Theorem efgtval 19664

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 19663	. . . . 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 494	. . . 4 ⊢ (𝑋 ∈ 𝑊 → (𝑇‘𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
7	6	oveqd 7385	. . 3 ⊢ (𝑋 ∈ 𝑊 → (𝑁(𝑇‘𝑋)𝐴) = (𝑁(𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))𝐴))
8		oteq1 4840	. . . . . 6 ⊢ (𝑎 = 𝑁 → ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩ = ⟨𝑁, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)
9		oteq2 4841	. . . . . 6 ⊢ (𝑎 = 𝑁 → ⟨𝑁, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩ = ⟨𝑁, 𝑁, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)
10	8, 9	eqtrd 2772	. . . . 5 ⊢ (𝑎 = 𝑁 → ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩ = ⟨𝑁, 𝑁, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)
11	10	oveq2d 7384	. . . 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 6842	. . . . . . 7 ⊢ (𝑏 = 𝐴 → (𝑀‘𝑏) = (𝑀‘𝐴))
14	12, 13	s2eqd 14798	. . . . . 6 ⊢ (𝑏 = 𝐴 → ⟨“𝑏(𝑀‘𝑏)”⟩ = ⟨“𝐴(𝑀‘𝐴)”⟩)
15	14	oteq3d 4845	. . . . 5 ⊢ (𝑏 = 𝐴 → ⟨𝑁, 𝑁, ⟨“𝑏(𝑀‘𝑏)”⟩⟩ = ⟨𝑁, 𝑁, ⟨“𝐴(𝑀‘𝐴)”⟩⟩)
16	15	oveq2d 7384	. . . 4 ⊢ (𝑏 = 𝐴 → (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) = (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝐴(𝑀‘𝐴)”⟩⟩))
17		eqid 2737	. . . 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 7401	. . . 4 ⊢ (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝐴(𝑀‘𝐴)”⟩⟩) ∈ V
19	11, 16, 17, 18	ovmpo 7528	. . 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 2792	. 2 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑁 ∈ (0...(♯‘𝑋)) ∧ 𝐴 ∈ (𝐼 × 2o))) → (𝑁(𝑇‘𝑋)𝐴) = (𝑋 splice ⟨𝑁, 𝑁, ⟨“𝐴(𝑀‘𝐴)”⟩⟩))
21	20	3impb 1115	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 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ∖ cdif 3900  ⟨cop 4588  ⟨cotp 4590   ↦ cmpt 5181   I cid 5526   × cxp 5630  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  1_oc1o 8400  2_oc2o 8401  0cc0 11038  ...cfz 13435  ♯chash 14265  Word cword 14448   splice csplice 14684  ⟨“cs2 14776   ~_FG cefg 19647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-ot 4591  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-n0 12414  df-z 12501  df-uz 12764  df-fz 13436  df-fzo 13583  df-hash 14266  df-word 14449  df-concat 14506  df-s1 14532  df-substr 14577  df-pfx 14607  df-splice 14685  df-s2 14783

This theorem is referenced by:  efginvrel2  19668  efgredleme  19684  efgredlemc  19686  efgcpbllemb  19696