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Mirrors > Home > MPE Home > Th. List > efgs1 | Structured version Visualization version GIF version |
Description: A singleton of an irreducible word is an extension sequence. (Contributed by Mario Carneiro, 27-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 |
---|---|
efgs1 | ⊢ (𝐴 ∈ 𝐷 → 〈“𝐴”〉 ∈ dom 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifi 4154 | . . . . 5 ⊢ (𝐴 ∈ (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) → 𝐴 ∈ 𝑊) | |
2 | efgred.d | . . . . 5 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) | |
3 | 1, 2 | eleq2s 2862 | . . . 4 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ 𝑊) |
4 | 3 | s1cld 14651 | . . 3 ⊢ (𝐴 ∈ 𝐷 → 〈“𝐴”〉 ∈ Word 𝑊) |
5 | s1nz 14655 | . . 3 ⊢ 〈“𝐴”〉 ≠ ∅ | |
6 | eldifsn 4811 | . . 3 ⊢ (〈“𝐴”〉 ∈ (Word 𝑊 ∖ {∅}) ↔ (〈“𝐴”〉 ∈ Word 𝑊 ∧ 〈“𝐴”〉 ≠ ∅)) | |
7 | 4, 5, 6 | sylanblrc 589 | . 2 ⊢ (𝐴 ∈ 𝐷 → 〈“𝐴”〉 ∈ (Word 𝑊 ∖ {∅})) |
8 | s1fv 14658 | . . 3 ⊢ (𝐴 ∈ 𝐷 → (〈“𝐴”〉‘0) = 𝐴) | |
9 | id 22 | . . 3 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ 𝐷) | |
10 | 8, 9 | eqeltrd 2844 | . 2 ⊢ (𝐴 ∈ 𝐷 → (〈“𝐴”〉‘0) ∈ 𝐷) |
11 | s1len 14654 | . . . . . 6 ⊢ (♯‘〈“𝐴”〉) = 1 | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ 𝐷 → (♯‘〈“𝐴”〉) = 1) |
13 | 12 | oveq2d 7464 | . . . 4 ⊢ (𝐴 ∈ 𝐷 → (1..^(♯‘〈“𝐴”〉)) = (1..^1)) |
14 | fzo0 13740 | . . . 4 ⊢ (1..^1) = ∅ | |
15 | 13, 14 | eqtrdi 2796 | . . 3 ⊢ (𝐴 ∈ 𝐷 → (1..^(♯‘〈“𝐴”〉)) = ∅) |
16 | rzal 4532 | . . 3 ⊢ ((1..^(♯‘〈“𝐴”〉)) = ∅ → ∀𝑖 ∈ (1..^(♯‘〈“𝐴”〉))(〈“𝐴”〉‘𝑖) ∈ ran (𝑇‘(〈“𝐴”〉‘(𝑖 − 1)))) | |
17 | 15, 16 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝐷 → ∀𝑖 ∈ (1..^(♯‘〈“𝐴”〉))(〈“𝐴”〉‘𝑖) ∈ ran (𝑇‘(〈“𝐴”〉‘(𝑖 − 1)))) |
18 | efgval.w | . . 3 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
19 | efgval.r | . . 3 ⊢ ∼ = ( ~FG ‘𝐼) | |
20 | efgval2.m | . . 3 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) | |
21 | efgval2.t | . . 3 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) | |
22 | efgred.s | . . 3 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) | |
23 | 18, 19, 20, 21, 2, 22 | efgsdm 19772 | . 2 ⊢ (〈“𝐴”〉 ∈ dom 𝑆 ↔ (〈“𝐴”〉 ∈ (Word 𝑊 ∖ {∅}) ∧ (〈“𝐴”〉‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘〈“𝐴”〉))(〈“𝐴”〉‘𝑖) ∈ ran (𝑇‘(〈“𝐴”〉‘(𝑖 − 1))))) |
24 | 7, 10, 17, 23 | syl3anbrc 1343 | 1 ⊢ (𝐴 ∈ 𝐷 → 〈“𝐴”〉 ∈ dom 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 {crab 3443 ∖ cdif 3973 ∅c0 4352 {csn 4648 〈cop 4654 〈cotp 4656 ∪ ciun 5015 ↦ cmpt 5249 I cid 5592 × cxp 5698 dom cdm 5700 ran crn 5701 ‘cfv 6573 (class class class)co 7448 ∈ cmpo 7450 1oc1o 8515 2oc2o 8516 0cc0 11184 1c1 11185 − cmin 11520 ...cfz 13567 ..^cfzo 13711 ♯chash 14379 Word cword 14562 〈“cs1 14643 splice csplice 14797 〈“cs2 14890 ~FG cefg 19748 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-n0 12554 df-z 12640 df-uz 12904 df-fz 13568 df-fzo 13712 df-hash 14380 df-word 14563 df-s1 14644 |
This theorem is referenced by: efgsfo 19781 |
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