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Theorem efgtlen 18060
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 18056 . . . . . . 7 (𝑋𝑊 → ((𝑇𝑋) = (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) ∧ (𝑇𝑋):((0...(#‘𝑋)) × (𝐼 × 2𝑜))⟶𝑊))
65simpld 475 . . . . . 6 (𝑋𝑊 → (𝑇𝑋) = (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
76rneqd 5313 . . . . 5 (𝑋𝑊 → ran (𝑇𝑋) = ran (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
87eleq2d 2684 . . . 4 (𝑋𝑊 → (𝐴 ∈ ran (𝑇𝑋) ↔ 𝐴 ∈ ran (𝑎 ∈ (0...(#‘𝑋)), 𝑏 ∈ (𝐼 × 2𝑜) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩))))
9 eqid 2621 . . . . 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 6632 . . . . 5 (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) ∈ V
119, 10elrnmpt2 6726 . . . 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 6213 . . . . . . . . 9 ( I ‘Word (𝐼 × 2𝑜)) ⊆ Word (𝐼 × 2𝑜)
141, 13eqsstri 3614 . . . . . . . 8 𝑊 ⊆ Word (𝐼 × 2𝑜)
15 simpl 473 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑋𝑊)
1614, 15sseldi 3581 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑋 ∈ Word (𝐼 × 2𝑜))
17 elfzuz 12280 . . . . . . . . 9 (𝑎 ∈ (0...(#‘𝑋)) → 𝑎 ∈ (ℤ‘0))
1817ad2antrl 763 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 ∈ (ℤ‘0))
19 eluzfz2b 12292 . . . . . . . 8 (𝑎 ∈ (ℤ‘0) ↔ 𝑎 ∈ (0...𝑎))
2018, 19sylib 208 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 ∈ (0...𝑎))
21 simprl 793 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 ∈ (0...(#‘𝑋)))
22 simprr 795 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑏 ∈ (𝐼 × 2𝑜))
233efgmf 18047 . . . . . . . . . 10 𝑀:(𝐼 × 2𝑜)⟶(𝐼 × 2𝑜)
2423ffvelrni 6314 . . . . . . . . 9 (𝑏 ∈ (𝐼 × 2𝑜) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
2522, 24syl 17 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑀𝑏) ∈ (𝐼 × 2𝑜))
2622, 25s2cld 13552 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ⟨“𝑏(𝑀𝑏)”⟩ ∈ Word (𝐼 × 2𝑜))
2716, 20, 21, 26spllen 13442 . . . . . 6 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) = ((#‘𝑋) + ((#‘⟨“𝑏(𝑀𝑏)”⟩) − (𝑎𝑎))))
28 s2len 13570 . . . . . . . . . 10 (#‘⟨“𝑏(𝑀𝑏)”⟩) = 2
2928a1i 11 . . . . . . . . 9 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘⟨“𝑏(𝑀𝑏)”⟩) = 2)
30 eluzelcn 11643 . . . . . . . . . . 11 (𝑎 ∈ (ℤ‘0) → 𝑎 ∈ ℂ)
3118, 30syl 17 . . . . . . . . . 10 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → 𝑎 ∈ ℂ)
3231subidd 10324 . . . . . . . . 9 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (𝑎𝑎) = 0)
3329, 32oveq12d 6622 . . . . . . . 8 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((#‘⟨“𝑏(𝑀𝑏)”⟩) − (𝑎𝑎)) = (2 − 0))
34 2cn 11035 . . . . . . . . 9 2 ∈ ℂ
3534subid1i 10297 . . . . . . . 8 (2 − 0) = 2
3633, 35syl6eq 2671 . . . . . . 7 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((#‘⟨“𝑏(𝑀𝑏)”⟩) − (𝑎𝑎)) = 2)
3736oveq2d 6620 . . . . . 6 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → ((#‘𝑋) + ((#‘⟨“𝑏(𝑀𝑏)”⟩) − (𝑎𝑎))) = ((#‘𝑋) + 2))
3827, 37eqtrd 2655 . . . . 5 ((𝑋𝑊 ∧ (𝑎 ∈ (0...(#‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2𝑜))) → (#‘(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)) = ((#‘𝑋) + 2))
39 fveq2 6148 . . . . . 6 (𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩) → (#‘𝐴) = (#‘(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀𝑏)”⟩⟩)))
4039eqeq1d 2623 . . . . 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 3031 . . 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 445 1 ((𝑋𝑊𝐴 ∈ ran (𝑇𝑋)) → (#‘𝐴) = ((#‘𝑋) + 2))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wrex 2908  cdif 3552  cop 4154  cotp 4156  cmpt 4673   I cid 4984   × cxp 5072  ran crn 5075  wf 5843  cfv 5847  (class class class)co 6604  cmpt2 6606  1𝑜c1o 7498  2𝑜c2o 7499  cc 9878  0cc0 9880   + caddc 9883  cmin 10210  2c2 11014  cuz 11631  ...cfz 12268  #chash 13057  Word cword 13230   splice csplice 13235  ⟨“cs2 13523   ~FG cefg 18040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-ot 4157  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-concat 13240  df-s1 13241  df-substr 13242  df-splice 13243  df-s2 13530
This theorem is referenced by:  efgsfo  18073  efgredlemg  18076  efgredlemd  18078  efgredlem  18081  frgpnabllem1  18197
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