Theorem efgi2 18851

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 6670	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐴 → (𝑇‘𝑎) = (𝑇‘𝐴))
2	1	rneqd 5808	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → ran (𝑇‘𝑎) = ran (𝑇‘𝐴))
3		eceq1 8327	. . . . . . . . . 10 ⊢ (𝑎 = 𝐴 → [𝑎]𝑟 = [𝐴]𝑟)
4	2, 3	sseq12d 4000	. . . . . . . . 9 ⊢ (𝑎 = 𝐴 → (ran (𝑇‘𝑎) ⊆ [𝑎]𝑟 ↔ ran (𝑇‘𝐴) ⊆ [𝐴]𝑟))
5	4	rspcv 3618	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑊 → (∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟 → ran (𝑇‘𝐴) ⊆ [𝐴]𝑟))
6	5	adantr 483	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ ran (𝑇‘𝐴)) → (∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟 → ran (𝑇‘𝐴) ⊆ [𝐴]𝑟))
7		ssel 3961	. . . . . . . . 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 485	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ [𝐴]𝑟 ∧ 𝐴 ∈ 𝑊) → 𝐵 ∈ [𝐴]𝑟)
10		elecg 8332	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ [𝐴]𝑟 ∧ 𝐴 ∈ 𝑊) → (𝐵 ∈ [𝐴]𝑟 ↔ 𝐴𝑟𝐵))
11	9, 10	mpbid 234	. . . . . . . . . 10 ⊢ ((𝐵 ∈ [𝐴]𝑟 ∧ 𝐴 ∈ 𝑊) → 𝐴𝑟𝐵)
12		df-br 5067	. . . . . . . . . 10 ⊢ (𝐴𝑟𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑟)
13	11, 12	sylib 220	. . . . . . . . 9 ⊢ ((𝐵 ∈ [𝐴]𝑟 ∧ 𝐴 ∈ 𝑊) → ⟨𝐴, 𝐵⟩ ∈ 𝑟)
14	13	expcom 416	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑊 → (𝐵 ∈ [𝐴]𝑟 → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
15	8, 14	sylan9r 511	. . . . . . 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 493	. . . . 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 5356	. . . . 5 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
20	19	elintab 4887	. . . 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 236	. . 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 18850	. . 3 ⊢ ∼ = ∩ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎 ∈ 𝑊 ran (𝑇‘𝑎) ⊆ [𝑎]𝑟)}
27	21, 26	eleqtrrdi 2924	. 2 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ ran (𝑇‘𝐴)) → ⟨𝐴, 𝐵⟩ ∈ ∼ )
28		df-br 5067	. 2 ⊢ (𝐴 ∼ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ∼ )
29	27, 28	sylibr 236	1 ⊢ ((𝐴 ∈ 𝑊 ∧ 𝐵 ∈ ran (𝑇‘𝐴)) → 𝐴 ∼ 𝐵)

Syntax hints:   → wi 4   ∧ wa 398  ∀wal 1535   = wceq 1537   ∈ wcel 2114  {cab 2799  ∀wral 3138   ∖ cdif 3933   ⊆ wss 3936  ⟨cop 4573  ⟨cotp 4575  ∩ cint 4876   class class class wbr 5066   ↦ cmpt 5146   I cid 5459   × cxp 5553  ran crn 5556  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158  1_oc1o 8095  2_oc2o 8096   Er wer 8286  [cec 8287  0cc0 10537  ...cfz 12893  ♯chash 13691  Word cword 13862   splice csplice 14111  ⟨“cs2 14203   ~_FG cefg 18832

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-1cn 10595  ax-icn 10596  ax-addcl 10597  ax-addrcl 10598  ax-mulcl 10599  ax-mulrcl 10600  ax-mulcom 10601  ax-addass 10602  ax-mulass 10603  ax-distr 10604  ax-i2m1 10605  ax-1ne0 10606  ax-1rid 10607  ax-rnegex 10608  ax-rrecex 10609  ax-cnre 10610  ax-pre-lttri 10611  ax-pre-lttrn 10612  ax-pre-ltadd 10613  ax-pre-mulgt0 10614

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-ot 4576  df-uni 4839  df-int 4877  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-2o 8103  df-oadd 8106  df-er 8289  df-ec 8291  df-map 8408  df-en 8510  df-dom 8511  df-sdom 8512  df-fin 8513  df-card 9368  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680  df-le 10681  df-sub 10872  df-neg 10873  df-nn 11639  df-n0 11899  df-z 11983  df-uz 12245  df-fz 12894  df-fzo 13035  df-hash 13692  df-word 13863  df-concat 13923  df-s1 13950  df-substr 14003  df-pfx 14033  df-splice 14112  df-s2 14210  df-efg 18835

This theorem is referenced by:  efginvrel2  18853  efgsrel  18860  efgcpbllemb  18881