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Theorem efgtlen 19688

Description: Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.)

**Hypotheses**
Ref	Expression
efgval.w	⊢ 𝑊 = ( I ‘Word (𝐼 × 2_o))
efgval.r	⊢ ∼ = ( ~_FG ‘𝐼)
efgval2.m	⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)
efgval2.t	⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (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 × 2_o))
2		efgval.r	. . . . . . . 8 ⊢ ∼ = ( ~_FG ‘𝐼)
3		efgval2.m	. . . . . . . 8 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2_o ↦ ⟨𝑦, (1_o ∖ 𝑧)⟩)
4		efgval2.t	. . . . . . . 8 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2_o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
5	1, 2, 3, 4	efgtf 19684	. . . . . . 7 ⊢ (𝑋 ∈ 𝑊 → ((𝑇‘𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2_o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ∧ (𝑇‘𝑋):((0...(♯‘𝑋)) × (𝐼 × 2_o))⟶𝑊))
6	5	simpld 493	. . . . . 6 ⊢ (𝑋 ∈ 𝑊 → (𝑇‘𝑋) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2_o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
7	6	rneqd 5944	. . . . 5 ⊢ (𝑋 ∈ 𝑊 → ran (𝑇‘𝑋) = ran (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2_o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
8	7	eleq2d 2815	. . . 4 ⊢ (𝑋 ∈ 𝑊 → (𝐴 ∈ ran (𝑇‘𝑋) ↔ 𝐴 ∈ ran (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2_o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))))
9		eqid 2728	. . . . 5 ⊢ (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2_o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) = (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2_o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
10		ovex 7459	. . . . 5 ⊢ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) ∈ V
11	9, 10	elrnmpo 7563	. . . 4 ⊢ (𝐴 ∈ ran (𝑎 ∈ (0...(♯‘𝑋)), 𝑏 ∈ (𝐼 × 2_o) ↦ (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) ↔ ∃𝑎 ∈ (0...(♯‘𝑋))∃𝑏 ∈ (𝐼 × 2_o)𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩))
12	8, 11	bitrdi 286	. . 3 ⊢ (𝑋 ∈ 𝑊 → (𝐴 ∈ ran (𝑇‘𝑋) ↔ ∃𝑎 ∈ (0...(♯‘𝑋))∃𝑏 ∈ (𝐼 × 2_o)𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)))
13		fviss 6980	. . . . . . . . 9 ⊢ ( I ‘Word (𝐼 × 2_o)) ⊆ Word (𝐼 × 2_o)
14	1, 13	eqsstri 4016	. . . . . . . 8 ⊢ 𝑊 ⊆ Word (𝐼 × 2_o)
15		simpl 481	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2_o))) → 𝑋 ∈ 𝑊)
16	14, 15	sselid 3980	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2_o))) → 𝑋 ∈ Word (𝐼 × 2_o))
17		elfzuz 13537	. . . . . . . . 9 ⊢ (𝑎 ∈ (0...(♯‘𝑋)) → 𝑎 ∈ (ℤ_≥‘0))
18	17	ad2antrl 726	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2_o))) → 𝑎 ∈ (ℤ_≥‘0))
19		eluzfz2b 13550	. . . . . . . 8 ⊢ (𝑎 ∈ (ℤ_≥‘0) ↔ 𝑎 ∈ (0...𝑎))
20	18, 19	sylib 217	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2_o))) → 𝑎 ∈ (0...𝑎))
21		simprl 769	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2_o))) → 𝑎 ∈ (0...(♯‘𝑋)))
22		simprr 771	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2_o))) → 𝑏 ∈ (𝐼 × 2_o))
23	3	efgmf 19675	. . . . . . . . . 10 ⊢ 𝑀:(𝐼 × 2_o)⟶(𝐼 × 2_o)
24	23	ffvelcdmi 7098	. . . . . . . . 9 ⊢ (𝑏 ∈ (𝐼 × 2_o) → (𝑀‘𝑏) ∈ (𝐼 × 2_o))
25	22, 24	syl 17	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (𝑀‘𝑏) ∈ (𝐼 × 2_o))
26	22, 25	s2cld 14862	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ⟨“𝑏(𝑀‘𝑏)”⟩ ∈ Word (𝐼 × 2_o))
27	16, 20, 21, 26	spllen 14744	. . . . . 6 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (♯‘(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) = ((♯‘𝑋) + ((♯‘⟨“𝑏(𝑀‘𝑏)”⟩) − (𝑎 − 𝑎))))
28		s2len 14880	. . . . . . . . . 10 ⊢ (♯‘⟨“𝑏(𝑀‘𝑏)”⟩) = 2
29	28	a1i 11	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (♯‘⟨“𝑏(𝑀‘𝑏)”⟩) = 2)
30		eluzelcn 12872	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (ℤ_≥‘0) → 𝑎 ∈ ℂ)
31	18, 30	syl 17	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2_o))) → 𝑎 ∈ ℂ)
32	31	subidd 11597	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (𝑎 − 𝑎) = 0)
33	29, 32	oveq12d 7444	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ((♯‘⟨“𝑏(𝑀‘𝑏)”⟩) − (𝑎 − 𝑎)) = (2 − 0))
34		2cn 12325	. . . . . . . . 9 ⊢ 2 ∈ ℂ
35	34	subid1i 11570	. . . . . . . 8 ⊢ (2 − 0) = 2
36	33, 35	eqtrdi 2784	. . . . . . 7 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ((♯‘⟨“𝑏(𝑀‘𝑏)”⟩) − (𝑎 − 𝑎)) = 2)
37	36	oveq2d 7442	. . . . . 6 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2_o))) → ((♯‘𝑋) + ((♯‘⟨“𝑏(𝑀‘𝑏)”⟩) − (𝑎 − 𝑎))) = ((♯‘𝑋) + 2))
38	27, 37	eqtrd 2768	. . . . 5 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (♯‘(𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩)) = ((♯‘𝑋) + 2))
39		fveqeq2 6911	. . . . 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 246	. . . 4 ⊢ ((𝑋 ∈ 𝑊 ∧ (𝑎 ∈ (0...(♯‘𝑋)) ∧ 𝑏 ∈ (𝐼 × 2_o))) → (𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) → (♯‘𝐴) = ((♯‘𝑋) + 2)))
41	40	rexlimdvva 3209	. . 3 ⊢ (𝑋 ∈ 𝑊 → (∃𝑎 ∈ (0...(♯‘𝑋))∃𝑏 ∈ (𝐼 × 2_o)𝐴 = (𝑋 splice ⟨𝑎, 𝑎, ⟨“𝑏(𝑀‘𝑏)”⟩⟩) → (♯‘𝐴) = ((♯‘𝑋) + 2)))
42	12, 41	sylbid 239	. 2 ⊢ (𝑋 ∈ 𝑊 → (𝐴 ∈ ran (𝑇‘𝑋) → (♯‘𝐴) = ((♯‘𝑋) + 2)))
43	42	imp 405	1 ⊢ ((𝑋 ∈ 𝑊 ∧ 𝐴 ∈ ran (𝑇‘𝑋)) → (♯‘𝐴) = ((♯‘𝑋) + 2))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3067 ∖ cdif 3946 ⟨cop 4638 ⟨cotp 4640 ↦ cmpt 5235 I cid 5579 × cxp 5680 ran crn 5683 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 ∈ cmpo 7428 1_oc1o 8486 2_oc2o 8487 ℂcc 11144 0cc0 11146 + caddc 11149 − cmin 11482 2c2 12305 ℤ_≥cuz 12860 ...cfz 13524 ♯chash 14329 Word cword 14504 splice csplice 14739 ⟨“cs2 14832 ~_FG cefg 19668

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-ot 4641 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-2o 8494 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-fzo 13668 df-hash 14330 df-word 14505 df-concat 14561 df-s1 14586 df-substr 14631 df-pfx 14661 df-splice 14740 df-s2 14839

This theorem is referenced by: efgsfo 19701 efgredlemg 19704 efgredlemd 19706 efgredlem 19709 frgpnabllem1 19835

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