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Mirrors > Home > MPE Home > Th. List > efgredlemg | Structured version Visualization version GIF version |
Description: Lemma for efgred 18161. (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 6256 | . . . . . 6 ⊢ ( I ‘Word (𝐼 × 2𝑜)) ⊆ Word (𝐼 × 2𝑜) | |
3 | 1, 2 | eqsstri 3635 | . . . . 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 18154 | . . . . . 6 ⊢ (𝜑 → ((𝐴‘𝐾) ∈ 𝑊 ∧ (𝐵‘𝐿) ∈ 𝑊)) |
17 | 16 | simpld 475 | . . . . 5 ⊢ (𝜑 → (𝐴‘𝐾) ∈ 𝑊) |
18 | 3, 17 | sseldi 3601 | . . . 4 ⊢ (𝜑 → (𝐴‘𝐾) ∈ Word (𝐼 × 2𝑜)) |
19 | lencl 13324 | . . . 4 ⊢ ((𝐴‘𝐾) ∈ Word (𝐼 × 2𝑜) → (#‘(𝐴‘𝐾)) ∈ ℕ0) | |
20 | 18, 19 | syl 17 | . . 3 ⊢ (𝜑 → (#‘(𝐴‘𝐾)) ∈ ℕ0) |
21 | 20 | nn0cnd 11353 | . 2 ⊢ (𝜑 → (#‘(𝐴‘𝐾)) ∈ ℂ) |
22 | 16 | simprd 479 | . . . . 5 ⊢ (𝜑 → (𝐵‘𝐿) ∈ 𝑊) |
23 | 3, 22 | sseldi 3601 | . . . 4 ⊢ (𝜑 → (𝐵‘𝐿) ∈ Word (𝐼 × 2𝑜)) |
24 | lencl 13324 | . . . 4 ⊢ ((𝐵‘𝐿) ∈ Word (𝐼 × 2𝑜) → (#‘(𝐵‘𝐿)) ∈ ℕ0) | |
25 | 23, 24 | syl 17 | . . 3 ⊢ (𝜑 → (#‘(𝐵‘𝐿)) ∈ ℕ0) |
26 | 25 | nn0cnd 11353 | . 2 ⊢ (𝜑 → (#‘(𝐵‘𝐿)) ∈ ℂ) |
27 | 2cnd 11093 | . 2 ⊢ (𝜑 → 2 ∈ ℂ) | |
28 | 1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 | efgredlema 18153 | . . . . . . 7 ⊢ (𝜑 → (((#‘𝐴) − 1) ∈ ℕ ∧ ((#‘𝐵) − 1) ∈ ℕ)) |
29 | 28 | simpld 475 | . . . . . 6 ⊢ (𝜑 → ((#‘𝐴) − 1) ∈ ℕ) |
30 | 1, 4, 5, 6, 7, 8 | efgsdmi 18145 | . . . . . 6 ⊢ ((𝐴 ∈ dom 𝑆 ∧ ((#‘𝐴) − 1) ∈ ℕ) → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘(((#‘𝐴) − 1) − 1)))) |
31 | 10, 29, 30 | syl2anc 693 | . . . . 5 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘(((#‘𝐴) − 1) − 1)))) |
32 | 14 | fveq2i 6194 | . . . . . . 7 ⊢ (𝐴‘𝐾) = (𝐴‘(((#‘𝐴) − 1) − 1)) |
33 | 32 | fveq2i 6194 | . . . . . 6 ⊢ (𝑇‘(𝐴‘𝐾)) = (𝑇‘(𝐴‘(((#‘𝐴) − 1) − 1))) |
34 | 33 | rneqi 5352 | . . . . 5 ⊢ ran (𝑇‘(𝐴‘𝐾)) = ran (𝑇‘(𝐴‘(((#‘𝐴) − 1) − 1))) |
35 | 31, 34 | syl6eleqr 2712 | . . . 4 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘𝐾))) |
36 | 1, 4, 5, 6 | efgtlen 18139 | . . . 4 ⊢ (((𝐴‘𝐾) ∈ 𝑊 ∧ (𝑆‘𝐴) ∈ ran (𝑇‘(𝐴‘𝐾))) → (#‘(𝑆‘𝐴)) = ((#‘(𝐴‘𝐾)) + 2)) |
37 | 17, 35, 36 | syl2anc 693 | . . 3 ⊢ (𝜑 → (#‘(𝑆‘𝐴)) = ((#‘(𝐴‘𝐾)) + 2)) |
38 | 28 | simprd 479 | . . . . . . 7 ⊢ (𝜑 → ((#‘𝐵) − 1) ∈ ℕ) |
39 | 1, 4, 5, 6, 7, 8 | efgsdmi 18145 | . . . . . . 7 ⊢ ((𝐵 ∈ dom 𝑆 ∧ ((#‘𝐵) − 1) ∈ ℕ) → (𝑆‘𝐵) ∈ ran (𝑇‘(𝐵‘(((#‘𝐵) − 1) − 1)))) |
40 | 11, 38, 39 | syl2anc 693 | . . . . . 6 ⊢ (𝜑 → (𝑆‘𝐵) ∈ ran (𝑇‘(𝐵‘(((#‘𝐵) − 1) − 1)))) |
41 | 12, 40 | eqeltrd 2701 | . . . . 5 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐵‘(((#‘𝐵) − 1) − 1)))) |
42 | 15 | fveq2i 6194 | . . . . . . 7 ⊢ (𝐵‘𝐿) = (𝐵‘(((#‘𝐵) − 1) − 1)) |
43 | 42 | fveq2i 6194 | . . . . . 6 ⊢ (𝑇‘(𝐵‘𝐿)) = (𝑇‘(𝐵‘(((#‘𝐵) − 1) − 1))) |
44 | 43 | rneqi 5352 | . . . . 5 ⊢ ran (𝑇‘(𝐵‘𝐿)) = ran (𝑇‘(𝐵‘(((#‘𝐵) − 1) − 1))) |
45 | 41, 44 | syl6eleqr 2712 | . . . 4 ⊢ (𝜑 → (𝑆‘𝐴) ∈ ran (𝑇‘(𝐵‘𝐿))) |
46 | 1, 4, 5, 6 | efgtlen 18139 | . . . 4 ⊢ (((𝐵‘𝐿) ∈ 𝑊 ∧ (𝑆‘𝐴) ∈ ran (𝑇‘(𝐵‘𝐿))) → (#‘(𝑆‘𝐴)) = ((#‘(𝐵‘𝐿)) + 2)) |
47 | 22, 45, 46 | syl2anc 693 | . . 3 ⊢ (𝜑 → (#‘(𝑆‘𝐴)) = ((#‘(𝐵‘𝐿)) + 2)) |
48 | 37, 47 | eqtr3d 2658 | . 2 ⊢ (𝜑 → ((#‘(𝐴‘𝐾)) + 2) = ((#‘(𝐵‘𝐿)) + 2)) |
49 | 21, 26, 27, 48 | addcan2ad 10242 | 1 ⊢ (𝜑 → (#‘(𝐴‘𝐾)) = (#‘(𝐵‘𝐿))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 ∖ cdif 3571 ∅c0 3915 {csn 4177 〈cop 4183 〈cotp 4185 ∪ ciun 4520 class class class wbr 4653 ↦ cmpt 4729 I cid 5023 × cxp 5112 dom cdm 5114 ran crn 5115 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 1𝑜c1o 7553 2𝑜c2o 7554 0cc0 9936 1c1 9937 + caddc 9939 < clt 10074 − cmin 10266 ℕcn 11020 2c2 11070 ℕ0cn0 11292 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 splice csplice 13296 〈“cs2 13586 ~FG cefg 18119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301 df-s1 13302 df-substr 13303 df-splice 13304 df-s2 13593 |
This theorem is referenced by: efgredleme 18156 |
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