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Mirrors > Home > MPE Home > Th. List > efgredlemf | Structured version Visualization version GIF version |
Description: Lemma for efgredleme 18470. (Contributed by Mario Carneiro, 4-Jun-2016.) |
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))) |
efgredlem.1 | ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))) |
efgredlem.2 | ⊢ (𝜑 → 𝐴 ∈ dom 𝑆) |
efgredlem.3 | ⊢ (𝜑 → 𝐵 ∈ dom 𝑆) |
efgredlem.4 | ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵)) |
efgredlem.5 | ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0)) |
efgredlemb.k | ⊢ 𝐾 = (((♯‘𝐴) − 1) − 1) |
efgredlemb.l | ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1) |
Ref | Expression |
---|---|
efgredlemf | ⊢ (𝜑 → ((𝐴‘𝐾) ∈ 𝑊 ∧ (𝐵‘𝐿) ∈ 𝑊)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efgredlem.2 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ dom 𝑆) | |
2 | efgval.w | . . . . . . . 8 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2𝑜)) | |
3 | efgval.r | . . . . . . . 8 ⊢ ∼ = ( ~FG ‘𝐼) | |
4 | efgval2.m | . . . . . . . 8 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2𝑜 ↦ 〈𝑦, (1𝑜 ∖ 𝑧)〉) | |
5 | efgval2.t | . . . . . . . 8 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) | |
6 | efgred.d | . . . . . . . 8 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) | |
7 | efgred.s | . . . . . . . 8 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) | |
8 | 2, 3, 4, 5, 6, 7 | efgsdm 18456 | . . . . . . 7 ⊢ (𝐴 ∈ dom 𝑆 ↔ (𝐴 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐴‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐴))(𝐴‘𝑖) ∈ ran (𝑇‘(𝐴‘(𝑖 − 1))))) |
9 | 8 | simp1bi 1176 | . . . . . 6 ⊢ (𝐴 ∈ dom 𝑆 → 𝐴 ∈ (Word 𝑊 ∖ {∅})) |
10 | 1, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Word 𝑊 ∖ {∅})) |
11 | 10 | eldifad 3781 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Word 𝑊) |
12 | wrdf 13539 | . . . 4 ⊢ (𝐴 ∈ Word 𝑊 → 𝐴:(0..^(♯‘𝐴))⟶𝑊) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴:(0..^(♯‘𝐴))⟶𝑊) |
14 | fzossfz 12743 | . . . . 5 ⊢ (0..^((♯‘𝐴) − 1)) ⊆ (0...((♯‘𝐴) − 1)) | |
15 | lencl 13553 | . . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑊 → (♯‘𝐴) ∈ ℕ0) | |
16 | 11, 15 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ0) |
17 | 16 | nn0zd 11770 | . . . . . 6 ⊢ (𝜑 → (♯‘𝐴) ∈ ℤ) |
18 | fzoval 12726 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℤ → (0..^(♯‘𝐴)) = (0...((♯‘𝐴) − 1))) | |
19 | 17, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → (0..^(♯‘𝐴)) = (0...((♯‘𝐴) − 1))) |
20 | 14, 19 | syl5sseqr 3850 | . . . 4 ⊢ (𝜑 → (0..^((♯‘𝐴) − 1)) ⊆ (0..^(♯‘𝐴))) |
21 | efgredlemb.k | . . . . 5 ⊢ 𝐾 = (((♯‘𝐴) − 1) − 1) | |
22 | efgredlem.1 | . . . . . . . 8 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))) | |
23 | efgredlem.3 | . . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ dom 𝑆) | |
24 | efgredlem.4 | . . . . . . . 8 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵)) | |
25 | efgredlem.5 | . . . . . . . 8 ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0)) | |
26 | 2, 3, 4, 5, 6, 7, 22, 1, 23, 24, 25 | efgredlema 18467 | . . . . . . 7 ⊢ (𝜑 → (((♯‘𝐴) − 1) ∈ ℕ ∧ ((♯‘𝐵) − 1) ∈ ℕ)) |
27 | 26 | simpld 489 | . . . . . 6 ⊢ (𝜑 → ((♯‘𝐴) − 1) ∈ ℕ) |
28 | fzo0end 12815 | . . . . . 6 ⊢ (((♯‘𝐴) − 1) ∈ ℕ → (((♯‘𝐴) − 1) − 1) ∈ (0..^((♯‘𝐴) − 1))) | |
29 | 27, 28 | syl 17 | . . . . 5 ⊢ (𝜑 → (((♯‘𝐴) − 1) − 1) ∈ (0..^((♯‘𝐴) − 1))) |
30 | 21, 29 | syl5eqel 2882 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (0..^((♯‘𝐴) − 1))) |
31 | 20, 30 | sseldd 3799 | . . 3 ⊢ (𝜑 → 𝐾 ∈ (0..^(♯‘𝐴))) |
32 | 13, 31 | ffvelrnd 6586 | . 2 ⊢ (𝜑 → (𝐴‘𝐾) ∈ 𝑊) |
33 | 2, 3, 4, 5, 6, 7 | efgsdm 18456 | . . . . . . 7 ⊢ (𝐵 ∈ dom 𝑆 ↔ (𝐵 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐵‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐵))(𝐵‘𝑖) ∈ ran (𝑇‘(𝐵‘(𝑖 − 1))))) |
34 | 33 | simp1bi 1176 | . . . . . 6 ⊢ (𝐵 ∈ dom 𝑆 → 𝐵 ∈ (Word 𝑊 ∖ {∅})) |
35 | 23, 34 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (Word 𝑊 ∖ {∅})) |
36 | 35 | eldifad 3781 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ Word 𝑊) |
37 | wrdf 13539 | . . . 4 ⊢ (𝐵 ∈ Word 𝑊 → 𝐵:(0..^(♯‘𝐵))⟶𝑊) | |
38 | 36, 37 | syl 17 | . . 3 ⊢ (𝜑 → 𝐵:(0..^(♯‘𝐵))⟶𝑊) |
39 | fzossfz 12743 | . . . . 5 ⊢ (0..^((♯‘𝐵) − 1)) ⊆ (0...((♯‘𝐵) − 1)) | |
40 | lencl 13553 | . . . . . . . 8 ⊢ (𝐵 ∈ Word 𝑊 → (♯‘𝐵) ∈ ℕ0) | |
41 | 36, 40 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ0) |
42 | 41 | nn0zd 11770 | . . . . . 6 ⊢ (𝜑 → (♯‘𝐵) ∈ ℤ) |
43 | fzoval 12726 | . . . . . 6 ⊢ ((♯‘𝐵) ∈ ℤ → (0..^(♯‘𝐵)) = (0...((♯‘𝐵) − 1))) | |
44 | 42, 43 | syl 17 | . . . . 5 ⊢ (𝜑 → (0..^(♯‘𝐵)) = (0...((♯‘𝐵) − 1))) |
45 | 39, 44 | syl5sseqr 3850 | . . . 4 ⊢ (𝜑 → (0..^((♯‘𝐵) − 1)) ⊆ (0..^(♯‘𝐵))) |
46 | efgredlemb.l | . . . . 5 ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1) | |
47 | 26 | simprd 490 | . . . . . 6 ⊢ (𝜑 → ((♯‘𝐵) − 1) ∈ ℕ) |
48 | fzo0end 12815 | . . . . . 6 ⊢ (((♯‘𝐵) − 1) ∈ ℕ → (((♯‘𝐵) − 1) − 1) ∈ (0..^((♯‘𝐵) − 1))) | |
49 | 47, 48 | syl 17 | . . . . 5 ⊢ (𝜑 → (((♯‘𝐵) − 1) − 1) ∈ (0..^((♯‘𝐵) − 1))) |
50 | 46, 49 | syl5eqel 2882 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ (0..^((♯‘𝐵) − 1))) |
51 | 45, 50 | sseldd 3799 | . . 3 ⊢ (𝜑 → 𝐿 ∈ (0..^(♯‘𝐵))) |
52 | 38, 51 | ffvelrnd 6586 | . 2 ⊢ (𝜑 → (𝐵‘𝐿) ∈ 𝑊) |
53 | 32, 52 | jca 508 | 1 ⊢ (𝜑 → ((𝐴‘𝐾) ∈ 𝑊 ∧ (𝐵‘𝐿) ∈ 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ∀wral 3089 {crab 3093 ∖ cdif 3766 ∅c0 4115 {csn 4368 〈cop 4374 〈cotp 4376 ∪ ciun 4710 class class class wbr 4843 ↦ cmpt 4922 I cid 5219 × cxp 5310 dom cdm 5312 ran crn 5313 ⟶wf 6097 ‘cfv 6101 (class class class)co 6878 ↦ cmpt2 6880 1𝑜c1o 7792 2𝑜c2o 7793 0cc0 10224 1c1 10225 < clt 10363 − cmin 10556 ℕcn 11312 ℕ0cn0 11580 ℤcz 11666 ...cfz 12580 ..^cfzo 12720 ♯chash 13370 Word cword 13534 splice csplice 13819 〈“cs2 13926 ~FG cefg 18432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-card 9051 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-n0 11581 df-z 11667 df-uz 11931 df-fz 12581 df-fzo 12721 df-hash 13371 df-word 13535 |
This theorem is referenced by: efgredlemg 18469 efgredleme 18470 |
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