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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 19703 | . . 3 ⊢ (𝐹 ∈ dom 𝑆 → (𝑆‘𝐹) = (𝐹‘((♯‘𝐹) − 1))) |
8 | 7 | adantr 479 | . 2 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → (𝑆‘𝐹) = (𝐹‘((♯‘𝐹) − 1))) |
9 | fveq2 6896 | . . . 4 ⊢ (𝑖 = ((♯‘𝐹) − 1) → (𝐹‘𝑖) = (𝐹‘((♯‘𝐹) − 1))) | |
10 | fvoveq1 7442 | . . . . . 6 ⊢ (𝑖 = ((♯‘𝐹) − 1) → (𝐹‘(𝑖 − 1)) = (𝐹‘(((♯‘𝐹) − 1) − 1))) | |
11 | 10 | fveq2d 6900 | . . . . 5 ⊢ (𝑖 = ((♯‘𝐹) − 1) → (𝑇‘(𝐹‘(𝑖 − 1))) = (𝑇‘(𝐹‘(((♯‘𝐹) − 1) − 1)))) |
12 | 11 | rneqd 5940 | . . . 4 ⊢ (𝑖 = ((♯‘𝐹) − 1) → ran (𝑇‘(𝐹‘(𝑖 − 1))) = ran (𝑇‘(𝐹‘(((♯‘𝐹) − 1) − 1)))) |
13 | 9, 12 | eleq12d 2819 | . . 3 ⊢ (𝑖 = ((♯‘𝐹) − 1) → ((𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1))) ↔ (𝐹‘((♯‘𝐹) − 1)) ∈ ran (𝑇‘(𝐹‘(((♯‘𝐹) − 1) − 1))))) |
14 | 1, 2, 3, 4, 5, 6 | efgsdm 19702 | . . . . 5 ⊢ (𝐹 ∈ dom 𝑆 ↔ (𝐹 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐹‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐹))(𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1))))) |
15 | 14 | simp3bi 1144 | . . . 4 ⊢ (𝐹 ∈ dom 𝑆 → ∀𝑖 ∈ (1..^(♯‘𝐹))(𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1)))) |
16 | 15 | adantr 479 | . . 3 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → ∀𝑖 ∈ (1..^(♯‘𝐹))(𝐹‘𝑖) ∈ ran (𝑇‘(𝐹‘(𝑖 − 1)))) |
17 | simpr 483 | . . . . . . 7 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → ((♯‘𝐹) − 1) ∈ ℕ) | |
18 | nnuz 12903 | . . . . . . 7 ⊢ ℕ = (ℤ≥‘1) | |
19 | 17, 18 | eleqtrdi 2835 | . . . . . 6 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → ((♯‘𝐹) − 1) ∈ (ℤ≥‘1)) |
20 | eluzfz1 13548 | . . . . . 6 ⊢ (((♯‘𝐹) − 1) ∈ (ℤ≥‘1) → 1 ∈ (1...((♯‘𝐹) − 1))) | |
21 | 19, 20 | syl 17 | . . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → 1 ∈ (1...((♯‘𝐹) − 1))) |
22 | 14 | simp1bi 1142 | . . . . . . . 8 ⊢ (𝐹 ∈ dom 𝑆 → 𝐹 ∈ (Word 𝑊 ∖ {∅})) |
23 | 22 | adantr 479 | . . . . . . 7 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → 𝐹 ∈ (Word 𝑊 ∖ {∅})) |
24 | 23 | eldifad 3956 | . . . . . 6 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → 𝐹 ∈ Word 𝑊) |
25 | lencl 14524 | . . . . . 6 ⊢ (𝐹 ∈ Word 𝑊 → (♯‘𝐹) ∈ ℕ0) | |
26 | nn0z 12621 | . . . . . 6 ⊢ ((♯‘𝐹) ∈ ℕ0 → (♯‘𝐹) ∈ ℤ) | |
27 | fzoval 13673 | . . . . . 6 ⊢ ((♯‘𝐹) ∈ ℤ → (1..^(♯‘𝐹)) = (1...((♯‘𝐹) − 1))) | |
28 | 24, 25, 26, 27 | 4syl 19 | . . . . 5 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → (1..^(♯‘𝐹)) = (1...((♯‘𝐹) − 1))) |
29 | 21, 28 | eleqtrrd 2828 | . . . 4 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → 1 ∈ (1..^(♯‘𝐹))) |
30 | fzoend 13763 | . . . 4 ⊢ (1 ∈ (1..^(♯‘𝐹)) → ((♯‘𝐹) − 1) ∈ (1..^(♯‘𝐹))) | |
31 | 29, 30 | syl 17 | . . 3 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → ((♯‘𝐹) − 1) ∈ (1..^(♯‘𝐹))) |
32 | 13, 16, 31 | rspcdva 3607 | . 2 ⊢ ((𝐹 ∈ dom 𝑆 ∧ ((♯‘𝐹) − 1) ∈ ℕ) → (𝐹‘((♯‘𝐹) − 1)) ∈ ran (𝑇‘(𝐹‘(((♯‘𝐹) − 1) − 1)))) |
33 | 8, 32 | eqeltrd 2825 | 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 394 = wceq 1533 ∈ wcel 2098 ∀wral 3050 {crab 3418 ∖ cdif 3941 ∅c0 4322 {csn 4630 〈cop 4636 〈cotp 4638 ∪ ciun 4997 ↦ cmpt 5232 I cid 5575 × cxp 5676 dom cdm 5678 ran crn 5679 ‘cfv 6549 (class class class)co 7419 ∈ cmpo 7421 1oc1o 8480 2oc2o 8481 0cc0 11145 1c1 11146 − cmin 11481 ℕcn 12250 ℕ0cn0 12510 ℤcz 12596 ℤ≥cuz 12860 ...cfz 13524 ..^cfzo 13667 ♯chash 14330 Word cword 14505 splice csplice 14740 〈“cs2 14833 ~FG cefg 19678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9969 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-hash 14331 df-word 14506 |
This theorem is referenced by: efgs1b 19708 efgredlemg 19714 efgredlemd 19716 efgredlem 19719 |
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