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Mirrors > Home > MPE Home > Th. List > efgredlemf | Structured version Visualization version GIF version |
Description: Lemma for efgredleme 19735. (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 19722 | . . . . . . 7 ⊢ (𝐴 ∈ dom 𝑆 ↔ (𝐴 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐴‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐴))(𝐴‘𝑖) ∈ ran (𝑇‘(𝐴‘(𝑖 − 1))))) |
9 | 8 | simp1bi 1142 | . . . . . 6 ⊢ (𝐴 ∈ dom 𝑆 → 𝐴 ∈ (Word 𝑊 ∖ {∅})) |
10 | 1, 9 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Word 𝑊 ∖ {∅})) |
11 | 10 | eldifad 3959 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Word 𝑊) |
12 | wrdf 14520 | . . . 4 ⊢ (𝐴 ∈ Word 𝑊 → 𝐴:(0..^(♯‘𝐴))⟶𝑊) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ (𝜑 → 𝐴:(0..^(♯‘𝐴))⟶𝑊) |
14 | fzossfz 13697 | . . . . 5 ⊢ (0..^((♯‘𝐴) − 1)) ⊆ (0...((♯‘𝐴) − 1)) | |
15 | lencl 14534 | . . . . . . . 8 ⊢ (𝐴 ∈ Word 𝑊 → (♯‘𝐴) ∈ ℕ0) | |
16 | 11, 15 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐴) ∈ ℕ0) |
17 | 16 | nn0zd 12628 | . . . . . 6 ⊢ (𝜑 → (♯‘𝐴) ∈ ℤ) |
18 | fzoval 13679 | . . . . . 6 ⊢ ((♯‘𝐴) ∈ ℤ → (0..^(♯‘𝐴)) = (0...((♯‘𝐴) − 1))) | |
19 | 17, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → (0..^(♯‘𝐴)) = (0...((♯‘𝐴) − 1))) |
20 | 14, 19 | sseqtrrid 4033 | . . . 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 19732 | . . . . . . 7 ⊢ (𝜑 → (((♯‘𝐴) − 1) ∈ ℕ ∧ ((♯‘𝐵) − 1) ∈ ℕ)) |
27 | 26 | simpld 493 | . . . . . 6 ⊢ (𝜑 → ((♯‘𝐴) − 1) ∈ ℕ) |
28 | fzo0end 13770 | . . . . . 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 2830 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ (0..^((♯‘𝐴) − 1))) |
31 | 20, 30 | sseldd 3980 | . . 3 ⊢ (𝜑 → 𝐾 ∈ (0..^(♯‘𝐴))) |
32 | 13, 31 | ffvelcdmd 7089 | . 2 ⊢ (𝜑 → (𝐴‘𝐾) ∈ 𝑊) |
33 | 2, 3, 4, 5, 6, 7 | efgsdm 19722 | . . . . . . 7 ⊢ (𝐵 ∈ dom 𝑆 ↔ (𝐵 ∈ (Word 𝑊 ∖ {∅}) ∧ (𝐵‘0) ∈ 𝐷 ∧ ∀𝑖 ∈ (1..^(♯‘𝐵))(𝐵‘𝑖) ∈ ran (𝑇‘(𝐵‘(𝑖 − 1))))) |
34 | 33 | simp1bi 1142 | . . . . . 6 ⊢ (𝐵 ∈ dom 𝑆 → 𝐵 ∈ (Word 𝑊 ∖ {∅})) |
35 | 23, 34 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (Word 𝑊 ∖ {∅})) |
36 | 35 | eldifad 3959 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ Word 𝑊) |
37 | wrdf 14520 | . . . 4 ⊢ (𝐵 ∈ Word 𝑊 → 𝐵:(0..^(♯‘𝐵))⟶𝑊) | |
38 | 36, 37 | syl 17 | . . 3 ⊢ (𝜑 → 𝐵:(0..^(♯‘𝐵))⟶𝑊) |
39 | fzossfz 13697 | . . . . 5 ⊢ (0..^((♯‘𝐵) − 1)) ⊆ (0...((♯‘𝐵) − 1)) | |
40 | lencl 14534 | . . . . . . . 8 ⊢ (𝐵 ∈ Word 𝑊 → (♯‘𝐵) ∈ ℕ0) | |
41 | 36, 40 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (♯‘𝐵) ∈ ℕ0) |
42 | 41 | nn0zd 12628 | . . . . . 6 ⊢ (𝜑 → (♯‘𝐵) ∈ ℤ) |
43 | fzoval 13679 | . . . . . 6 ⊢ ((♯‘𝐵) ∈ ℤ → (0..^(♯‘𝐵)) = (0...((♯‘𝐵) − 1))) | |
44 | 42, 43 | syl 17 | . . . . 5 ⊢ (𝜑 → (0..^(♯‘𝐵)) = (0...((♯‘𝐵) − 1))) |
45 | 39, 44 | sseqtrrid 4033 | . . . 4 ⊢ (𝜑 → (0..^((♯‘𝐵) − 1)) ⊆ (0..^(♯‘𝐵))) |
46 | efgredlemb.l | . . . . 5 ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1) | |
47 | fzo0end 13770 | . . . . . 6 ⊢ (((♯‘𝐵) − 1) ∈ ℕ → (((♯‘𝐵) − 1) − 1) ∈ (0..^((♯‘𝐵) − 1))) | |
48 | 26, 47 | simpl2im 502 | . . . . 5 ⊢ (𝜑 → (((♯‘𝐵) − 1) − 1) ∈ (0..^((♯‘𝐵) − 1))) |
49 | 46, 48 | eqeltrid 2830 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ (0..^((♯‘𝐵) − 1))) |
50 | 45, 49 | sseldd 3980 | . . 3 ⊢ (𝜑 → 𝐿 ∈ (0..^(♯‘𝐵))) |
51 | 38, 50 | ffvelcdmd 7089 | . 2 ⊢ (𝜑 → (𝐵‘𝐿) ∈ 𝑊) |
52 | 32, 51 | jca 510 | 1 ⊢ (𝜑 → ((𝐴‘𝐾) ∈ 𝑊 ∧ (𝐵‘𝐿) ∈ 𝑊)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ∀wral 3051 {crab 3420 ∖ cdif 3944 ∅c0 4323 {csn 4624 〈cop 4630 〈cotp 4632 ∪ ciun 4994 class class class wbr 5144 ↦ cmpt 5227 I cid 5570 × cxp 5671 dom cdm 5673 ran crn 5674 ⟶wf 6540 ‘cfv 6544 (class class class)co 7414 ∈ cmpo 7416 1oc1o 8479 2oc2o 8480 0cc0 11147 1c1 11148 < clt 11287 − cmin 11483 ℕcn 12256 ℕ0cn0 12516 ℤcz 12602 ...cfz 13530 ..^cfzo 13673 ♯chash 14340 Word cword 14515 splice csplice 14750 〈“cs2 14843 ~FG cefg 19698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-int 4948 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9973 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-nn 12257 df-2 12319 df-n0 12517 df-z 12603 df-uz 12867 df-fz 13531 df-fzo 13674 df-hash 14341 df-word 14516 |
This theorem is referenced by: efgredlemg 19734 efgredleme 19735 |
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