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Theorem efgtlen 18360
Description: Value of the free group construction. (Contributed by Mario Carneiro, 27-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
efgtlen ((𝑋𝑊𝐴 ∈ ran (𝑇𝑋)) → (♯‘𝐴) = ((♯‘𝑋) + 2))
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)   𝑋(𝑦,𝑧,𝑤,𝑣,𝑛)

Proof of Theorem efgtlen
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . . . . . . 8 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
2 efgval.r . . . . . . . 8 = ( ~FG𝐼)
3 efgval2.m . . . . . . . 8 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
4 efgval2.t . . . . . . . 8 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
51, 2, 3, 4efgtf 18356 . . . . . . 7 (𝑋𝑊 → ((𝑇𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∧ (𝑇𝑋):((0...(♯‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊))
65simpld 477 . . . . . 6 (𝑋𝑊 → (𝑇𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
76rneqd 5509 . . . . 5 (𝑋𝑊 → ran (𝑇𝑋) = ran (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
87eleq2d 2826 . . . 4 (𝑋𝑊 → (𝐴 ∈ ran (𝑇𝑋) ↔ 𝐴 ∈ ran (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))))
9 eqid 2761 . . . . 5 (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
10 ovex 6843 . . . . 5 (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ V
119, 10elrnmpt2 6940 . . . 4 (𝐴 ∈ ran (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ↔ ∃𝑎 ∈ (0...(♯‘𝑋))∃𝑏 ∈ (𝐼 × 2𝑜)𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))
128, 11syl6bb 276 . . 3 (𝑋𝑊 → (𝐴 ∈ ran (𝑇𝑋) ↔ ∃𝑎 ∈ (0...(♯‘𝑋))∃𝑏 ∈ (𝐼 × 2𝑜)𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
13 fviss 6420 . . . . . . . . 9 ( I ‘Word (𝐼 × 2𝑜)) ⊆ Word (𝐼 × 2𝑜)
141, 13eqsstri 3777 . . . . . . . 8 𝑊 ⊆ Word (𝐼 × 2𝑜)
15 simpl 474 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑋𝑊)
1614, 15sseldi 3743 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑋 ∈ Word (𝐼 × 2𝑜))
17 elfzuz 12552 . . . . . . . . 9 (𝑎 ∈ (0...(♯‘𝑋)) → 𝑎 ∈ (ℤ‘0))
1817ad2antrl 766 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 ∈ (ℤ‘0))
19 eluzfz2b 12564 . . . . . . . 8 (𝑎 ∈ (ℤ‘0) ↔ 𝑎 ∈ (0...𝑎))
2018, 19sylib 208 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 ∈ (0...𝑎))
21 simprl 811 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 ∈ (0...(♯‘𝑋)))
22 simprr 813 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑏 ∈ (𝐼 × 2𝑜))
233efgmf 18347 . . . . . . . . . 10 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)
2423ffvelrni 6523 . . . . . . . . 9 (𝑏 ∈ (𝐼 × 2𝑜) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
2522, 24syl 17 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
2622, 25s2cld 13837 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜))
2716, 20, 21, 26spllen 13726 . . . . . 6 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (♯‘(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) = ((♯‘𝑋) + ((♯‘⟨“𝑏(𝑀𝑏)”⟩) − (𝑎𝑎))))
28 s2len 13855 . . . . . . . . . 10 (♯‘⟨“𝑏(𝑀𝑏)”⟩) = 2
2928a1i 11 . . . . . . . . 9 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (♯‘⟨“𝑏(𝑀𝑏)”⟩) = 2)
30 eluzelcn 11912 . . . . . . . . . . 11 (𝑎 ∈ (ℤ‘0) → 𝑎 ∈ ℂ)
3118, 30syl 17 . . . . . . . . . 10 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 ∈ ℂ)
3231subidd 10593 . . . . . . . . 9 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎𝑎) = 0)
3329, 32oveq12d 6833 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((♯‘⟨“𝑏(𝑀𝑏)”⟩) − (𝑎𝑎)) = (2 − 0))
34 2cn 11304 . . . . . . . . 9 2 ∈ ℂ
3534subid1i 10566 . . . . . . . 8 (2 − 0) = 2
3633, 35syl6eq 2811 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((♯‘⟨“𝑏(𝑀𝑏)”⟩) − (𝑎𝑎)) = 2)
3736oveq2d 6831 . . . . . 6 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((♯‘𝑋) + ((♯‘⟨“𝑏(𝑀𝑏)”⟩) − (𝑎𝑎))) = ((♯‘𝑋) + 2))
3827, 37eqtrd 2795 . . . . 5 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (♯‘(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) = ((♯‘𝑋) + 2))
39 fveq2 6354 . . . . . 6 (𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) → (♯‘𝐴) = (♯‘(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
4039eqeq1d 2763 . . . . 5 (𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) → ((♯‘𝐴) = ((♯‘𝑋) + 2) ↔ (♯‘(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) = ((♯‘𝑋) + 2)))
4138, 40syl5ibrcom 237 . . . 4 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) → (♯‘𝐴) = ((♯‘𝑋) + 2)))
4241rexlimdvva 3177 . . 3 (𝑋𝑊 → (∃𝑎 ∈ (0...(♯‘𝑋))∃𝑏 ∈ (𝐼 × 2𝑜)𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) → (♯‘𝐴) = ((♯‘𝑋) + 2)))
4312, 42sylbid 230 . 2 (𝑋𝑊 → (𝐴 ∈ ran (𝑇𝑋) → (♯‘𝐴) = ((♯‘𝑋) + 2)))
4443imp 444 1 ((𝑋𝑊𝐴 ∈ ran (𝑇𝑋)) → (♯‘𝐴) = ((♯‘𝑋) + 2))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2140  wrex 3052  cdif 3713  cop 4328  cotp 4330  cmpt 4882   I cid 5174   × cxp 5265  ran crn 5268  wf 6046  cfv 6050  (class class class)co 6815  cmpt2 6817  1𝑜c1o 7724  2𝑜c2o 7725  cc 10147  0cc0 10149   + caddc 10152  cmin 10479  2c2 11283  cuz 11900  ...cfz 12540  chash 13332  Word cword 13498   splice csplice 13503  ⟨“cs2 13807   ~FG cefg 18340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116  ax-cnex 10205  ax-resscn 10206  ax-1cn 10207  ax-icn 10208  ax-addcl 10209  ax-addrcl 10210  ax-mulcl 10211  ax-mulrcl 10212  ax-mulcom 10213  ax-addass 10214  ax-mulass 10215  ax-distr 10216  ax-i2m1 10217  ax-1ne0 10218  ax-1rid 10219  ax-rnegex 10220  ax-rrecex 10221  ax-cnre 10222  ax-pre-lttri 10223  ax-pre-lttrn 10224  ax-pre-ltadd 10225  ax-pre-mulgt0 10226
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-nel 3037  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-ot 4331  df-uni 4590  df-int 4629  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-pred 5842  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-ov 6818  df-oprab 6819  df-mpt2 6820  df-om 7233  df-1st 7335  df-2nd 7336  df-wrecs 7578  df-recs 7639  df-rdg 7677  df-1o 7731  df-2o 7732  df-oadd 7735  df-er 7914  df-map 8028  df-pm 8029  df-en 8125  df-dom 8126  df-sdom 8127  df-fin 8128  df-card 8976  df-pnf 10289  df-mnf 10290  df-xr 10291  df-ltxr 10292  df-le 10293  df-sub 10481  df-neg 10482  df-nn 11234  df-2 11292  df-n0 11506  df-z 11591  df-uz 11901  df-fz 12541  df-fzo 12681  df-hash 13333  df-word 13506  df-concat 13508  df-s1 13509  df-substr 13510  df-splice 13511  df-s2 13814
This theorem is referenced by:  efgsfo  18373  efgredlemg  18376  efgredlemd  18378  efgredlem  18381  frgpnabllem1  18497
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