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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 3694 | . . . . 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 2706 | . . . 4 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ 𝑊) |
4 | 3 | s1cld 13236 | . . 3 ⊢ (𝐴 ∈ 𝐷 → 〈“𝐴”〉 ∈ Word 𝑊) |
5 | s1nz 13239 | . . . 4 ⊢ 〈“𝐴”〉 ≠ ∅ | |
6 | eldifsn 4260 | . . . 4 ⊢ (〈“𝐴”〉 ∈ (Word 𝑊 ∖ {∅}) ↔ (〈“𝐴”〉 ∈ Word 𝑊 ∧ 〈“𝐴”〉 ≠ ∅)) | |
7 | 5, 6 | mpbiran2 956 | . . 3 ⊢ (〈“𝐴”〉 ∈ (Word 𝑊 ∖ {∅}) ↔ 〈“𝐴”〉 ∈ Word 𝑊) |
8 | 4, 7 | sylibr 223 | . 2 ⊢ (𝐴 ∈ 𝐷 → 〈“𝐴”〉 ∈ (Word 𝑊 ∖ {∅})) |
9 | s1fv 13243 | . . 3 ⊢ (𝐴 ∈ 𝐷 → (〈“𝐴”〉‘0) = 𝐴) | |
10 | id 22 | . . 3 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ 𝐷) | |
11 | 9, 10 | eqeltrd 2688 | . 2 ⊢ (𝐴 ∈ 𝐷 → (〈“𝐴”〉‘0) ∈ 𝐷) |
12 | s1len 13238 | . . . . . 6 ⊢ (#‘〈“𝐴”〉) = 1 | |
13 | 12 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ 𝐷 → (#‘〈“𝐴”〉) = 1) |
14 | 13 | oveq2d 6565 | . . . 4 ⊢ (𝐴 ∈ 𝐷 → (1..^(#‘〈“𝐴”〉)) = (1..^1)) |
15 | fzo0 12361 | . . . 4 ⊢ (1..^1) = ∅ | |
16 | 14, 15 | syl6eq 2660 | . . 3 ⊢ (𝐴 ∈ 𝐷 → (1..^(#‘〈“𝐴”〉)) = ∅) |
17 | rzal 4025 | . . 3 ⊢ ((1..^(#‘〈“𝐴”〉)) = ∅ → ∀𝑖 ∈ (1..^(#‘〈“𝐴”〉))(〈“𝐴”〉‘𝑖) ∈ ran (𝑇‘(〈“𝐴”〉‘(𝑖 − 1)))) | |
18 | 16, 17 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝐷 → ∀𝑖 ∈ (1..^(#‘〈“𝐴”〉))(〈“𝐴”〉‘𝑖) ∈ ran (𝑇‘(〈“𝐴”〉‘(𝑖 − 1)))) |
19 | efgval.w | . . 3 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2𝑜)) | |
20 | efgval.r | . . 3 ⊢ ∼ = ( ~FG ‘𝐼) | |
21 | efgval2.m | . . 3 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ 〈𝑦, (1𝑜 ∖ 𝑧)〉) | |
22 | efgval2.t | . . 3 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) | |
23 | efgred.s | . . 3 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1))) | |
24 | 19, 20, 21, 22, 2, 23 | efgsdm 17966 | . 2 ⊢ (〈“𝐴”〉 ∈ dom 𝑆 ↔ (〈“𝐴”〉 ∈ (Word 𝑊 ∖ {∅}) ∧ (〈“𝐴”〉‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(#‘〈“𝐴”〉))(〈“𝐴”〉‘𝑖) ∈ ran (𝑇‘(〈“𝐴”〉‘(𝑖 − 1))))) |
25 | 8, 11, 18, 24 | syl3anbrc 1239 | 1 ⊢ (𝐴 ∈ 𝐷 → 〈“𝐴”〉 ∈ dom 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 {crab 2900 ∖ cdif 3537 ∅c0 3874 {csn 4125 〈cop 4131 〈cotp 4133 ∪ ciun 4455 ↦ cmpt 4643 I cid 4948 × cxp 5036 dom cdm 5038 ran crn 5039 ‘cfv 5804 (class class class)co 6549 ↦ cmpt2 6551 1𝑜c1o 7440 2𝑜c2o 7441 0cc0 9815 1c1 9816 − cmin 10145 ...cfz 12197 ..^cfzo 12334 #chash 12979 Word cword 13146 〈“cs1 13149 splice csplice 13151 〈“cs2 13437 ~FG cefg 17942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-s1 13157 |
This theorem is referenced by: efgsfo 17975 |
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