Theorem efgtval 19754

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	⊢ ∼ = ( ~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)”⟩⟩)))

Assertion

Ref	Expression
efgtval	⊢ ((𝑋 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝑋)) ∧ 𝐴 ∈ (𝐼 × 2o)) → (𝑁(𝑇‘𝑋)𝐴) = (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝐴(𝑀‘𝐴)”⟩⟩))

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.

Step	Hyp	Ref	Expression
1		efgval.w	. . . . . 6 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))
2		efgval.r	. . . . . 6 ⊢ ∼ = ( ~FG ‘𝐼)
3		efgval2.m	. . . . . 6 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
4		efgval2.t	. . . . . 6 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
5	1, 2, 3, 4	efgtf 19753	. . . . 5 ⊢ (𝑋 ∈ 𝑊 → ((𝑇‘𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∧ (𝑇‘𝑋):((0...(♯‘𝑋)) × (𝐼 × 2o))⟶𝑊))
6	5	simpld 498	. . . 4 ⊢ (𝑋 ∈ 𝑊 → (𝑇‘𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
7	6	oveqd 7408	. . 3 ⊢ (𝑋 ∈ 𝑊 → (𝑁(𝑇‘𝑋)𝐴) = (𝑁(𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))𝐴))
8		oteq1 4837	. . . . . 6 ⊢ (𝑎 = 𝑁 → ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩ = ⟨𝑁, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)
9		oteq2 4838	. . . . . 6 ⊢ (𝑎 = 𝑁 → ⟨𝑁, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩ = ⟨𝑁, 𝑁, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)
10	8, 9	eqtrd 2796	. . . . 5 ⊢ (𝑎 = 𝑁 → ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩ = ⟨𝑁, 𝑁, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)
11	10	oveq2d 7407	. . . 4 ⊢ (𝑎 = 𝑁 → (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
12		id 22	. . . . . . 7 ⊢ (𝑏 = 𝐴 → 𝑏 = 𝐴)
13		fveq2 6862	. . . . . . 7 ⊢ (𝑏 = 𝐴 → (𝑀‘𝑏) = (𝑀‘𝐴))
14	12, 13	s2eqd 14870	. . . . . 6 ⊢ (𝑏 = 𝐴 → ⟨“𝑏(𝑀‘𝑏)”⟩ = ⟨“𝐴(𝑀‘𝐴)”⟩)
15	14	oteq3d 4842	. . . . 5 ⊢ (𝑏 = 𝐴 → ⟨𝑁, 𝑁, ⟨“𝑏(𝑀‘𝑏)”⟩⟩ = ⟨𝑁, 𝑁, ⟨“𝐴(𝑀‘𝐴)”⟩⟩)
16	15	oveq2d 7407	. . . 4 ⊢ (𝑏 = 𝐴 → (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝐴(𝑀‘𝐴)”⟩⟩))
17		eqid 2761	. . . 4 ⊢ (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
18		ovex 7424	. . . 4 ⊢ (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝐴(𝑀‘𝐴)”⟩⟩) ∈ V
19	11, 16, 17, 18	ovmpo 7551	. . 3 ⊢ ((𝑁 ∈ (0...(♯‘𝑋)) ∧ 𝐴 ∈ (𝐼 × 2o)) → (𝑁(𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))𝐴) = (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝐴(𝑀‘𝐴)”⟩⟩))
20	7, 19	sylan9eq 2816	. 2 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑁 ∈ (0...(♯‘𝑋)) ∧ 𝐴 ∈ (𝐼 × 2o))) → (𝑁(𝑇‘𝑋)𝐴) = (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝐴(𝑀‘𝐴)”⟩⟩))
21	20	3impb 1126	1 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝑁 ∈ (0...(♯‘𝑋)) ∧ 𝐴 ∈ (𝐼 × 2o)) → (𝑁(𝑇‘𝑋)𝐴) = (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝐴(𝑀‘𝐴)”⟩⟩))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141   ∖ cdif 3899  ⟨cop 4585  ⟨cotp 4587   ↦ cmpt 5178   I cid 5537   × cxp 5641  ⟶wf 6512  ‘cfv 6516  (class class class)co 7391   ∈ cmpo 7393  1_oc1o 8424  2_oc2o 8425  0cc0 11067  ...cfz 13506  ♯chash 14337  Word cword 14520   splice csplice 14756  ⟨“cs2 14848   ~_FG cefg 19737

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-ot 4588  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-1o 8431  df-2o 8432  df-er 8672  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-card 9891  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-nn 12205  df-n0 12476  df-z 12563  df-uz 12834  df-fz 13507  df-fzo 13654  df-hash 14338  df-word 14521  df-concat 14578  df-s1 14604  df-substr 14649  df-pfx 14679  df-splice 14757  df-s2 14855

This theorem is referenced by:  efginvrel2  19758  efgredleme  19774  efgredlemc  19776  efgcpbllemb  19786