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Theorem efgi2 19593
Description: Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
Hypotheses
Ref Expression
efgval.w π‘Š = ( I β€˜Word (𝐼 Γ— 2o))
efgval.r ∼ = ( ~FG β€˜πΌ)
efgval2.m 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)
efgval2.t 𝑇 = (𝑣 ∈ π‘Š ↦ (𝑛 ∈ (0...(β™―β€˜π‘£)), 𝑀 ∈ (𝐼 Γ— 2o) ↦ (𝑣 splice βŸ¨π‘›, 𝑛, βŸ¨β€œπ‘€(π‘€β€˜π‘€)β€βŸ©βŸ©)))
Assertion
Ref Expression
efgi2 ((𝐴 ∈ π‘Š ∧ 𝐡 ∈ ran (π‘‡β€˜π΄)) β†’ 𝐴 ∼ 𝐡)
Distinct variable groups:   𝑦,𝑧   𝑣,𝑛,𝑀,𝑦,𝑧   𝑛,𝑀,𝑣,𝑀   𝑛,π‘Š,𝑣,𝑀,𝑦,𝑧   𝑦, ∼ ,𝑧   𝑛,𝐼,𝑣,𝑀,𝑦,𝑧
Allowed substitution hints:   𝐴(𝑦,𝑧,𝑀,𝑣,𝑛)   𝐡(𝑦,𝑧,𝑀,𝑣,𝑛)   ∼ (𝑀,𝑣,𝑛)   𝑇(𝑦,𝑧,𝑀,𝑣,𝑛)   𝑀(𝑦,𝑧)

Proof of Theorem efgi2
Dummy variables π‘Ž π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6892 . . . . . . . . . . 11 (π‘Ž = 𝐴 β†’ (π‘‡β€˜π‘Ž) = (π‘‡β€˜π΄))
21rneqd 5938 . . . . . . . . . 10 (π‘Ž = 𝐴 β†’ ran (π‘‡β€˜π‘Ž) = ran (π‘‡β€˜π΄))
3 eceq1 8741 . . . . . . . . . 10 (π‘Ž = 𝐴 β†’ [π‘Ž]π‘Ÿ = [𝐴]π‘Ÿ)
42, 3sseq12d 4016 . . . . . . . . 9 (π‘Ž = 𝐴 β†’ (ran (π‘‡β€˜π‘Ž) βŠ† [π‘Ž]π‘Ÿ ↔ ran (π‘‡β€˜π΄) βŠ† [𝐴]π‘Ÿ))
54rspcv 3609 . . . . . . . 8 (𝐴 ∈ π‘Š β†’ (βˆ€π‘Ž ∈ π‘Š ran (π‘‡β€˜π‘Ž) βŠ† [π‘Ž]π‘Ÿ β†’ ran (π‘‡β€˜π΄) βŠ† [𝐴]π‘Ÿ))
65adantr 482 . . . . . . 7 ((𝐴 ∈ π‘Š ∧ 𝐡 ∈ ran (π‘‡β€˜π΄)) β†’ (βˆ€π‘Ž ∈ π‘Š ran (π‘‡β€˜π‘Ž) βŠ† [π‘Ž]π‘Ÿ β†’ ran (π‘‡β€˜π΄) βŠ† [𝐴]π‘Ÿ))
7 ssel 3976 . . . . . . . . 9 (ran (π‘‡β€˜π΄) βŠ† [𝐴]π‘Ÿ β†’ (𝐡 ∈ ran (π‘‡β€˜π΄) β†’ 𝐡 ∈ [𝐴]π‘Ÿ))
87com12 32 . . . . . . . 8 (𝐡 ∈ ran (π‘‡β€˜π΄) β†’ (ran (π‘‡β€˜π΄) βŠ† [𝐴]π‘Ÿ β†’ 𝐡 ∈ [𝐴]π‘Ÿ))
9 simpl 484 . . . . . . . . . . 11 ((𝐡 ∈ [𝐴]π‘Ÿ ∧ 𝐴 ∈ π‘Š) β†’ 𝐡 ∈ [𝐴]π‘Ÿ)
10 elecg 8746 . . . . . . . . . . 11 ((𝐡 ∈ [𝐴]π‘Ÿ ∧ 𝐴 ∈ π‘Š) β†’ (𝐡 ∈ [𝐴]π‘Ÿ ↔ π΄π‘Ÿπ΅))
119, 10mpbid 231 . . . . . . . . . 10 ((𝐡 ∈ [𝐴]π‘Ÿ ∧ 𝐴 ∈ π‘Š) β†’ π΄π‘Ÿπ΅)
12 df-br 5150 . . . . . . . . . 10 (π΄π‘Ÿπ΅ ↔ ⟨𝐴, 𝐡⟩ ∈ π‘Ÿ)
1311, 12sylib 217 . . . . . . . . 9 ((𝐡 ∈ [𝐴]π‘Ÿ ∧ 𝐴 ∈ π‘Š) β†’ ⟨𝐴, 𝐡⟩ ∈ π‘Ÿ)
1413expcom 415 . . . . . . . 8 (𝐴 ∈ π‘Š β†’ (𝐡 ∈ [𝐴]π‘Ÿ β†’ ⟨𝐴, 𝐡⟩ ∈ π‘Ÿ))
158, 14sylan9r 510 . . . . . . 7 ((𝐴 ∈ π‘Š ∧ 𝐡 ∈ ran (π‘‡β€˜π΄)) β†’ (ran (π‘‡β€˜π΄) βŠ† [𝐴]π‘Ÿ β†’ ⟨𝐴, 𝐡⟩ ∈ π‘Ÿ))
166, 15syld 47 . . . . . 6 ((𝐴 ∈ π‘Š ∧ 𝐡 ∈ ran (π‘‡β€˜π΄)) β†’ (βˆ€π‘Ž ∈ π‘Š ran (π‘‡β€˜π‘Ž) βŠ† [π‘Ž]π‘Ÿ β†’ ⟨𝐴, 𝐡⟩ ∈ π‘Ÿ))
1716adantld 492 . . . . 5 ((𝐴 ∈ π‘Š ∧ 𝐡 ∈ ran (π‘‡β€˜π΄)) β†’ ((π‘Ÿ Er π‘Š ∧ βˆ€π‘Ž ∈ π‘Š ran (π‘‡β€˜π‘Ž) βŠ† [π‘Ž]π‘Ÿ) β†’ ⟨𝐴, 𝐡⟩ ∈ π‘Ÿ))
1817alrimiv 1931 . . . 4 ((𝐴 ∈ π‘Š ∧ 𝐡 ∈ ran (π‘‡β€˜π΄)) β†’ βˆ€π‘Ÿ((π‘Ÿ Er π‘Š ∧ βˆ€π‘Ž ∈ π‘Š ran (π‘‡β€˜π‘Ž) βŠ† [π‘Ž]π‘Ÿ) β†’ ⟨𝐴, 𝐡⟩ ∈ π‘Ÿ))
19 opex 5465 . . . . 5 ⟨𝐴, 𝐡⟩ ∈ V
2019elintab 4963 . . . 4 (⟨𝐴, 𝐡⟩ ∈ ∩ {π‘Ÿ ∣ (π‘Ÿ Er π‘Š ∧ βˆ€π‘Ž ∈ π‘Š ran (π‘‡β€˜π‘Ž) βŠ† [π‘Ž]π‘Ÿ)} ↔ βˆ€π‘Ÿ((π‘Ÿ Er π‘Š ∧ βˆ€π‘Ž ∈ π‘Š ran (π‘‡β€˜π‘Ž) βŠ† [π‘Ž]π‘Ÿ) β†’ ⟨𝐴, 𝐡⟩ ∈ π‘Ÿ))
2118, 20sylibr 233 . . 3 ((𝐴 ∈ π‘Š ∧ 𝐡 ∈ ran (π‘‡β€˜π΄)) β†’ ⟨𝐴, 𝐡⟩ ∈ ∩ {π‘Ÿ ∣ (π‘Ÿ Er π‘Š ∧ βˆ€π‘Ž ∈ π‘Š ran (π‘‡β€˜π‘Ž) βŠ† [π‘Ž]π‘Ÿ)})
22 efgval.w . . . 4 π‘Š = ( I β€˜Word (𝐼 Γ— 2o))
23 efgval.r . . . 4 ∼ = ( ~FG β€˜πΌ)
24 efgval2.m . . . 4 𝑀 = (𝑦 ∈ 𝐼, 𝑧 ∈ 2o ↦ βŸ¨π‘¦, (1o βˆ– 𝑧)⟩)
25 efgval2.t . . . 4 𝑇 = (𝑣 ∈ π‘Š ↦ (𝑛 ∈ (0...(β™―β€˜π‘£)), 𝑀 ∈ (𝐼 Γ— 2o) ↦ (𝑣 splice βŸ¨π‘›, 𝑛, βŸ¨β€œπ‘€(π‘€β€˜π‘€)β€βŸ©βŸ©)))
2622, 23, 24, 25efgval2 19592 . . 3 ∼ = ∩ {π‘Ÿ ∣ (π‘Ÿ Er π‘Š ∧ βˆ€π‘Ž ∈ π‘Š ran (π‘‡β€˜π‘Ž) βŠ† [π‘Ž]π‘Ÿ)}
2721, 26eleqtrrdi 2845 . 2 ((𝐴 ∈ π‘Š ∧ 𝐡 ∈ ran (π‘‡β€˜π΄)) β†’ ⟨𝐴, 𝐡⟩ ∈ ∼ )
28 df-br 5150 . 2 (𝐴 ∼ 𝐡 ↔ ⟨𝐴, 𝐡⟩ ∈ ∼ )
2927, 28sylibr 233 1 ((𝐴 ∈ π‘Š ∧ 𝐡 ∈ ran (π‘‡β€˜π΄)) β†’ 𝐴 ∼ 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3062   βˆ– cdif 3946   βŠ† wss 3949  βŸ¨cop 4635  βŸ¨cotp 4637  βˆ© cint 4951   class class class wbr 5149   ↦ cmpt 5232   I cid 5574   Γ— cxp 5675  ran crn 5678  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  1oc1o 8459  2oc2o 8460   Er wer 8700  [cec 8701  0cc0 11110  ...cfz 13484  β™―chash 14290  Word cword 14464   splice csplice 14699  βŸ¨β€œcs2 14792   ~FG cefg 19574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-ot 4638  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-2o 8467  df-er 8703  df-ec 8705  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-concat 14521  df-s1 14546  df-substr 14591  df-pfx 14621  df-splice 14700  df-s2 14799  df-efg 19577
This theorem is referenced by:  efginvrel2  19595  efgsrel  19602  efgcpbllemb  19623
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