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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 4017 | . . . . 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 2851 | . . . 4 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ 𝑊) |
4 | 3 | s1cld 14046 | . . 3 ⊢ (𝐴 ∈ 𝐷 → 〈“𝐴”〉 ∈ Word 𝑊) |
5 | s1nz 14050 | . . 3 ⊢ 〈“𝐴”〉 ≠ ∅ | |
6 | eldifsn 4675 | . . 3 ⊢ (〈“𝐴”〉 ∈ (Word 𝑊 ∖ {∅}) ↔ (〈“𝐴”〉 ∈ Word 𝑊 ∧ 〈“𝐴”〉 ≠ ∅)) | |
7 | 4, 5, 6 | sylanblrc 593 | . 2 ⊢ (𝐴 ∈ 𝐷 → 〈“𝐴”〉 ∈ (Word 𝑊 ∖ {∅})) |
8 | s1fv 14053 | . . 3 ⊢ (𝐴 ∈ 𝐷 → (〈“𝐴”〉‘0) = 𝐴) | |
9 | id 22 | . . 3 ⊢ (𝐴 ∈ 𝐷 → 𝐴 ∈ 𝐷) | |
10 | 8, 9 | eqeltrd 2833 | . 2 ⊢ (𝐴 ∈ 𝐷 → (〈“𝐴”〉‘0) ∈ 𝐷) |
11 | s1len 14049 | . . . . . 6 ⊢ (♯‘〈“𝐴”〉) = 1 | |
12 | 11 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ 𝐷 → (♯‘〈“𝐴”〉) = 1) |
13 | 12 | oveq2d 7186 | . . . 4 ⊢ (𝐴 ∈ 𝐷 → (1..^(♯‘〈“𝐴”〉)) = (1..^1)) |
14 | fzo0 13152 | . . . 4 ⊢ (1..^1) = ∅ | |
15 | 13, 14 | eqtrdi 2789 | . . 3 ⊢ (𝐴 ∈ 𝐷 → (1..^(♯‘〈“𝐴”〉)) = ∅) |
16 | rzal 4395 | . . 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 18974 | . 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 1344 | 1 ⊢ (𝐴 ∈ 𝐷 → 〈“𝐴”〉 ∈ dom 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ∀wral 3053 {crab 3057 ∖ cdif 3840 ∅c0 4211 {csn 4516 〈cop 4522 〈cotp 4524 ∪ ciun 4881 ↦ cmpt 5110 I cid 5428 × cxp 5523 dom cdm 5525 ran crn 5526 ‘cfv 6339 (class class class)co 7170 ∈ cmpo 7172 1oc1o 8124 2oc2o 8125 0cc0 10615 1c1 10616 − cmin 10948 ...cfz 12981 ..^cfzo 13124 ♯chash 13782 Word cword 13955 〈“cs1 14038 splice csplice 14200 〈“cs2 14292 ~FG cefg 18950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 df-fzo 13125 df-hash 13783 df-word 13956 df-s1 14039 |
This theorem is referenced by: efgsfo 18983 |
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