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Theorem efgtlen 17962
 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 17958 . . . . . . 7 (𝑋𝑊 → ((𝑇𝑋) = (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∧ (𝑇𝑋):((0...(#‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊))
65simpld 474 . . . . . 6 (𝑋𝑊 → (𝑇𝑋) = (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
76rneqd 5274 . . . . 5 (𝑋𝑊 → ran (𝑇𝑋) = ran (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
87eleq2d 2673 . . . 4 (𝑋𝑊 → (𝐴 ∈ ran (𝑇𝑋) ↔ 𝐴 ∈ ran (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))))
9 eqid 2610 . . . . 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 6577 . . . . 5 (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ V
119, 10elrnmpt2 6671 . . . 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 275 . . 3 (𝑋𝑊 → (𝐴 ∈ ran (𝑇𝑋) ↔ ∃𝑎 ∈ (0...(#‘𝑋))∃𝑏 ∈ (𝐼 × 2𝑜)𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
13 fviss 6166 . . . . . . . . 9 ( I ‘Word (𝐼 × 2𝑜)) ⊆ Word (𝐼 × 2𝑜)
141, 13eqsstri 3598 . . . . . . . 8 𝑊 ⊆ Word (𝐼 × 2𝑜)
15 simpl 472 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑋𝑊)
1614, 15sseldi 3566 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑋 ∈ Word (𝐼 × 2𝑜))
17 elfzuz 12209 . . . . . . . . 9 (𝑎 ∈ (0...(#‘𝑋)) → 𝑎 ∈ (ℤ‘0))
1817ad2antrl 760 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 ∈ (ℤ‘0))
19 eluzfz2b 12221 . . . . . . . 8 (𝑎 ∈ (ℤ‘0) ↔ 𝑎 ∈ (0...𝑎))
2018, 19sylib 207 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 ∈ (0...𝑎))
21 simprl 790 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 ∈ (0...(#‘𝑋)))
22 simprr 792 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑏 ∈ (𝐼 × 2𝑜))
233efgmf 17949 . . . . . . . . . 10 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)
2423ffvelrni 6266 . . . . . . . . 9 (𝑏 ∈ (𝐼 × 2𝑜) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
2522, 24syl 17 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
2622, 25s2cld 13466 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜))
2716, 20, 21, 26spllen 13356 . . . . . 6 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) = ((#‘𝑋) + ((#‘⟨“𝑏(𝑀𝑏)”⟩) − (𝑎𝑎))))
28 s2len 13484 . . . . . . . . . 10 (#‘⟨“𝑏(𝑀𝑏)”⟩) = 2
2928a1i 11 . . . . . . . . 9 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘⟨“𝑏(𝑀𝑏)”⟩) = 2)
30 eluzelcn 11575 . . . . . . . . . . 11 (𝑎 ∈ (ℤ‘0) → 𝑎 ∈ ℂ)
3118, 30syl 17 . . . . . . . . . 10 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 ∈ ℂ)
3231subidd 10259 . . . . . . . . 9 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎𝑎) = 0)
3329, 32oveq12d 6567 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((#‘⟨“𝑏(𝑀𝑏)”⟩) − (𝑎𝑎)) = (2 − 0))
34 2cn 10968 . . . . . . . . 9 2 ∈ ℂ
3534subid1i 10232 . . . . . . . 8 (2 − 0) = 2
3633, 35syl6eq 2660 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((#‘⟨“𝑏(𝑀𝑏)”⟩) − (𝑎𝑎)) = 2)
3736oveq2d 6565 . . . . . 6 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((#‘𝑋) + ((#‘⟨“𝑏(𝑀𝑏)”⟩) − (𝑎𝑎))) = ((#‘𝑋) + 2))
3827, 37eqtrd 2644 . . . . 5 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) = ((#‘𝑋) + 2))
39 fveq2 6103 . . . . . 6 (𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) → (#‘𝐴) = (#‘(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
4039eqeq1d 2612 . . . . 5 (𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) → ((#‘𝐴) = ((#‘𝑋) + 2) ↔ (#‘(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) = ((#‘𝑋) + 2)))
4138, 40syl5ibrcom 236 . . . 4 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) → (#‘𝐴) = ((#‘𝑋) + 2)))
4241rexlimdvva 3020 . . 3 (𝑋𝑊 → (∃𝑎 ∈ (0...(#‘𝑋))∃𝑏 ∈ (𝐼 × 2𝑜)𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) → (#‘𝐴) = ((#‘𝑋) + 2)))
4312, 42sylbid 229 . 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 1475   ∈ wcel 1977  ∃wrex 2897   ∖ cdif 3537  ⟨cop 4131  ⟨cotp 4133   ↦ cmpt 4643   I cid 4948   × cxp 5036  ran crn 5039  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1𝑜c1o 7440  2𝑜c2o 7441  ℂcc 9813  0cc0 9815   + caddc 9818   − cmin 10145  2c2 10947  ℤ≥cuz 11563  ...cfz 12197  #chash 12979  Word cword 13146   splice csplice 13151  ⟨“cs2 13437   ~FG cefg 17942 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-substr 13158  df-splice 13159  df-s2 13444 This theorem is referenced by:  efgsfo  17975  efgredlemg  17978  efgredlemd  17980  efgredlem  17983  frgpnabllem1  18099
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