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Mirrors > Home > MPE Home > Th. List > efgredlemg | Structured version Visualization version GIF version |
Description: Lemma for efgred 18868. (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 6736 | . . . . . 6 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o) | |
3 | 1, 2 | eqsstri 4001 | . . . . 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 18861 | . . . . . 6 ⊢ (𝜑 → ((𝐴‘𝐾) ∈ 𝑊 ∧ (𝐵‘𝐿) ∈ 𝑊)) |
17 | 16 | simpld 497 | . . . . 5 ⊢ (𝜑 → (𝐴‘𝐾) ∈ 𝑊) |
18 | 3, 17 | sseldi 3965 | . . . 4 ⊢ (𝜑 → (𝐴‘𝐾) ∈ Word (𝐼 × 2o)) |
19 | lencl 13877 | . . . 4 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2o) → (♯‘(𝐴‘𝐾)) ∈ ℕ0) | |
20 | 18, 19 | syl 17 | . . 3 ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) ∈ ℕ0) |
21 | 20 | nn0cnd 11951 | . 2 ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) ∈ ℂ) |
22 | 16 | simprd 498 | . . . . 5 ⊢ (𝜑 → (𝐵‘𝐿) ∈ 𝑊) |
23 | 3, 22 | sseldi 3965 | . . . 4 ⊢ (𝜑 → (𝐵‘𝐿) ∈ Word (𝐼 × 2o)) |
24 | lencl 13877 | . . . 4 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2o) → (♯‘(𝐵‘𝐿)) ∈ ℕ0) | |
25 | 23, 24 | syl 17 | . . 3 ⊢ (𝜑 → (♯‘(𝐵‘𝐿)) ∈ ℕ0) |
26 | 25 | nn0cnd 11951 | . 2 ⊢ (𝜑 → (♯‘(𝐵‘𝐿)) ∈ ℂ) |
27 | 2cnd 11709 | . 2 ⊢ (𝜑 → 2 ∈ ℂ) | |
28 | 1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | efgredlema 18860 | . . . . . . 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 18852 | . . . . . 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 6668 | . . . . . . 7 ⊢ (𝐴‘𝐾) = (𝐴‘(((♯‘𝐴) − 1) − 1)) |
33 | 32 | fveq2i 6668 | . . . . . 6 ⊢ (𝑇‘(𝐴‘𝐾)) = (𝑇‘(𝐴‘(((♯‘𝐴) − 1) − 1))) |
34 | 33 | rneqi 5802 | . . . . 5 ⊢ ran (𝑇‘(𝐴‘𝐾)) = ran (𝑇‘(𝐴‘(((♯‘𝐴) − 1) − 1))) |
35 | 31, 34 | eleqtrrdi 2924 | . . . 4 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘𝐾))) |
36 | 1, 4, 5, 6 | efgtlen 18846 | . . . 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 18852 | . . . . . . 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 2913 | . . . . 5 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1)))) |
42 | 15 | fveq2i 6668 | . . . . . . 7 ⊢ (𝐵‘𝐿) = (𝐵‘(((♯‘𝐵) − 1) − 1)) |
43 | 42 | fveq2i 6668 | . . . . . 6 ⊢ (𝑇‘(𝐵‘𝐿)) = (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1))) |
44 | 43 | rneqi 5802 | . . . . 5 ⊢ ran (𝑇‘(𝐵‘𝐿)) = ran (𝑇‘(𝐵‘(((♯‘𝐵) − 1) − 1))) |
45 | 41, 44 | eleqtrrdi 2924 | . . . 4 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐵‘𝐿))) |
46 | 1, 4, 5, 6 | efgtlen 18846 | . . . 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 2858 | . 2 ⊢ (𝜑 → ((♯‘(𝐴‘𝐾)) + 2) = ((♯‘(𝐵‘𝐿)) + 2)) |
49 | 21, 26, 27, 48 | addcan2ad 10840 | 1 ⊢ (𝜑 → (♯‘(𝐴‘𝐾)) = (♯‘(𝐵‘𝐿))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 {crab 3142 ∖ cdif 3933 ∅c0 4291 {csn 4561 〈cop 4567 〈cotp 4569 ∪ ciun 4912 class class class wbr 5059 ↦ cmpt 5139 I cid 5454 × cxp 5548 dom cdm 5550 ran crn 5551 ‘cfv 6350 (class class class)co 7150 ∈ cmpo 7152 1oc1o 8089 2oc2o 8090 0cc0 10531 1c1 10532 + caddc 10534 < clt 10669 − cmin 10864 ℕcn 11632 2c2 11686 ℕ0cn0 11891 ...cfz 12886 ..^cfzo 13027 ♯chash 13684 Word cword 13855 splice csplice 14105 〈“cs2 14197 ~FG cefg 18826 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-ot 4570 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-fzo 13028 df-hash 13685 df-word 13856 df-concat 13917 df-s1 13944 df-substr 13997 df-pfx 14027 df-splice 14106 df-s2 14204 |
This theorem is referenced by: efgredleme 18863 |
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