Theorem efgi2 19698

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 6834	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑇‘𝑎) = (𝑇‘𝐴))
2	1	rneqd 5887	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ran (𝑇‘𝑎) = ran (𝑇‘𝐴))
3		eceq1 8680	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → [𝑎]𝑟 = [𝐴]𝑟)
4	2, 3	sseq12d 3955	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (ran (𝑇‘𝑎) ⊆ [𝑎]𝑟 ↔ ran (𝑇‘𝐴) ⊆ [𝐴]𝑟))
5	4	rspcv 3563	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑊 → (∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟 → ran (𝑇‘𝐴) ⊆ [𝐴]𝑟))
6	5	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ ran (𝑇‘𝐴)) → (∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟 → ran (𝑇‘𝐴) ⊆ [𝐴]𝑟))
7		ssel 3916	. . . . . . . . 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 483	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ [𝐴]𝑟 ∧ 𝐴 ∈ 𝑊) → 𝐵 ∈ [𝐴]𝑟)
10		elecg 8685	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ [𝐴]𝑟 ∧ 𝐴 ∈ 𝑊) → (𝐵 ∈ [𝐴]𝑟 ↔ 𝐴𝑟𝐵))
11	9, 10	mpbid 233	. . . . . . . . . 10 ⊢ ((𝐵 ∈ [𝐴]𝑟 ∧ 𝐴 ∈ 𝑊) → 𝐴𝑟𝐵)
12		df-br 5080	. . . . . . . . . 10 ⊢ (𝐴𝑟𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑟)
13	11, 12	sylib 219	. . . . . . . . 9 ⊢ ((𝐵 ∈ [𝐴]𝑟 ∧ 𝐴 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ 𝑟)
14	13	expcom 414	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑊 → (𝐵 ∈ [𝐴]𝑟 → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
15	8, 14	sylan9r 513	. . . . . . 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 491	. . . . 5 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ ran (𝑇‘𝐴)) → ((𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟) → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
18	17	alrimiv 1934	. . . 4 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ ran (𝑇‘𝐴)) → ∀𝑟((𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟) → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
19		opex 5410	. . . . 5 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
20	19	elintab 4896	. . . 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 235	. . 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 19697	. . 3 ⊢ ∼ = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟)}
27	21, 26	eleqtrrdi 2851	. 2 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ ran (𝑇‘𝐴)) → ⟨𝐴, 𝐵⟩ ∈ ∼ )
28		df-br 5080	. 2 ⊢ (𝐴 ∼ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∼ )
29	27, 28	sylibr 235	1 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ ran (𝑇‘𝐴)) → 𝐴 ∼ 𝐵)

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1545   = wceq 1547   ∈ wcel 2119  {cab 2718  ∀wral 3054   ∖ cdif 3887   ⊆ wss 3890  ⟨cop 4568  ⟨cotp 4570  ∩ cint 4884   class class class wbr 5079   ↦ cmpt 5160   I cid 5519   × cxp 5623  ran crn 5626  ‘cfv 6492  (class class class)co 7363   ∈ cmpo 7365  1_oc1o 8395  2_oc2o 8396   Er wer 8637  [cec 8638  0cc0 11036  ...cfz 13459  ♯chash 14290  Word cword 14473   splice csplice 14709  ⟨“cs2 14801   ~_FG cefg 19679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685  ax-cnex 11092  ax-resscn 11093  ax-1cn 11094  ax-icn 11095  ax-addcl 11096  ax-addrcl 11097  ax-mulcl 11098  ax-mulrcl 11099  ax-mulcom 11100  ax-addass 11101  ax-mulass 11102  ax-distr 11103  ax-i2m1 11104  ax-1ne0 11105  ax-1rid 11106  ax-rnegex 11107  ax-rrecex 11108  ax-cnre 11109  ax-pre-lttri 11110  ax-pre-lttrn 11111  ax-pre-ltadd 11112  ax-pre-mulgt0 11113

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-nel 3040  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-ot 4571  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-er 8640  df-ec 8642  df-map 8772  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9861  df-pnf 11179  df-mnf 11180  df-xr 11181  df-ltxr 11182  df-le 11183  df-sub 11377  df-neg 11378  df-nn 12173  df-n0 12436  df-z 12523  df-uz 12787  df-fz 13460  df-fzo 13607  df-hash 14291  df-word 14474  df-concat 14531  df-s1 14557  df-substr 14602  df-pfx 14632  df-splice 14710  df-s2 14808  df-efg 19682

This theorem is referenced by:  efginvrel2  19700  efgsrel  19707  efgcpbllemb  19728