Theorem efgi2 19641

Description: Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-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
efgi2	⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ ran (𝑇‘𝐴)) → 𝐴 ∼ 𝐵)

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)   𝐵(𝑦,𝑧,𝑤,𝑣,𝑛)   ∼ (𝑤,𝑣,𝑛)   𝑇(𝑦,𝑧,𝑤,𝑣,𝑛)   𝑀(𝑦,𝑧)

Proof of Theorem efgi2

Dummy variables 𝑎 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6830	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑇‘𝑎) = (𝑇‘𝐴))
2	1	rneqd 5884	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ran (𝑇‘𝑎) = ran (𝑇‘𝐴))
3		eceq1 8669	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → [𝑎]𝑟 = [𝐴]𝑟)
4	2, 3	sseq12d 3964	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (ran (𝑇‘𝑎) ⊆ [𝑎]𝑟 ↔ ran (𝑇‘𝐴) ⊆ [𝐴]𝑟))
5	4	rspcv 3569	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑊 → (∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟 → ran (𝑇‘𝐴) ⊆ [𝐴]𝑟))
6	5	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ ran (𝑇‘𝐴)) → (∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟 → ran (𝑇‘𝐴) ⊆ [𝐴]𝑟))
7		ssel 3924	. . . . . . . . 9 ⊢ (ran (𝑇‘𝐴) ⊆ [𝐴]𝑟 → (𝐵 ∈ ran (𝑇‘𝐴) → 𝐵 ∈ [𝐴]𝑟))
8	7	com12 32	. . . . . . . 8 ⊢ (𝐵 ∈ ran (𝑇‘𝐴) → (ran (𝑇‘𝐴) ⊆ [𝐴]𝑟 → 𝐵 ∈ [𝐴]𝑟))
9		simpl 482	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ [𝐴]𝑟 ∧ 𝐴 ∈ 𝑊) → 𝐵 ∈ [𝐴]𝑟)
10		elecg 8674	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ [𝐴]𝑟 ∧ 𝐴 ∈ 𝑊) → (𝐵 ∈ [𝐴]𝑟 ↔ 𝐴𝑟𝐵))
11	9, 10	mpbid 232	. . . . . . . . . 10 ⊢ ((𝐵 ∈ [𝐴]𝑟 ∧ 𝐴 ∈ 𝑊) → 𝐴𝑟𝐵)
12		df-br 5096	. . . . . . . . . 10 ⊢ (𝐴𝑟𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑟)
13	11, 12	sylib 218	. . . . . . . . 9 ⊢ ((𝐵 ∈ [𝐴]𝑟 ∧ 𝐴 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ 𝑟)
14	13	expcom 413	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑊 → (𝐵 ∈ [𝐴]𝑟 → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
15	8, 14	sylan9r 508	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ ran (𝑇‘𝐴)) → (ran (𝑇‘𝐴) ⊆ [𝐴]𝑟 → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
16	6, 15	syld 47	. . . . . 6 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ ran (𝑇‘𝐴)) → (∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟 → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
17	16	adantld 490	. . . . 5 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ ran (𝑇‘𝐴)) → ((𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟) → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
18	17	alrimiv 1928	. . . 4 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ ran (𝑇‘𝐴)) → ∀𝑟((𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟) → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
19		opex 5409	. . . . 5 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
20	19	elintab 4911	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ ∈ ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟)} ↔ ∀𝑟((𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟) → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
21	18, 20	sylibr 234	. . 3 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ ran (𝑇‘𝐴)) → ⟨𝐴, 𝐵⟩ ∈ ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟)})
22		efgval.w	. . . 4 ⊢ 𝑊 = ( I ‘Word (𝐼 × 2o))
23		efgval.r	. . . 4 ⊢ ∼ = ( ~FG ‘𝐼)
24		efgval2.m	. . . 4 ⊢ 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ ⟨𝑦, (1o ∖ 𝑧)⟩)
25		efgval2.t	. . . 4 ⊢ 𝑇 = (𝑣 ∈ 𝑊 ↦ (𝑛 ∈ (0...(♯‘𝑣)), 𝑤 ∈ (𝐼 × 2o) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀‘𝑤)”⟩⟩)))
26	22, 23, 24, 25	efgval2 19640	. . 3 ⊢ ∼ = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟)}
27	21, 26	eleqtrrdi 2844	. 2 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ ran (𝑇‘𝐴)) → ⟨𝐴, 𝐵⟩ ∈ ∼ )
28		df-br 5096	. 2 ⊢ (𝐴 ∼ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∼ )
29	27, 28	sylibr 234	1 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ ran (𝑇‘𝐴)) → 𝐴 ∼ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1539   = wceq 1541   ∈ wcel 2113  {cab 2711  ∀wral 3048   ∖ cdif 3895   ⊆ wss 3898  ⟨cop 4583  ⟨cotp 4585  ∩ cint 4899   class class class wbr 5095   ↦ cmpt 5176   I cid 5515   × cxp 5619  ran crn 5622  ‘cfv 6488  (class class class)co 7354   ∈ cmpo 7356  1_oc1o 8386  2_oc2o 8387   Er wer 8627  [cec 8628  0cc0 11015  ...cfz 13411  ♯chash 14241  Word cword 14424   splice csplice 14660  ⟨“cs2 14752   ~_FG cefg 19622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676  ax-cnex 11071  ax-resscn 11072  ax-1cn 11073  ax-icn 11074  ax-addcl 11075  ax-addrcl 11076  ax-mulcl 11077  ax-mulrcl 11078  ax-mulcom 11079  ax-addass 11080  ax-mulass 11081  ax-distr 11082  ax-i2m1 11083  ax-1ne0 11084  ax-1rid 11085  ax-rnegex 11086  ax-rrecex 11087  ax-cnre 11088  ax-pre-lttri 11089  ax-pre-lttrn 11090  ax-pre-ltadd 11091  ax-pre-mulgt0 11092

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-ot 4586  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7311  df-ov 7357  df-oprab 7358  df-mpo 7359  df-om 7805  df-1st 7929  df-2nd 7930  df-frecs 8219  df-wrecs 8250  df-recs 8299  df-rdg 8337  df-1o 8393  df-2o 8394  df-er 8630  df-ec 8632  df-map 8760  df-en 8878  df-dom 8879  df-sdom 8880  df-fin 8881  df-card 9841  df-pnf 11157  df-mnf 11158  df-xr 11159  df-ltxr 11160  df-le 11161  df-sub 11355  df-neg 11356  df-nn 12135  df-n0 12391  df-z 12478  df-uz 12741  df-fz 13412  df-fzo 13559  df-hash 14242  df-word 14425  df-concat 14482  df-s1 14508  df-substr 14553  df-pfx 14583  df-splice 14661  df-s2 14759  df-efg 19625

This theorem is referenced by:  efginvrel2  19643  efgsrel  19650  efgcpbllemb  19671