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Mirrors > Home > MPE Home > Th. List > efgredlemf | Structured version Visualization version GIF version |
Description: Lemma for efgredleme 18871. (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 18858 | . . . . . . 7 ⊢ (𝐴 ∈ dom 𝑆 ↔ (𝐴 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐴‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐴))(𝐴‘𝑖) ∈ ran (𝑇‘(𝐴‘(𝑖 − 1))))) |
9 | 8 | simp1bi 1141 | . . . . . 6 ⊢ (𝐴 ∈ dom 𝑆 → 𝐴 ∈ (Word 𝑊 ∖ {∅})) |
10 | 1, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Word 𝑊 ∖ {∅})) |
11 | 10 | eldifad 3950 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Word 𝑊) |
12 | wrdf 13869 | . . . 4 ⊢ (𝐴 ∈ Word 𝑊 → 𝐴:(0..^(♯‘𝐴))⟶𝑊) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴:(0..^(♯‘𝐴))⟶𝑊) |
14 | fzossfz 13059 | . . . . 5 ⊢ (0..^((♯‘𝐴) − 1)) ⊆ (0...((♯‘𝐴) − 1)) | |
15 | lencl 13885 | . . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑊 → (♯‘𝐴) ∈ ℕ0) | |
16 | 11, 15 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ0) |
17 | 16 | nn0zd 12088 | . . . . . 6 ⊢ (𝜑 → (♯‘𝐴) ∈ ℤ) |
18 | fzoval 13042 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℤ → (0..^(♯‘𝐴)) = (0...((♯‘𝐴) − 1))) | |
19 | 17, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → (0..^(♯‘𝐴)) = (0...((♯‘𝐴) − 1))) |
20 | 14, 19 | sseqtrrid 4022 | . . . 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 18868 | . . . . . . 7 ⊢ (𝜑 → (((♯‘𝐴) − 1) ∈ ℕ ∧ ((♯‘𝐵) − 1) ∈ ℕ)) |
27 | 26 | simpld 497 | . . . . . 6 ⊢ (𝜑 → ((♯‘𝐴) − 1) ∈ ℕ) |
28 | fzo0end 13132 | . . . . . 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 | eqeltrid 2919 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (0..^((♯‘𝐴) − 1))) |
31 | 20, 30 | sseldd 3970 | . . 3 ⊢ (𝜑 → 𝐾 ∈ (0..^(♯‘𝐴))) |
32 | 13, 31 | ffvelrnd 6854 | . 2 ⊢ (𝜑 → (𝐴‘𝐾) ∈ 𝑊) |
33 | 2, 3, 4, 5, 6, 7 | efgsdm 18858 | . . . . . . 7 ⊢ (𝐵 ∈ dom 𝑆 ↔ (𝐵 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐵‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐵))(𝐵‘𝑖) ∈ ran (𝑇‘(𝐵‘(𝑖 − 1))))) |
34 | 33 | simp1bi 1141 | . . . . . 6 ⊢ (𝐵 ∈ dom 𝑆 → 𝐵 ∈ (Word 𝑊 ∖ {∅})) |
35 | 23, 34 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (Word 𝑊 ∖ {∅})) |
36 | 35 | eldifad 3950 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ Word 𝑊) |
37 | wrdf 13869 | . . . 4 ⊢ (𝐵 ∈ Word 𝑊 → 𝐵:(0..^(♯‘𝐵))⟶𝑊) | |
38 | 36, 37 | syl 17 | . . 3 ⊢ (𝜑 → 𝐵:(0..^(♯‘𝐵))⟶𝑊) |
39 | fzossfz 13059 | . . . . 5 ⊢ (0..^((♯‘𝐵) − 1)) ⊆ (0...((♯‘𝐵) − 1)) | |
40 | lencl 13885 | . . . . . . . 8 ⊢ (𝐵 ∈ Word 𝑊 → (♯‘𝐵) ∈ ℕ0) | |
41 | 36, 40 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ0) |
42 | 41 | nn0zd 12088 | . . . . . 6 ⊢ (𝜑 → (♯‘𝐵) ∈ ℤ) |
43 | fzoval 13042 | . . . . . 6 ⊢ ((♯‘𝐵) ∈ ℤ → (0..^(♯‘𝐵)) = (0...((♯‘𝐵) − 1))) | |
44 | 42, 43 | syl 17 | . . . . 5 ⊢ (𝜑 → (0..^(♯‘𝐵)) = (0...((♯‘𝐵) − 1))) |
45 | 39, 44 | sseqtrrid 4022 | . . . 4 ⊢ (𝜑 → (0..^((♯‘𝐵) − 1)) ⊆ (0..^(♯‘𝐵))) |
46 | efgredlemb.l | . . . . 5 ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1) | |
47 | fzo0end 13132 | . . . . . 6 ⊢ (((♯‘𝐵) − 1) ∈ ℕ → (((♯‘𝐵) − 1) − 1) ∈ (0..^((♯‘𝐵) − 1))) | |
48 | 26, 47 | simpl2im 506 | . . . . 5 ⊢ (𝜑 → (((♯‘𝐵) − 1) − 1) ∈ (0..^((♯‘𝐵) − 1))) |
49 | 46, 48 | eqeltrid 2919 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ (0..^((♯‘𝐵) − 1))) |
50 | 45, 49 | sseldd 3970 | . . 3 ⊢ (𝜑 → 𝐿 ∈ (0..^(♯‘𝐵))) |
51 | 38, 50 | ffvelrnd 6854 | . 2 ⊢ (𝜑 → (𝐵‘𝐿) ∈ 𝑊) |
52 | 32, 51 | jca 514 | 1 ⊢ (𝜑 → ((𝐴‘𝐾) ∈ 𝑊 ∧ (𝐵‘𝐿) ∈ 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 {crab 3144 ∖ cdif 3935 ∅c0 4293 {csn 4569 〈cop 4575 〈cotp 4577 ∪ ciun 4921 class class class wbr 5068 ↦ cmpt 5148 I cid 5461 × cxp 5555 dom cdm 5557 ran crn 5558 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 1oc1o 8097 2oc2o 8098 0cc0 10539 1c1 10540 < clt 10677 − cmin 10872 ℕcn 11640 ℕ0cn0 11900 ℤcz 11984 ...cfz 12895 ..^cfzo 13036 ♯chash 13693 Word cword 13864 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-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 |
This theorem is referenced by: efgredlemg 18870 efgredleme 18871 |
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