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Mirrors > Home > MPE Home > Th. List > efgredlemg | Structured version Visualization version GIF version |
Description: Lemma for efgred 18877. (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) |
efgredlemb.p | ⊢ (𝜑 → 𝑃 ∈ (0...(♯‘(𝐴‘𝐾)))) |
efgredlemb.q | ⊢ (𝜑 → 𝑄 ∈ (0...(♯‘(𝐵‘𝐿)))) |
efgredlemb.u | ⊢ (𝜑 → 𝑈 ∈ (𝐼 × 2o)) |
efgredlemb.v | ⊢ (𝜑 → 𝑉 ∈ (𝐼 × 2o)) |
efgredlemb.6 | ⊢ (𝜑 → (𝑆‘𝐴) = (𝑃(𝑇‘(𝐴‘𝐾))𝑈)) |
efgredlemb.7 | ⊢ (𝜑 → (𝑆‘𝐵) = (𝑄(𝑇‘(𝐵‘𝐿))𝑉)) |
Ref | Expression |
---|---|
efgredlemg | ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) = (♯‘(𝐵‘𝐿))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efgval.w | . . . . . 6 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o)) | |
2 | fviss 6744 | . . . . . 6 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o) | |
3 | 1, 2 | eqsstri 4004 | . . . . 5 ⊢ 𝑊 ⊆ Word (𝐼 × 2o) |
4 | efgval.r | . . . . . . 7 ⊢ ∼ = ( ~FG ‘𝐼) | |
5 | efgval2.m | . . . . . . 7 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ 〈𝑦, (1o ∖ 𝑧)〉) | |
6 | efgval2.t | . . . . . . 7 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice 〈𝑛, 𝑛, 〈“𝑤(𝑀‘𝑤)”〉〉))) | |
7 | efgred.d | . . . . . . 7 ⊢ 𝐷 = (𝑊 ∖ ∪ 𝑥 ∈ 𝑊 ran (𝑇‘𝑥)) | |
8 | efgred.s | . . . . . . 7 ⊢ 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(♯‘𝑡))(𝑡‘𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((♯‘𝑚) − 1))) | |
9 | efgredlem.1 | . . . . . . 7 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑆∀𝑏 ∈ dom 𝑆((♯‘(𝑆‘𝑎)) < (♯‘(𝑆‘𝐴)) → ((𝑆‘𝑎) = (𝑆‘𝑏) → (𝑎‘0) = (𝑏‘0)))) | |
10 | efgredlem.2 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ dom 𝑆) | |
11 | efgredlem.3 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ dom 𝑆) | |
12 | efgredlem.4 | . . . . . . 7 ⊢ (𝜑 → (𝑆‘𝐴) = (𝑆‘𝐵)) | |
13 | efgredlem.5 | . . . . . . 7 ⊢ (𝜑 → ¬ (𝐴‘0) = (𝐵‘0)) | |
14 | efgredlemb.k | . . . . . . 7 ⊢ 𝐾 = (((♯‘𝐴) − 1) − 1) | |
15 | efgredlemb.l | . . . . . . 7 ⊢ 𝐿 = (((♯‘𝐵) − 1) − 1) | |
16 | 1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 | efgredlemf 18870 | . . . . . 6 ⊢ (𝜑 → ((𝐴‘𝐾) ∈ 𝑊 ∧ (𝐵‘𝐿) ∈ 𝑊)) |
17 | 16 | simpld 497 | . . . . 5 ⊢ (𝜑 → (𝐴‘𝐾) ∈ 𝑊) |
18 | 3, 17 | sseldi 3968 | . . . 4 ⊢ (𝜑 → (𝐴‘𝐾) ∈ Word (𝐼 × 2o)) |
19 | lencl 13886 | . . . 4 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2o) → (♯‘(𝐴‘𝐾)) ∈ ℕ0) | |
20 | 18, 19 | syl 17 | . . 3 ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) ∈ ℕ0) |
21 | 20 | nn0cnd 11960 | . 2 ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) ∈ ℂ) |
22 | 16 | simprd 498 | . . . . 5 ⊢ (𝜑 → (𝐵‘𝐿) ∈ 𝑊) |
23 | 3, 22 | sseldi 3968 | . . . 4 ⊢ (𝜑 → (𝐵‘𝐿) ∈ Word (𝐼 × 2o)) |
24 | lencl 13886 | . . . 4 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2o) → (♯‘(𝐵‘𝐿)) ∈ ℕ0) | |
25 | 23, 24 | syl 17 | . . 3 ⊢ (𝜑 → (♯‘(𝐵‘𝐿)) ∈ ℕ0) |
26 | 25 | nn0cnd 11960 | . 2 ⊢ (𝜑 → (♯‘(𝐵‘𝐿)) ∈ ℂ) |
27 | 2cnd 11718 | . 2 ⊢ (𝜑 → 2 ∈ ℂ) | |
28 | 1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | efgredlema 18869 | . . . . . . 7 ⊢ (𝜑 → (((♯‘𝐴) − 1) ∈ ℕ ∧ ((♯‘𝐵) − 1) ∈ ℕ)) |
29 | 28 | simpld 497 | . . . . . 6 ⊢ (𝜑 → ((♯‘𝐴) − 1) ∈ ℕ) |
30 | 1, 4, 5, 6, 7, 8 | efgsdmi 18861 | . . . . . 6 ⊢ ((𝐴 ∈ dom 𝑆 ∧ ((♯‘𝐴) − 1) ∈ ℕ) → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘(((♯‘𝐴) − 1) − 1)))) |
31 | 10, 29, 30 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘(((♯‘𝐴) − 1) − 1)))) |
32 | 14 | fveq2i 6676 | . . . . . . 7 ⊢ (𝐴‘𝐾) = (𝐴‘(((♯‘𝐴) − 1) − 1)) |
33 | 32 | fveq2i 6676 | . . . . . 6 ⊢ (𝑇‘(𝐴‘𝐾)) = (𝑇‘(𝐴‘(((♯‘𝐴) − 1) − 1))) |
34 | 33 | rneqi 5810 | . . . . 5 ⊢ ran (𝑇‘(𝐴‘𝐾)) = ran (𝑇‘(𝐴‘(((♯‘𝐴) − 1) − 1))) |
35 | 31, 34 | eleqtrrdi 2927 | . . . 4 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘𝐾))) |
36 | 1, 4, 5, 6 | efgtlen 18855 | . . . 4 ⊢ (((𝐴‘𝐾) ∈ 𝑊 ∧ (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘𝐾))) → (♯‘(𝑆‘𝐴)) = ((♯‘(𝐴‘𝐾)) + 2)) |
37 | 17, 35, 36 | syl2anc 586 | . . 3 ⊢ (𝜑 → (♯‘(𝑆‘𝐴)) = ((♯‘(𝐴‘𝐾)) + 2)) |
38 | 28 | simprd 498 | . . . . . . 7 ⊢ (𝜑 → ((♯‘𝐵) − 1) ∈ ℕ) |
39 | 1, 4, 5, 6, 7, 8 | efgsdmi 18861 | . . . . . . 7 ⊢ ((𝐵 ∈ dom 𝑆 ∧ ((♯‘𝐵) − 1) ∈ ℕ) → (𝑆‘𝐵) ∈ ran (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1)))) |
40 | 11, 38, 39 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → (𝑆‘𝐵) ∈ ran (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1)))) |
41 | 12, 40 | eqeltrd 2916 | . . . . 5 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1)))) |
42 | 15 | fveq2i 6676 | . . . . . . 7 ⊢ (𝐵‘𝐿) = (𝐵‘(((♯‘𝐵) − 1) − 1)) |
43 | 42 | fveq2i 6676 | . . . . . 6 ⊢ (𝑇‘(𝐵‘𝐿)) = (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1))) |
44 | 43 | rneqi 5810 | . . . . 5 ⊢ ran (𝑇‘(𝐵‘𝐿)) = ran (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1))) |
45 | 41, 44 | eleqtrrdi 2927 | . . . 4 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐵‘𝐿))) |
46 | 1, 4, 5, 6 | efgtlen 18855 | . . . 4 ⊢ (((𝐵‘𝐿) ∈ 𝑊 ∧ (𝑆‘𝐴) ∈ ran (𝑇‘(𝐵‘𝐿))) → (♯‘(𝑆‘𝐴)) = ((♯‘(𝐵‘𝐿)) + 2)) |
47 | 22, 45, 46 | syl2anc 586 | . . 3 ⊢ (𝜑 → (♯‘(𝑆‘𝐴)) = ((♯‘(𝐵‘𝐿)) + 2)) |
48 | 37, 47 | eqtr3d 2861 | . 2 ⊢ (𝜑 → ((♯‘(𝐴‘𝐾)) + 2) = ((♯‘(𝐵‘𝐿)) + 2)) |
49 | 21, 26, 27, 48 | addcan2ad 10849 | 1 ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) = (♯‘(𝐵‘𝐿))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3141 {crab 3145 ∖ cdif 3936 ∅c0 4294 {csn 4570 〈cop 4576 〈cotp 4578 ∪ ciun 4922 class class class wbr 5069 ↦ cmpt 5149 I cid 5462 × cxp 5556 dom cdm 5558 ran crn 5559 ‘cfv 6358 (class class class)co 7159 ∈ cmpo 7161 1oc1o 8098 2oc2o 8099 0cc0 10540 1c1 10541 + caddc 10543 < clt 10678 − cmin 10873 ℕcn 11641 2c2 11695 ℕ0cn0 11900 ...cfz 12895 ..^cfzo 13036 ♯chash 13693 Word cword 13864 splice csplice 14114 〈“cs2 14206 ~FG cefg 18835 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-ot 4579 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-concat 13926 df-s1 13953 df-substr 14006 df-pfx 14036 df-splice 14115 df-s2 14213 |
This theorem is referenced by: efgredleme 18872 |
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