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Theorem efgredlemb 18083
Description: The reduced word that forms the base of the sequence in efgsval 18068 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-2015.)
Hypotheses
Ref Expression
efgval.w 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
efgval.r = ( ~FG𝐼)
efgval2.m 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
efgval2.t 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
efgred.d 𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))
efgred.s 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))
efgredlem.1 (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))
efgredlem.2 (𝜑𝐴 ∈ dom 𝑆)
efgredlem.3 (𝜑𝐵 ∈ dom 𝑆)
efgredlem.4 (𝜑 → (𝑆𝐴) = (𝑆𝐵))
efgredlem.5 (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))
efgredlemb.k 𝐾 = (((#‘𝐴) − 1) − 1)
efgredlemb.l 𝐿 = (((#‘𝐵) − 1) − 1)
efgredlemb.p (𝜑𝑃 ∈ (0...(#‘(𝐴𝐾))))
efgredlemb.q (𝜑𝑄 ∈ (0...(#‘(𝐵𝐿))))
efgredlemb.u (𝜑𝑈 ∈ (𝐼 × 2𝑜))
efgredlemb.v (𝜑𝑉 ∈ (𝐼 × 2𝑜))
efgredlemb.6 (𝜑 → (𝑆𝐴) = (𝑃(𝑇‘(𝐴𝐾))𝑈))
efgredlemb.7 (𝜑 → (𝑆𝐵) = (𝑄(𝑇‘(𝐵𝐿))𝑉))
efgredlemb.8 (𝜑 → ¬ (𝐴𝐾) = (𝐵𝐿))
Assertion
Ref Expression
efgredlemb ¬ 𝜑
Distinct variable groups:   𝑎,𝑏,𝐴   𝑦,𝑎,𝑧,𝑏   𝐿,𝑎,𝑏   𝐾,𝑎,𝑏   𝑡,𝑛,𝑣,𝑤,𝑦,𝑧,𝑃   𝑚,𝑎,𝑛,𝑡,𝑣,𝑤,𝑥,𝑀,𝑏   𝑈,𝑛,𝑣,𝑤,𝑦,𝑧   𝑘,𝑎,𝑇,𝑏,𝑚,𝑡,𝑥   𝑛,𝑉,𝑣,𝑤,𝑦,𝑧   𝑄,𝑛,𝑡,𝑣,𝑤,𝑦,𝑧   𝑊,𝑎,𝑏   𝑘,𝑛,𝑣,𝑤,𝑦,𝑧,𝑊,𝑚,𝑡,𝑥   ,𝑎,𝑏,𝑚,𝑡,𝑥,𝑦,𝑧   𝐵,𝑎,𝑏   𝑆,𝑎,𝑏   𝐼,𝑎,𝑏,𝑚,𝑛,𝑡,𝑣,𝑤,𝑥,𝑦,𝑧   𝐷,𝑎,𝑏,𝑚,𝑡
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛,𝑎,𝑏)   𝐴(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛)   𝐵(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛)   𝐷(𝑥,𝑦,𝑧,𝑤,𝑣,𝑘,𝑛)   𝑃(𝑥,𝑘,𝑚,𝑎,𝑏)   𝑄(𝑥,𝑘,𝑚,𝑎,𝑏)   (𝑤,𝑣,𝑘,𝑛)   𝑆(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛)   𝑇(𝑦,𝑧,𝑤,𝑣,𝑛)   𝑈(𝑥,𝑡,𝑘,𝑚,𝑎,𝑏)   𝐼(𝑘)   𝐾(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛)   𝐿(𝑥,𝑦,𝑧,𝑤,𝑣,𝑡,𝑘,𝑚,𝑛)   𝑀(𝑦,𝑧,𝑘)   𝑉(𝑥,𝑡,𝑘,𝑚,𝑎,𝑏)

Proof of Theorem efgredlemb
StepHypRef Expression
1 efgval.w . . . . 5 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
2 efgval.r . . . . 5 = ( ~FG𝐼)
3 efgval2.m . . . . 5 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
4 efgval2.t . . . . 5 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
5 efgred.d . . . . 5 𝐷 = (𝑊 𝑥𝑊 ran (𝑇𝑥))
6 efgred.s . . . . 5 𝑆 = (𝑚 ∈ {𝑡 ∈ (Word 𝑊 ∖ {∅}) ∣ ((𝑡‘0) ∈ 𝐷 ∧ ∀𝑘 ∈ (1..^(#‘𝑡))(𝑡𝑘) ∈ ran (𝑇‘(𝑡‘(𝑘 − 1))))} ↦ (𝑚‘((#‘𝑚) − 1)))
7 efgredlem.1 . . . . . 6 (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))
8 efgredlem.4 . . . . . . 7 (𝜑 → (𝑆𝐴) = (𝑆𝐵))
9 fveq2 6150 . . . . . . . . . 10 ((𝑆𝐴) = (𝑆𝐵) → (#‘(𝑆𝐴)) = (#‘(𝑆𝐵)))
109breq2d 4627 . . . . . . . . 9 ((𝑆𝐴) = (𝑆𝐵) → ((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) ↔ (#‘(𝑆𝑎)) < (#‘(𝑆𝐵))))
1110imbi1d 331 . . . . . . . 8 ((𝑆𝐴) = (𝑆𝐵) → (((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ((#‘(𝑆𝑎)) < (#‘(𝑆𝐵)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0)))))
12112ralbidv 2983 . . . . . . 7 ((𝑆𝐴) = (𝑆𝐵) → (∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐵)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0)))))
138, 12syl 17 . . . . . 6 (𝜑 → (∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐴)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))) ↔ ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐵)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0)))))
147, 13mpbid 222 . . . . 5 (𝜑 → ∀𝑎 ∈ dom 𝑆𝑏 ∈ dom 𝑆((#‘(𝑆𝑎)) < (#‘(𝑆𝐵)) → ((𝑆𝑎) = (𝑆𝑏) → (𝑎‘0) = (𝑏‘0))))
15 efgredlem.3 . . . . 5 (𝜑𝐵 ∈ dom 𝑆)
16 efgredlem.2 . . . . 5 (𝜑𝐴 ∈ dom 𝑆)
178eqcomd 2627 . . . . 5 (𝜑 → (𝑆𝐵) = (𝑆𝐴))
18 efgredlem.5 . . . . . 6 (𝜑 → ¬ (𝐴‘0) = (𝐵‘0))
19 eqcom 2628 . . . . . 6 ((𝐴‘0) = (𝐵‘0) ↔ (𝐵‘0) = (𝐴‘0))
2018, 19sylnib 318 . . . . 5 (𝜑 → ¬ (𝐵‘0) = (𝐴‘0))
21 efgredlemb.l . . . . 5 𝐿 = (((#‘𝐵) − 1) − 1)
22 efgredlemb.k . . . . 5 𝐾 = (((#‘𝐴) − 1) − 1)
23 efgredlemb.q . . . . 5 (𝜑𝑄 ∈ (0...(#‘(𝐵𝐿))))
24 efgredlemb.p . . . . 5 (𝜑𝑃 ∈ (0...(#‘(𝐴𝐾))))
25 efgredlemb.v . . . . 5 (𝜑𝑉 ∈ (𝐼 × 2𝑜))
26 efgredlemb.u . . . . 5 (𝜑𝑈 ∈ (𝐼 × 2𝑜))
27 efgredlemb.7 . . . . 5 (𝜑 → (𝑆𝐵) = (𝑄(𝑇‘(𝐵𝐿))𝑉))
28 efgredlemb.6 . . . . 5 (𝜑 → (𝑆𝐴) = (𝑃(𝑇‘(𝐴𝐾))𝑈))
29 efgredlemb.8 . . . . . 6 (𝜑 → ¬ (𝐴𝐾) = (𝐵𝐿))
30 eqcom 2628 . . . . . 6 ((𝐴𝐾) = (𝐵𝐿) ↔ (𝐵𝐿) = (𝐴𝐾))
3129, 30sylnib 318 . . . . 5 (𝜑 → ¬ (𝐵𝐿) = (𝐴𝐾))
321, 2, 3, 4, 5, 6, 14, 15, 16, 17, 20, 21, 22, 23, 24, 25, 26, 27, 28, 31efgredlemc 18082 . . . 4 (𝜑 → (𝑄 ∈ (ℤ𝑃) → (𝐵‘0) = (𝐴‘0)))
3332, 19syl6ibr 242 . . 3 (𝜑 → (𝑄 ∈ (ℤ𝑃) → (𝐴‘0) = (𝐵‘0)))
341, 2, 3, 4, 5, 6, 7, 16, 15, 8, 18, 22, 21, 24, 23, 26, 25, 28, 27, 29efgredlemc 18082 . . 3 (𝜑 → (𝑃 ∈ (ℤ𝑄) → (𝐴‘0) = (𝐵‘0)))
35 elfzelz 12287 . . . . 5 (𝑃 ∈ (0...(#‘(𝐴𝐾))) → 𝑃 ∈ ℤ)
3624, 35syl 17 . . . 4 (𝜑𝑃 ∈ ℤ)
37 elfzelz 12287 . . . . 5 (𝑄 ∈ (0...(#‘(𝐵𝐿))) → 𝑄 ∈ ℤ)
3823, 37syl 17 . . . 4 (𝜑𝑄 ∈ ℤ)
39 uztric 11656 . . . 4 ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℤ) → (𝑄 ∈ (ℤ𝑃) ∨ 𝑃 ∈ (ℤ𝑄)))
4036, 38, 39syl2anc 692 . . 3 (𝜑 → (𝑄 ∈ (ℤ𝑃) ∨ 𝑃 ∈ (ℤ𝑄)))
4133, 34, 40mpjaod 396 . 2 (𝜑 → (𝐴‘0) = (𝐵‘0))
4241, 18pm2.65i 185 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911  cdif 3553  c0 3893  {csn 4150  cop 4156  cotp 4158   ciun 4487   class class class wbr 4615  cmpt 4675   I cid 4986   × cxp 5074  dom cdm 5076  ran crn 5077  cfv 5849  (class class class)co 6607  cmpt2 6609  1𝑜c1o 7501  2𝑜c2o 7502  0cc0 9883  1c1 9884   < clt 10021  cmin 10213  cz 11324  cuz 11634  ...cfz 12271  ..^cfzo 12409  #chash 13060  Word cword 13233   splice csplice 13238  ⟨“cs2 13526   ~FG cefg 18043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-ot 4159  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-2o 7509  df-oadd 7512  df-er 7690  df-map 7807  df-pm 7808  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-card 8712  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-nn 10968  df-2 11026  df-n0 11240  df-z 11325  df-uz 11635  df-rp 11780  df-fz 12272  df-fzo 12410  df-hash 13061  df-word 13241  df-concat 13243  df-s1 13244  df-substr 13245  df-splice 13246  df-s2 13533
This theorem is referenced by:  efgredlem  18084
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