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Theorem efgi2 17961
 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 = ( ~FG𝐼)
efgval2.m 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
efgval2.t 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
Assertion
Ref Expression
efgi2 ((𝐴𝑊𝐵 ∈ ran (𝑇𝐴)) → 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.
StepHypRef Expression
1 fveq2 6103 . . . . . . . . . . 11 (𝑎 = 𝐴 → (𝑇𝑎) = (𝑇𝐴))
21rneqd 5274 . . . . . . . . . 10 (𝑎 = 𝐴 → ran (𝑇𝑎) = ran (𝑇𝐴))
3 eceq1 7669 . . . . . . . . . 10 (𝑎 = 𝐴 → [𝑎]𝑟 = [𝐴]𝑟)
42, 3sseq12d 3597 . . . . . . . . 9 (𝑎 = 𝐴 → (ran (𝑇𝑎) ⊆ [𝑎]𝑟 ↔ ran (𝑇𝐴) ⊆ [𝐴]𝑟))
54rspcv 3278 . . . . . . . 8 (𝐴𝑊 → (∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟 → ran (𝑇𝐴) ⊆ [𝐴]𝑟))
65adantr 480 . . . . . . 7 ((𝐴𝑊𝐵 ∈ ran (𝑇𝐴)) → (∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟 → ran (𝑇𝐴) ⊆ [𝐴]𝑟))
7 ssel 3562 . . . . . . . . 9 (ran (𝑇𝐴) ⊆ [𝐴]𝑟 → (𝐵 ∈ ran (𝑇𝐴) → 𝐵 ∈ [𝐴]𝑟))
87com12 32 . . . . . . . 8 (𝐵 ∈ ran (𝑇𝐴) → (ran (𝑇𝐴) ⊆ [𝐴]𝑟𝐵 ∈ [𝐴]𝑟))
9 simpl 472 . . . . . . . . . . 11 ((𝐵 ∈ [𝐴]𝑟𝐴𝑊) → 𝐵 ∈ [𝐴]𝑟)
10 elecg 7672 . . . . . . . . . . 11 ((𝐵 ∈ [𝐴]𝑟𝐴𝑊) → (𝐵 ∈ [𝐴]𝑟𝐴𝑟𝐵))
119, 10mpbid 221 . . . . . . . . . 10 ((𝐵 ∈ [𝐴]𝑟𝐴𝑊) → 𝐴𝑟𝐵)
12 df-br 4584 . . . . . . . . . 10 (𝐴𝑟𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑟)
1311, 12sylib 207 . . . . . . . . 9 ((𝐵 ∈ [𝐴]𝑟𝐴𝑊) → ⟨𝐴, 𝐵⟩ ∈ 𝑟)
1413expcom 450 . . . . . . . 8 (𝐴𝑊 → (𝐵 ∈ [𝐴]𝑟 → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
158, 14sylan9r 688 . . . . . . 7 ((𝐴𝑊𝐵 ∈ ran (𝑇𝐴)) → (ran (𝑇𝐴) ⊆ [𝐴]𝑟 → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
166, 15syld 46 . . . . . 6 ((𝐴𝑊𝐵 ∈ ran (𝑇𝐴)) → (∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟 → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
1716adantld 482 . . . . 5 ((𝐴𝑊𝐵 ∈ ran (𝑇𝐴)) → ((𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟) → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
1817alrimiv 1842 . . . 4 ((𝐴𝑊𝐵 ∈ ran (𝑇𝐴)) → ∀𝑟((𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟) → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
19 opex 4859 . . . . 5 𝐴, 𝐵⟩ ∈ V
2019elintab 4422 . . . 4 (⟨𝐴, 𝐵⟩ ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)} ↔ ∀𝑟((𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟) → ⟨𝐴, 𝐵⟩ ∈ 𝑟))
2118, 20sylibr 223 . . 3 ((𝐴𝑊𝐵 ∈ ran (𝑇𝐴)) → ⟨𝐴, 𝐵⟩ ∈ {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)})
22 efgval.w . . . 4 𝑊 = ( I ‘Word (𝐼 × 2𝑜))
23 efgval.r . . . 4 = ( ~FG𝐼)
24 efgval2.m . . . 4 𝑀 = (𝑦𝐼, 𝑧 ∈ 2𝑜 ↦ ⟨𝑦, (1𝑜𝑧)⟩)
25 efgval2.t . . . 4 𝑇 = (𝑣𝑊 ↦ (𝑛 ∈ (0...(#‘𝑣)), 𝑤 ∈ (𝐼 × 2𝑜) ↦ (𝑣 splice ⟨𝑛, 𝑛, ⟨“𝑤(𝑀𝑤)”⟩⟩)))
2622, 23, 24, 25efgval2 17960 . . 3 = {𝑟 ∣ (𝑟 Er 𝑊 ∧ ∀𝑎𝑊 ran (𝑇𝑎) ⊆ [𝑎]𝑟)}
2721, 26syl6eleqr 2699 . 2 ((𝐴𝑊𝐵 ∈ ran (𝑇𝐴)) → ⟨𝐴, 𝐵⟩ ∈ )
28 df-br 4584 . 2 (𝐴 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ )
2927, 28sylibr 223 1 ((𝐴𝑊𝐵 ∈ ran (𝑇𝐴)) → 𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896   ∖ cdif 3537   ⊆ wss 3540  ⟨cop 4131  ⟨cotp 4133  ∩ cint 4410   class class class wbr 4583   ↦ cmpt 4643   I cid 4948   × cxp 5036  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1𝑜c1o 7440  2𝑜c2o 7441   Er wer 7626  [cec 7627  0cc0 9815  ...cfz 12197  #chash 12979  Word cword 13146   splice csplice 13151  ⟨“cs2 13437   ~FG cefg 17942 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-ot 4134  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-ec 7631  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-substr 13158  df-splice 13159  df-s2 13444  df-efg 17945 This theorem is referenced by:  efginvrel2  17963  efgsrel  17970  efgcpbllemb  17991
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