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Theorem efgi2 19418
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 6819 . . . . . . . . . . 11 (π‘Ž = 𝐴 β†’ (π‘‡β€˜π‘Ž) = (π‘‡β€˜π΄))
21rneqd 5873 . . . . . . . . . 10 (π‘Ž = 𝐴 β†’ ran (π‘‡β€˜π‘Ž) = ran (π‘‡β€˜π΄))
3 eceq1 8599 . . . . . . . . . 10 (π‘Ž = 𝐴 β†’ [π‘Ž]π‘Ÿ = [𝐴]π‘Ÿ)
42, 3sseq12d 3964 . . . . . . . . 9 (π‘Ž = 𝐴 β†’ (ran (π‘‡β€˜π‘Ž) βŠ† [π‘Ž]π‘Ÿ ↔ ran (π‘‡β€˜π΄) βŠ† [𝐴]π‘Ÿ))
54rspcv 3566 . . . . . . . 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 481 . . . . . . 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 3924 . . . . . . . . 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 483 . . . . . . . . . . 11 ((𝐡 ∈ [𝐴]π‘Ÿ ∧ 𝐴 ∈ π‘Š) β†’ 𝐡 ∈ [𝐴]π‘Ÿ)
10 elecg 8604 . . . . . . . . . . 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 5090 . . . . . . . . . 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 414 . . . . . . . 8 (𝐴 ∈ π‘Š β†’ (𝐡 ∈ [𝐴]π‘Ÿ β†’ ⟨𝐴, 𝐡⟩ ∈ π‘Ÿ))
158, 14sylan9r 509 . . . . . . 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 491 . . . . 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 1929 . . . 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 5403 . . . . 5 ⟨𝐴, 𝐡⟩ ∈ V
2019elintab 4904 . . . 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 19417 . . 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 2848 . 2 ((𝐴 ∈ π‘Š ∧ 𝐡 ∈ ran (π‘‡β€˜π΄)) β†’ ⟨𝐴, 𝐡⟩ ∈ ∼ )
28 df-br 5090 . 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 396  βˆ€wal 1538   = wceq 1540   ∈ wcel 2105  {cab 2713  βˆ€wral 3061   βˆ– cdif 3894   βŠ† wss 3897  βŸ¨cop 4578  βŸ¨cotp 4580  βˆ© cint 4893   class class class wbr 5089   ↦ cmpt 5172   I cid 5511   Γ— cxp 5612  ran crn 5615  β€˜cfv 6473  (class class class)co 7329   ∈ cmpo 7331  1oc1o 8352  2oc2o 8353   Er wer 8558  [cec 8559  0cc0 10964  ...cfz 13332  β™―chash 14137  Word cword 14309   splice csplice 14552  βŸ¨β€œcs2 14645   ~FG cefg 19399
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5226  ax-sep 5240  ax-nul 5247  ax-pow 5305  ax-pr 5369  ax-un 7642  ax-cnex 11020  ax-resscn 11021  ax-1cn 11022  ax-icn 11023  ax-addcl 11024  ax-addrcl 11025  ax-mulcl 11026  ax-mulrcl 11027  ax-mulcom 11028  ax-addass 11029  ax-mulass 11030  ax-distr 11031  ax-i2m1 11032  ax-1ne0 11033  ax-1rid 11034  ax-rnegex 11035  ax-rrecex 11036  ax-cnre 11037  ax-pre-lttri 11038  ax-pre-lttrn 11039  ax-pre-ltadd 11040  ax-pre-mulgt0 11041
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3727  df-csb 3843  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3916  df-nul 4269  df-if 4473  df-pw 4548  df-sn 4573  df-pr 4575  df-op 4579  df-ot 4581  df-uni 4852  df-int 4894  df-iun 4940  df-br 5090  df-opab 5152  df-mpt 5173  df-tr 5207  df-id 5512  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5569  df-we 5571  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6232  df-ord 6299  df-on 6300  df-lim 6301  df-suc 6302  df-iota 6425  df-fun 6475  df-fn 6476  df-f 6477  df-f1 6478  df-fo 6479  df-f1o 6480  df-fv 6481  df-riota 7286  df-ov 7332  df-oprab 7333  df-mpo 7334  df-om 7773  df-1st 7891  df-2nd 7892  df-frecs 8159  df-wrecs 8190  df-recs 8264  df-rdg 8303  df-1o 8359  df-2o 8360  df-er 8561  df-ec 8563  df-map 8680  df-en 8797  df-dom 8798  df-sdom 8799  df-fin 8800  df-card 9788  df-pnf 11104  df-mnf 11105  df-xr 11106  df-ltxr 11107  df-le 11108  df-sub 11300  df-neg 11301  df-nn 12067  df-n0 12327  df-z 12413  df-uz 12676  df-fz 13333  df-fzo 13476  df-hash 14138  df-word 14310  df-concat 14366  df-s1 14392  df-substr 14444  df-pfx 14474  df-splice 14553  df-s2 14652  df-efg 19402
This theorem is referenced by:  efginvrel2  19420  efgsrel  19427  efgcpbllemb  19448
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