.) |
Ref | Expression |
---|---|
efgval.w | ⊢ 𝑊 = ( I ‘Word (𝐼 × 2𝑜)) |
efgval.r | ⊢ ∼ = ( ~FG ‘𝐼) |
efgval2.m | ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ 〈𝑦, (1𝑜 ∖ 𝑧)〉) |
efgval2.t | ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) |
efgred.d | ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) |
efgred.s | ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1))) |
Ref | Expression |
---|---|
efgsf | ⊢ 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . . . . 6 ⊢ (𝑚 = 𝑡 → 𝑚 = 𝑡) | |
2 | fveq2 6191 | . . . . . . 7 ⊢ (𝑚 = 𝑡 → (#‘𝑚) = (#‘𝑡)) | |
3 | 2 | oveq1d 6665 | . . . . . 6 ⊢ (𝑚 = 𝑡 → ((#‘𝑚) − 1) = ((#‘𝑡) − 1)) |
4 | 1, 3 | fveq12d 6197 | . . . . 5 ⊢ (𝑚 = 𝑡 → (𝑚‘((#‘𝑚) − 1)) = (𝑡‘((#‘𝑡) − 1))) |
5 | 4 | eleq1d 2686 | . . . 4 ⊢ (𝑚 = 𝑡 → ((𝑚‘((#‘𝑚) − 1)) ∈ 𝑊 ↔ (𝑡‘((#‘𝑡) − 1)) ∈ 𝑊)) |
6 | 5 | ralrab2 3372 | . . 3 ⊢ (∀𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} (𝑚‘((#‘𝑚) − 1)) ∈ 𝑊 ↔ ∀𝑡 ∈ (Word 𝑊 ∖ {∅})(((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1)))) → (𝑡‘((#‘𝑡) − 1)) ∈ 𝑊)) |
7 | eldifi 3732 | . . . . . 6 ⊢ (𝑡 ∈ (Word 𝑊 ∖ {∅}) → 𝑡 ∈ Word 𝑊) | |
8 | wrdf 13310 | . . . . . 6 ⊢ (𝑡 ∈ Word 𝑊 → 𝑡:(0..^(#‘𝑡))⟶𝑊) | |
9 | 7, 8 | syl 17 | . . . . 5 ⊢ (𝑡 ∈ (Word 𝑊 ∖ {∅}) → 𝑡:(0..^(#‘𝑡))⟶𝑊) |
10 | eldifsn 4317 | . . . . . . 7 ⊢ (𝑡 ∈ (Word 𝑊 ∖ {∅}) ↔ (𝑡 ∈ Word 𝑊 ∧ 𝑡 ≠ ∅)) | |
11 | lennncl 13325 | . . . . . . 7 ⊢ ((𝑡 ∈ Word 𝑊 ∧ 𝑡 ≠ ∅) → (#‘𝑡) ∈ ℕ) | |
12 | 10, 11 | sylbi 207 | . . . . . 6 ⊢ (𝑡 ∈ (Word 𝑊 ∖ {∅}) → (#‘𝑡) ∈ ℕ) |
13 | fzo0end 12560 | . . . . . 6 ⊢ ((#‘𝑡) ∈ ℕ → ((#‘𝑡) − 1) ∈ (0..^(#‘𝑡))) | |
14 | 12, 13 | syl 17 | . . . . 5 ⊢ (𝑡 ∈ (Word 𝑊 ∖ {∅}) → ((#‘𝑡) − 1) ∈ (0..^(#‘𝑡))) |
15 | 9, 14 | ffvelrnd 6360 | . . . 4 ⊢ (𝑡 ∈ (Word 𝑊 ∖ {∅}) → (𝑡‘((#‘𝑡) − 1)) ∈ 𝑊) |
16 | 15 | a1d 25 | . . 3 ⊢ (𝑡 ∈ (Word 𝑊 ∖ {∅}) → (((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1)))) → (𝑡‘((#‘𝑡) − 1)) ∈ 𝑊)) |
17 | 6, 16 | mprgbir 2927 | . 2 ⊢ ∀𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} (𝑚‘((#‘𝑚) − 1)) ∈ 𝑊 |
18 | efgred.s | . . 3 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1))) | |
19 | 18 | fmpt 6381 | . 2 ⊢ (∀𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} (𝑚‘((#‘𝑚) − 1)) ∈ 𝑊 ↔ 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊) |
20 | 17, 19 | mpbi 220 | 1 ⊢ 𝑆:{𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))}⟶𝑊 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 {crab 2916 ∖ cdif 3571 ∅c0 3915 {csn 4177 〈cop 4183 〈cotp 4185 ∪ ciun 4520 ↦ cmpt 4729 I cid 5023 × cxp 5112 ran crn 5115 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1𝑜c1o 7553 2𝑜c2o 7554 0cc0 9936 1c1 9937 − cmin 10266 ℕcn 11020 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 splice csplice 13296 〈“cs2 13586 ~FG cefg 18119 |
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-oadd 7564 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: efgsdm 18143 efgsval 18144 efgsp1 18150 efgsfo 18152 efgredleme 18156 efgred 18161 |
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