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Theorem efgtlen 18843
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 18839 . . . . . . 7 (𝑋𝑊 → ((𝑇𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∧ (𝑇𝑋):((0...(♯‘𝑋)) × (𝐼 × 2o))⟶𝑊))
65simpld 498 . . . . . 6 (𝑋𝑊 → (𝑇𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
76rneqd 5785 . . . . 5 (𝑋𝑊 → ran (𝑇𝑋) = ran (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
87eleq2d 2899 . . . 4 (𝑋𝑊 → (𝐴 ∈ ran (𝑇𝑋) ↔ 𝐴 ∈ ran (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))))
9 eqid 2822 . . . . 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 7173 . . . . 5 (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ V
119, 10elrnmpo 7271 . . . 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 290 . . 3 (𝑋𝑊 → (𝐴 ∈ ran (𝑇𝑋) ↔ ∃𝑎 ∈ (0...(♯‘𝑋))∃𝑏 ∈ (𝐼 × 2o)𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
13 fviss 6723 . . . . . . . . 9 ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
141, 13eqsstri 3976 . . . . . . . 8 𝑊 ⊆ Word (𝐼 × 2o)
15 simpl 486 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑋𝑊)
1614, 15sseldi 3940 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑋 ∈ Word (𝐼 × 2o))
17 elfzuz 12898 . . . . . . . . 9 (𝑎 ∈ (0...(♯‘𝑋)) → 𝑎 ∈ (ℤ‘0))
1817ad2antrl 727 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ (ℤ‘0))
19 eluzfz2b 12911 . . . . . . . 8 (𝑎 ∈ (ℤ‘0) ↔ 𝑎 ∈ (0...𝑎))
2018, 19sylib 221 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ (0...𝑎))
21 simprl 770 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ (0...(♯‘𝑋)))
22 simprr 772 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑏 ∈ (𝐼 × 2o))
233efgmf 18830 . . . . . . . . . 10 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
2423ffvelrni 6832 . . . . . . . . 9 (𝑏 ∈ (𝐼 × 2o) → (𝑀𝑏) ∈ (𝐼 × 2o))
2522, 24syl 17 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀𝑏) ∈ (𝐼 × 2o))
2622, 25s2cld 14224 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2o))
2716, 20, 21, 26spllen 14107 . . . . . 6 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) = ((♯‘𝑋) + ((♯‘⟨“𝑏(𝑀𝑏)”⟩) − (𝑎𝑎))))
28 s2len 14242 . . . . . . . . . 10 (♯‘⟨“𝑏(𝑀𝑏)”⟩) = 2
2928a1i 11 . . . . . . . . 9 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘⟨“𝑏(𝑀𝑏)”⟩) = 2)
30 eluzelcn 12243 . . . . . . . . . . 11 (𝑎 ∈ (ℤ‘0) → 𝑎 ∈ ℂ)
3118, 30syl 17 . . . . . . . . . 10 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ ℂ)
3231subidd 10974 . . . . . . . . 9 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎𝑎) = 0)
3329, 32oveq12d 7158 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘⟨“𝑏(𝑀𝑏)”⟩) − (𝑎𝑎)) = (2 − 0))
34 2cn 11700 . . . . . . . . 9 2 ∈ ℂ
3534subid1i 10947 . . . . . . . 8 (2 − 0) = 2
3633, 35syl6eq 2873 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘⟨“𝑏(𝑀𝑏)”⟩) − (𝑎𝑎)) = 2)
3736oveq2d 7156 . . . . . 6 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘𝑋) + ((♯‘⟨“𝑏(𝑀𝑏)”⟩) − (𝑎𝑎))) = ((♯‘𝑋) + 2))
3827, 37eqtrd 2857 . . . . 5 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) = ((♯‘𝑋) + 2))
39 fveqeq2 6661 . . . . 5 (𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) → ((♯‘𝐴) = ((♯‘𝑋) + 2) ↔ (♯‘(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) = ((♯‘𝑋) + 2)))
4038, 39syl5ibrcom 250 . . . 4 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) → (♯‘𝐴) = ((♯‘𝑋) + 2)))
4140rexlimdvva 3280 . . 3 (𝑋𝑊 → (∃𝑎 ∈ (0...(♯‘𝑋))∃𝑏 ∈ (𝐼 × 2o)𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) → (♯‘𝐴) = ((♯‘𝑋) + 2)))
4212, 41sylbid 243 . 2 (𝑋𝑊 → (𝐴 ∈ ran (𝑇𝑋) → (♯‘𝐴) = ((♯‘𝑋) + 2)))
4342imp 410 1 ((𝑋𝑊𝐴 ∈ ran (𝑇𝑋)) → (♯‘𝐴) = ((♯‘𝑋) + 2))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  wrex 3131  cdif 3905  cop 4545  cotp 4547  cmpt 5122   I cid 5436   × cxp 5530  ran crn 5533  wf 6330  cfv 6334  (class class class)co 7140  cmpo 7142  1oc1o 8082  2oc2o 8083  cc 10524  0cc0 10526   + caddc 10529  cmin 10859  2c2 11680  cuz 12231  ...cfz 12885  chash 13686  Word cword 13857   splice csplice 14102  ⟨“cs2 14194   ~FG cefg 18823
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-ot 4548  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-card 9356  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-fzo 13029  df-hash 13687  df-word 13858  df-concat 13914  df-s1 13941  df-substr 13994  df-pfx 14024  df-splice 14103  df-s2 14201
This theorem is referenced by:  efgsfo  18856  efgredlemg  18859  efgredlemd  18861  efgredlem  18864  frgpnabllem1  18984
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