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Theorem efgtval 18363
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 18362 . . . . 5 (𝑋𝑊 → ((𝑇𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∧ (𝑇𝑋):((0...(♯‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊))
65simpld 483 . . . 4 (𝑋𝑊 → (𝑇𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
76oveqd 6829 . . 3 (𝑋𝑊 → (𝑁(𝑇𝑋)𝐴) = (𝑁(𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))𝐴))
8 oteq1 4559 . . . . . 6 (𝑎 = 𝑁 → ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩ = ⟨𝑁, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)
9 oteq2 4560 . . . . . 6 (𝑎 = 𝑁 → ⟨𝑁, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩ = ⟨𝑁, 𝑁, ⟨“𝑏(𝑀𝑏)”⟩⟩)
108, 9eqtrd 2808 . . . . 5 (𝑎 = 𝑁 → ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩ = ⟨𝑁, 𝑁, ⟨“𝑏(𝑀𝑏)”⟩⟩)
1110oveq2d 6828 . . . 4 (𝑎 = 𝑁 → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) = (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝑏(𝑀𝑏)”⟩⟩))
12 id 22 . . . . . . 7 (𝑏 = 𝐴𝑏 = 𝐴)
13 fveq2 6348 . . . . . . 7 (𝑏 = 𝐴 → (𝑀𝑏) = (𝑀𝐴))
1412, 13s2eqd 13839 . . . . . 6 (𝑏 = 𝐴 → ⟨“𝑏(𝑀𝑏)”⟩ = ⟨“𝐴(𝑀𝐴)”⟩)
1514oteq3d 4564 . . . . 5 (𝑏 = 𝐴 → ⟨𝑁, 𝑁, ⟨“𝑏(𝑀𝑏)”⟩⟩ = ⟨𝑁, 𝑁, ⟨“𝐴(𝑀𝐴)”⟩⟩)
1615oveq2d 6828 . . . 4 (𝑏 = 𝐴 → (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝑏(𝑀𝑏)”⟩⟩) = (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝐴(𝑀𝐴)”⟩⟩))
17 eqid 2774 . . . 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 6844 . . . 4 (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝐴(𝑀𝐴)”⟩⟩) ∈ V
1911, 16, 17, 18ovmpt2 6964 . . 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 2828 . 2 ((𝑋𝑊 ∧ (𝑁 ∈ (0...(♯‘𝑋)) ∧ 𝐴 ∈ (𝐼 × 2𝑜))) → (𝑁(𝑇𝑋)𝐴) = (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝐴(𝑀𝐴)”⟩⟩))
21203impb 1134 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 383  w3a 1098   = wceq 1634  wcel 2148  cdif 3726  cop 4332  cotp 4334  cmpt 4876   I cid 5170   × cxp 5261  wf 6038  cfv 6042  (class class class)co 6812  cmpt2 6814  1𝑜c1o 7727  2𝑜c2o 7728  0cc0 10159  ...cfz 12555  chash 13343  Word cword 13509   splice csplice 13514  ⟨“cs2 13817   ~FG cefg 18346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-rep 4917  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117  ax-cnex 10215  ax-resscn 10216  ax-1cn 10217  ax-icn 10218  ax-addcl 10219  ax-addrcl 10220  ax-mulcl 10221  ax-mulrcl 10222  ax-mulcom 10223  ax-addass 10224  ax-mulass 10225  ax-distr 10226  ax-i2m1 10227  ax-1ne0 10228  ax-1rid 10229  ax-rnegex 10230  ax-rrecex 10231  ax-cnre 10232  ax-pre-lttri 10233  ax-pre-lttrn 10234  ax-pre-ltadd 10235  ax-pre-mulgt0 10236
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-ot 4335  df-uni 4586  df-int 4623  df-iun 4667  df-br 4798  df-opab 4860  df-mpt 4877  df-tr 4900  df-id 5171  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-we 5224  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-pred 5834  df-ord 5880  df-on 5881  df-lim 5882  df-suc 5883  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-riota 6773  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-om 7234  df-1st 7336  df-2nd 7337  df-wrecs 7580  df-recs 7642  df-rdg 7680  df-1o 7734  df-2o 7735  df-oadd 7738  df-er 7917  df-map 8032  df-pm 8033  df-en 8131  df-dom 8132  df-sdom 8133  df-fin 8134  df-card 8986  df-pnf 10299  df-mnf 10300  df-xr 10301  df-ltxr 10302  df-le 10303  df-sub 10491  df-neg 10492  df-nn 11244  df-n0 11517  df-z 11602  df-uz 11911  df-fz 12556  df-fzo 12696  df-hash 13344  df-word 13517  df-concat 13519  df-s1 13520  df-substr 13521  df-splice 13522  df-s2 13824
This theorem is referenced by:  efginvrel2  18367  efgredleme  18383  efgredlemc  18385  efgcpbllemb  18395
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