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Mirrors > Home > MPE Home > Th. List > efgsdmi | Structured version Visualization version GIF version |
Description: Property of the last link in the chain of extensions. (Contributed by Mario Carneiro, 29-Sep-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 |
---|---|
efgsdmi | ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((#‘𝐹) − 1) ∈ ℕ) → (𝑆‘𝐹) ∈ ran (𝑇‘(𝐹‘(((#‘𝐹) − 1) − 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efgval.w | . . . 4 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2𝑜)) | |
2 | efgval.r | . . . 4 ⊢ ∼ = ( ~FG ‘𝐼) | |
3 | efgval2.m | . . . 4 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ 〈𝑦, (1𝑜 ∖ 𝑧)〉) | |
4 | efgval2.t | . . . 4 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) | |
5 | efgred.d | . . . 4 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) | |
6 | efgred.s | . . . 4 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1))) | |
7 | 1, 2, 3, 4, 5, 6 | efgsval 18190 | . . 3 ⊢ (𝐹 ∈ dom 𝑆 → (𝑆‘𝐹) = (𝐹‘((#‘𝐹) − 1))) |
8 | 7 | adantr 480 | . 2 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((#‘𝐹) − 1) ∈ ℕ) → (𝑆‘𝐹) = (𝐹‘((#‘𝐹) − 1))) |
9 | simpr 476 | . . . . . . 7 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((#‘𝐹) − 1) ∈ ℕ) → ((#‘𝐹) − 1) ∈ ℕ) | |
10 | nnuz 11761 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
11 | 9, 10 | syl6eleq 2740 | . . . . . 6 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((#‘𝐹) − 1) ∈ ℕ) → ((#‘𝐹) − 1) ∈ (ℤ≥‘1)) |
12 | eluzfz1 12386 | . . . . . 6 ⊢ (((#‘𝐹) − 1) ∈ (ℤ≥‘1) → 1 ∈ (1...((#‘𝐹) − 1))) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((#‘𝐹) − 1) ∈ ℕ) → 1 ∈ (1...((#‘𝐹) − 1))) |
14 | 1, 2, 3, 4, 5, 6 | efgsdm 18189 | . . . . . . . . 9 ⊢ (𝐹 ∈ dom 𝑆 ↔ (𝐹 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐹‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(#‘𝐹))(𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1))))) |
15 | 14 | simp1bi 1096 | . . . . . . . 8 ⊢ (𝐹 ∈ dom 𝑆 → 𝐹 ∈ (Word 𝑊 ∖ {∅})) |
16 | 15 | adantr 480 | . . . . . . 7 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((#‘𝐹) − 1) ∈ ℕ) → 𝐹 ∈ (Word 𝑊 ∖ {∅})) |
17 | 16 | eldifad 3619 | . . . . . 6 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((#‘𝐹) − 1) ∈ ℕ) → 𝐹 ∈ Word 𝑊) |
18 | lencl 13356 | . . . . . 6 ⊢ (𝐹 ∈ Word 𝑊 → (#‘𝐹) ∈ ℕ0) | |
19 | nn0z 11438 | . . . . . 6 ⊢ ((#‘𝐹) ∈ ℕ0 → (#‘𝐹) ∈ ℤ) | |
20 | fzoval 12510 | . . . . . 6 ⊢ ((#‘𝐹) ∈ ℤ → (1..^(#‘𝐹)) = (1...((#‘𝐹) − 1))) | |
21 | 17, 18, 19, 20 | 4syl 19 | . . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((#‘𝐹) − 1) ∈ ℕ) → (1..^(#‘𝐹)) = (1...((#‘𝐹) − 1))) |
22 | 13, 21 | eleqtrrd 2733 | . . . 4 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((#‘𝐹) − 1) ∈ ℕ) → 1 ∈ (1..^(#‘𝐹))) |
23 | fzoend 12599 | . . . 4 ⊢ (1 ∈ (1..^(#‘𝐹)) → ((#‘𝐹) − 1) ∈ (1..^(#‘𝐹))) | |
24 | 22, 23 | syl 17 | . . 3 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((#‘𝐹) − 1) ∈ ℕ) → ((#‘𝐹) − 1) ∈ (1..^(#‘𝐹))) |
25 | 14 | simp3bi 1098 | . . . 4 ⊢ (𝐹 ∈ dom 𝑆 → ∀𝑖 ∈ (1..^(#‘𝐹))(𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1)))) |
26 | 25 | adantr 480 | . . 3 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((#‘𝐹) − 1) ∈ ℕ) → ∀𝑖 ∈ (1..^(#‘𝐹))(𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1)))) |
27 | fveq2 6229 | . . . . 5 ⊢ (𝑖 = ((#‘𝐹) − 1) → (𝐹‘𝑖) = (𝐹‘((#‘𝐹) − 1))) | |
28 | oveq1 6697 | . . . . . . . 8 ⊢ (𝑖 = ((#‘𝐹) − 1) → (𝑖 − 1) = (((#‘𝐹) − 1) − 1)) | |
29 | 28 | fveq2d 6233 | . . . . . . 7 ⊢ (𝑖 = ((#‘𝐹) − 1) → (𝐹‘(𝑖 − 1)) = (𝐹‘(((#‘𝐹) − 1) − 1))) |
30 | 29 | fveq2d 6233 | . . . . . 6 ⊢ (𝑖 = ((#‘𝐹) − 1) → (𝑇‘(𝐹‘(𝑖 − 1))) = (𝑇‘(𝐹‘(((#‘𝐹) − 1) − 1)))) |
31 | 30 | rneqd 5385 | . . . . 5 ⊢ (𝑖 = ((#‘𝐹) − 1) → ran (𝑇‘(𝐹‘(𝑖 − 1))) = ran (𝑇‘(𝐹‘(((#‘𝐹) − 1) − 1)))) |
32 | 27, 31 | eleq12d 2724 | . . . 4 ⊢ (𝑖 = ((#‘𝐹) − 1) → ((𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1))) ↔ (𝐹‘((#‘𝐹) − 1)) ∈ ran (𝑇‘(𝐹‘(((#‘𝐹) − 1) − 1))))) |
33 | 32 | rspcv 3336 | . . 3 ⊢ (((#‘𝐹) − 1) ∈ (1..^(#‘𝐹)) → (∀𝑖 ∈ (1..^(#‘𝐹))(𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1))) → (𝐹‘((#‘𝐹) − 1)) ∈ ran (𝑇‘(𝐹‘(((#‘𝐹) − 1) − 1))))) |
34 | 24, 26, 33 | sylc 65 | . 2 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((#‘𝐹) − 1) ∈ ℕ) → (𝐹‘((#‘𝐹) − 1)) ∈ ran (𝑇‘(𝐹‘(((#‘𝐹) − 1) − 1)))) |
35 | 8, 34 | eqeltrd 2730 | 1 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((#‘𝐹) − 1) ∈ ℕ) → (𝑆‘𝐹) ∈ ran (𝑇‘(𝐹‘(((#‘𝐹) − 1) − 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ∀wral 2941 {crab 2945 ∖ cdif 3604 ∅c0 3948 {csn 4210 〈cop 4216 〈cotp 4218 ∪ ciun 4552 ↦ cmpt 4762 I cid 5052 × cxp 5141 dom cdm 5143 ran crn 5144 ‘cfv 5926 (class class class)co 6690 ↦ cmpt2 6692 1𝑜c1o 7598 2𝑜c2o 7599 0cc0 9974 1c1 9975 − cmin 10304 ℕcn 11058 ℕ0cn0 11330 ℤcz 11415 ℤ≥cuz 11725 ...cfz 12364 ..^cfzo 12504 #chash 13157 Word cword 13323 splice csplice 13328 〈“cs2 13632 ~FG cefg 18165 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-card 8803 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-fzo 12505 df-hash 13158 df-word 13331 |
This theorem is referenced by: efgs1b 18195 efgredlemg 18201 efgredlemd 18203 efgredlem 18206 |
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