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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 4105 | . . . . 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 2933 | . . . 4 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ 𝑊) |
4 | 3 | s1cld 13959 | . . 3 ⊢ (𝐴 ∈ 𝐷 → 〈“𝐴”〉 ∈ Word 𝑊) |
5 | s1nz 13963 | . . 3 ⊢ 〈“𝐴”〉 ≠ ∅ | |
6 | eldifsn 4721 | . . 3 ⊢ (〈“𝐴”〉 ∈ (Word 𝑊 ∖ {∅}) ↔ (〈“𝐴”〉 ∈ Word 𝑊 ∧ 〈“𝐴”〉 ≠ ∅)) | |
7 | 4, 5, 6 | sylanblrc 592 | . 2 ⊢ (𝐴 ∈ 𝐷 → 〈“𝐴”〉 ∈ (Word 𝑊 ∖ {∅})) |
8 | s1fv 13966 | . . 3 ⊢ (𝐴 ∈ 𝐷 → (〈“𝐴”〉‘0) = 𝐴) | |
9 | id 22 | . . 3 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ 𝐷) | |
10 | 8, 9 | eqeltrd 2915 | . 2 ⊢ (𝐴 ∈ 𝐷 → (〈“𝐴”〉‘0) ∈ 𝐷) |
11 | s1len 13962 | . . . . . 6 ⊢ (♯‘〈“𝐴”〉) = 1 | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ 𝐷 → (♯‘〈“𝐴”〉) = 1) |
13 | 12 | oveq2d 7174 | . . . 4 ⊢ (𝐴 ∈ 𝐷 → (1..^(♯‘〈“𝐴”〉)) = (1..^1)) |
14 | fzo0 13064 | . . . 4 ⊢ (1..^1) = ∅ | |
15 | 13, 14 | syl6eq 2874 | . . 3 ⊢ (𝐴 ∈ 𝐷 → (1..^(♯‘〈“𝐴”〉)) = ∅) |
16 | rzal 4455 | . . 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 18858 | . 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 1339 | 1 ⊢ (𝐴 ∈ 𝐷 → 〈“𝐴”〉 ∈ dom 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∀wral 3140 {crab 3144 ∖ cdif 3935 ∅c0 4293 {csn 4569 〈cop 4575 〈cotp 4577 ∪ ciun 4921 ↦ cmpt 5148 I cid 5461 × cxp 5555 dom cdm 5557 ran crn 5558 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 1oc1o 8097 2oc2o 8098 0cc0 10539 1c1 10540 − cmin 10872 ...cfz 12895 ..^cfzo 13036 ♯chash 13693 Word cword 13864 〈“cs1 13951 splice csplice 14113 〈“cs2 14205 ~FG cefg 18834 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-s1 13952 |
This theorem is referenced by: efgsfo 18867 |
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