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Theorem efgtval 18057
Description: Value of the extension function, which maps a word (a representation of the group element as a sequence of elements and their inverses) to its direct extensions, defined as the original representation with an element and its inverse inserted somewhere in the string. (Contributed by Mario Carneiro, 29-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)”⟩⟩)))
Assertion
Ref Expression
efgtval ((𝑋𝑊𝑁 ∈ (0...(#‘𝑋)) ∧ 𝐴 ∈ (𝐼 × 2𝑜)) → (𝑁(𝑇𝑋)𝐴) = (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝐴(𝑀𝐴)”⟩⟩))
Distinct variable groups:   𝑦,𝑧   𝑣,𝑛,𝑤,𝑦,𝑧   𝑛,𝑀,𝑣,𝑤   𝑛,𝑊,𝑣,𝑤,𝑦,𝑧   𝑦, ,𝑧   𝑛,𝐼,𝑣,𝑤,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑦,𝑧,𝑤,𝑣,𝑛)   (𝑤,𝑣,𝑛)   𝑇(𝑦,𝑧,𝑤,𝑣,𝑛)   𝑀(𝑦,𝑧)   𝑁(𝑦,𝑧,𝑤,𝑣,𝑛)   𝑋(𝑦,𝑧,𝑤,𝑣,𝑛)

Proof of Theorem efgtval
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . . . . 6 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
2 efgval.r . . . . . 6 = ( ~FG𝐼)
3 efgval2.m . . . . . 6 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
4 efgval2.t . . . . . 6 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
51, 2, 3, 4efgtf 18056 . . . . 5 (𝑋𝑊 → ((𝑇𝑋) = (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∧ (𝑇𝑋):((0...(#‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊))
65simpld 475 . . . 4 (𝑋𝑊 → (𝑇𝑋) = (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
76oveqd 6621 . . 3 (𝑋𝑊 → (𝑁(𝑇𝑋)𝐴) = (𝑁(𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))𝐴))
8 oteq1 4379 . . . . . 6 (𝑎 = 𝑁 → ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩ = ⟨𝑁, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)
9 oteq2 4380 . . . . . 6 (𝑎 = 𝑁 → ⟨𝑁, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩ = ⟨𝑁, 𝑁, ⟨“𝑏(𝑀𝑏)”⟩⟩)
108, 9eqtrd 2655 . . . . 5 (𝑎 = 𝑁 → ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩ = ⟨𝑁, 𝑁, ⟨“𝑏(𝑀𝑏)”⟩⟩)
1110oveq2d 6620 . . . 4 (𝑎 = 𝑁 → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) = (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝑏(𝑀𝑏)”⟩⟩))
12 id 22 . . . . . . 7 (𝑏 = 𝐴𝑏 = 𝐴)
13 fveq2 6148 . . . . . . 7 (𝑏 = 𝐴 → (𝑀𝑏) = (𝑀𝐴))
1412, 13s2eqd 13545 . . . . . 6 (𝑏 = 𝐴 → ⟨“𝑏(𝑀𝑏)”⟩ = ⟨“𝐴(𝑀𝐴)”⟩)
1514oteq3d 4384 . . . . 5 (𝑏 = 𝐴 → ⟨𝑁, 𝑁, ⟨“𝑏(𝑀𝑏)”⟩⟩ = ⟨𝑁, 𝑁, ⟨“𝐴(𝑀𝐴)”⟩⟩)
1615oveq2d 6620 . . . 4 (𝑏 = 𝐴 → (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝑏(𝑀𝑏)”⟩⟩) = (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝐴(𝑀𝐴)”⟩⟩))
17 eqid 2621 . . . 4 (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) = (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
18 ovex 6632 . . . 4 (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝐴(𝑀𝐴)”⟩⟩) ∈ V
1911, 16, 17, 18ovmpt2 6749 . . 3 ((𝑁 ∈ (0...(#‘𝑋)) ∧ 𝐴 ∈ (𝐼 × 2𝑜)) → (𝑁(𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))𝐴) = (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝐴(𝑀𝐴)”⟩⟩))
207, 19sylan9eq 2675 . 2 ((𝑋𝑊 ∧ (𝑁 ∈ (0...(#‘𝑋)) ∧ 𝐴 ∈ (𝐼 × 2𝑜))) → (𝑁(𝑇𝑋)𝐴) = (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝐴(𝑀𝐴)”⟩⟩))
21203impb 1257 1 ((𝑋𝑊𝑁 ∈ (0...(#‘𝑋)) ∧ 𝐴 ∈ (𝐼 × 2𝑜)) → (𝑁(𝑇𝑋)𝐴) = (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝐴(𝑀𝐴)”⟩⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  cdif 3552  cop 4154  cotp 4156  cmpt 4673   I cid 4984   × cxp 5072  wf 5843  cfv 5847  (class class class)co 6604  cmpt2 6606  1𝑜c1o 7498  2𝑜c2o 7499  0cc0 9880  ...cfz 12268  #chash 13057  Word cword 13230   splice csplice 13235  ⟨“cs2 13523   ~FG cefg 18040
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
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 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-ot 4157  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-concat 13240  df-s1 13241  df-substr 13242  df-splice 13243  df-s2 13530
This theorem is referenced by:  efginvrel2  18061  efgredleme  18077  efgredlemc  18079  efgcpbllemb  18089
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