Theorem efgtlen 19605

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	⊢ ∼ = ( ~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
efgtlen	⊢ ((𝑋 ∈ 𝑊 ∧ 𝐴 ∈ ran (𝑇‘𝑋)) → (♯‘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.

Step	Hyp	Ref	Expression
1		efgval.w	. . . . . . . 8 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))
2		efgval.r	. . . . . . . 8 ⊢ ∼ = ( ~FG ‘𝐼)
3		efgval2.m	. . . . . . . 8 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
4		efgval2.t	. . . . . . . 8 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
5	1, 2, 3, 4	efgtf 19601	. . . . . . 7 ⊢ (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	. . . . . 6 ⊢ (𝑋 ∈ 𝑊 → (𝑇‘𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
7	6	rneqd 5880	. . . . 5 ⊢ (𝑋 ∈ 𝑊 → ran (𝑇‘𝑋) = ran (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
8	7	eleq2d 2814	. . . 4 ⊢ (𝑋 ∈ 𝑊 → (𝐴 ∈ ran (𝑇‘𝑋) ↔ 𝐴 ∈ ran (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))))
9		eqid 2729	. . . . 5 ⊢ (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)”⟩⟩))
10		ovex 7382	. . . . 5 ⊢ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ V
11	9, 10	elrnmpo 7485	. . . 4 ⊢ (𝐴 ∈ ran (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ↔ ∃𝑎 ∈ (0...(♯‘𝑋))∃𝑏 ∈ (𝐼 × 2o)𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
12	8, 11	bitrdi 287	. . 3 ⊢ (𝑋 ∈ 𝑊 → (𝐴 ∈ ran (𝑇‘𝑋) ↔ ∃𝑎 ∈ (0...(♯‘𝑋))∃𝑏 ∈ (𝐼 × 2o)𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
13		fviss 6900	. . . . . . . . 9 ⊢ ( I ‘Word (𝐼 × 2o)) ⊆ Word (𝐼 × 2o)
14	1, 13	eqsstri 3982	. . . . . . . 8 ⊢ 𝑊 ⊆ Word (𝐼 × 2o)
15		simpl 482	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑋 ∈ 𝑊)
16	14, 15	sselid 3933	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑋 ∈ Word (𝐼 × 2o))
17		elfzuz 13423	. . . . . . . . 9 ⊢ (𝑎 ∈ (0...(♯‘𝑋)) → 𝑎 ∈ (ℤ≥‘0))
18	17	ad2antrl 728	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ (ℤ≥‘0))
19		eluzfz2b 13436	. . . . . . . 8 ⊢ (𝑎 ∈ (ℤ≥‘0) ↔ 𝑎 ∈ (0...𝑎))
20	18, 19	sylib 218	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ (0...𝑎))
21		simprl 770	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ (0...(♯‘𝑋)))
22		simprr 772	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑏 ∈ (𝐼 × 2o))
23	3	efgmf 19592	. . . . . . . . . 10 ⊢ 𝑀:(𝐼 × 2o)⟶(𝐼 × 2o)
24	23	ffvelcdmi 7017	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝐼 × 2o) → (𝑀‘𝑏) ∈ (𝐼 × 2o))
25	22, 24	syl 17	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑀‘𝑏) ∈ (𝐼 × 2o))
26	22, 25	s2cld 14778	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2o))
27	16, 20, 21, 26	spllen 14660	. . . . . 6 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) = ((♯‘𝑋) + ((♯‘⟨“𝑏(𝑀‘𝑏)”⟩) − (𝑎 − 𝑎))))
28		s2len 14796	. . . . . . . . . 10 ⊢ (♯‘⟨“𝑏(𝑀‘𝑏)”⟩) = 2
29	28	a1i 11	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘⟨“𝑏(𝑀‘𝑏)”⟩) = 2)
30		eluzelcn 12747	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (ℤ≥‘0) → 𝑎 ∈ ℂ)
31	18, 30	syl 17	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → 𝑎 ∈ ℂ)
32	31	subidd 11463	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝑎 − 𝑎) = 0)
33	29, 32	oveq12d 7367	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘⟨“𝑏(𝑀‘𝑏)”⟩) − (𝑎 − 𝑎)) = (2 − 0))
34		2cn 12203	. . . . . . . . 9 ⊢ 2 ∈ ℂ
35	34	subid1i 11436	. . . . . . . 8 ⊢ (2 − 0) = 2
36	33, 35	eqtrdi 2780	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘⟨“𝑏(𝑀‘𝑏)”⟩) − (𝑎 − 𝑎)) = 2)
37	36	oveq2d 7365	. . . . . 6 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → ((♯‘𝑋) + ((♯‘⟨“𝑏(𝑀‘𝑏)”⟩) − (𝑎 − 𝑎))) = ((♯‘𝑋) + 2))
38	27, 37	eqtrd 2764	. . . . 5 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (♯‘(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) = ((♯‘𝑋) + 2))
39		fveqeq2 6831	. . . . 5 ⊢ (𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) → ((♯‘𝐴) = ((♯‘𝑋) + 2) ↔ (♯‘(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) = ((♯‘𝑋) + 2)))
40	38, 39	syl5ibrcom 247	. . . 4 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2o))) → (𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) → (♯‘𝐴) = ((♯‘𝑋) + 2)))
41	40	rexlimdvva 3186	. . 3 ⊢ (𝑋 ∈ 𝑊 → (∃𝑎 ∈ (0...(♯‘𝑋))∃𝑏 ∈ (𝐼 × 2o)𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) → (♯‘𝐴) = ((♯‘𝑋) + 2)))
42	12, 41	sylbid 240	. 2 ⊢ (𝑋 ∈ 𝑊 → (𝐴 ∈ ran (𝑇‘𝑋) → (♯‘𝐴) = ((♯‘𝑋) + 2)))
43	42	imp 406	1 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝐴 ∈ ran (𝑇‘𝑋)) → (♯‘𝐴) = ((♯‘𝑋) + 2))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3053   ∖ cdif 3900  ⟨cop 4583  ⟨cotp 4585   ↦ cmpt 5173   I cid 5513   × cxp 5617  ran crn 5620  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351  1_oc1o 8381  2_oc2o 8382  ℂcc 11007  0cc0 11009   + caddc 11012   − cmin 11347  2c2 12183  ℤ_≥cuz 12735  ...cfz 13410  ♯chash 14237  Word cword 14420   splice csplice 14655  ⟨“cs2 14748   ~_FG cefg 19585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-ot 4586  df-uni 4859  df-int 4897  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6249  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-frecs 8214  df-wrecs 8245  df-recs 8294  df-rdg 8332  df-1o 8388  df-2o 8389  df-er 8625  df-map 8755  df-en 8873  df-dom 8874  df-sdom 8875  df-fin 8876  df-card 9835  df-pnf 11151  df-mnf 11152  df-xr 11153  df-ltxr 11154  df-le 11155  df-sub 11349  df-neg 11350  df-nn 12129  df-2 12191  df-n0 12385  df-z 12472  df-uz 12736  df-fz 13411  df-fzo 13558  df-hash 14238  df-word 14421  df-concat 14478  df-s1 14503  df-substr 14548  df-pfx 14578  df-splice 14656  df-s2 14755

This theorem is referenced by:  efgsfo  19618  efgredlemg  19621  efgredlemd  19623  efgredlem  19626  frgpnabllem1  19752