Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mrsubff Structured version   Visualization version   GIF version

Theorem mrsubff 32759
Description: A substitution is a function from 𝑅 to 𝑅. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mrsubvr.v 𝑉 = (mVR‘𝑇)
mrsubvr.r 𝑅 = (mREx‘𝑇)
mrsubvr.s 𝑆 = (mRSubst‘𝑇)
Assertion
Ref Expression
mrsubff (𝑇𝑊𝑆:(𝑅pm 𝑉)⟶(𝑅m 𝑅))

Proof of Theorem mrsubff
Dummy variables 𝑒 𝑓 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6682 . . . . . . . . 9 (mCN‘𝑇) ∈ V
2 mrsubvr.v . . . . . . . . . 10 𝑉 = (mVR‘𝑇)
32fvexi 6683 . . . . . . . . 9 𝑉 ∈ V
41, 3unex 7468 . . . . . . . 8 ((mCN‘𝑇) ∪ 𝑉) ∈ V
5 eqid 2821 . . . . . . . . 9 (freeMnd‘((mCN‘𝑇) ∪ 𝑉)) = (freeMnd‘((mCN‘𝑇) ∪ 𝑉))
65frmdmnd 18023 . . . . . . . 8 (((mCN‘𝑇) ∪ 𝑉) ∈ V → (freeMnd‘((mCN‘𝑇) ∪ 𝑉)) ∈ Mnd)
74, 6mp1i 13 . . . . . . 7 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → (freeMnd‘((mCN‘𝑇) ∪ 𝑉)) ∈ Mnd)
8 simpr 487 . . . . . . . . 9 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → 𝑒𝑅)
9 eqid 2821 . . . . . . . . . . 11 (mCN‘𝑇) = (mCN‘𝑇)
10 mrsubvr.r . . . . . . . . . . 11 𝑅 = (mREx‘𝑇)
119, 2, 10mrexval 32748 . . . . . . . . . 10 (𝑇𝑊𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
1211ad2antrr 724 . . . . . . . . 9 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
138, 12eleqtrd 2915 . . . . . . . 8 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → 𝑒 ∈ Word ((mCN‘𝑇) ∪ 𝑉))
14 elpmi 8424 . . . . . . . . . . . . . 14 (𝑓 ∈ (𝑅pm 𝑉) → (𝑓:dom 𝑓𝑅 ∧ dom 𝑓𝑉))
1514simpld 497 . . . . . . . . . . . . 13 (𝑓 ∈ (𝑅pm 𝑉) → 𝑓:dom 𝑓𝑅)
1615ad3antlr 729 . . . . . . . . . . . 12 ((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) → 𝑓:dom 𝑓𝑅)
1716ffvelrnda 6850 . . . . . . . . . . 11 (((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ 𝑣 ∈ dom 𝑓) → (𝑓𝑣) ∈ 𝑅)
1812ad2antrr 724 . . . . . . . . . . 11 (((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ 𝑣 ∈ dom 𝑓) → 𝑅 = Word ((mCN‘𝑇) ∪ 𝑉))
1917, 18eleqtrd 2915 . . . . . . . . . 10 (((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ 𝑣 ∈ dom 𝑓) → (𝑓𝑣) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
20 simplr 767 . . . . . . . . . . 11 (((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ ¬ 𝑣 ∈ dom 𝑓) → 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉))
2120s1cld 13956 . . . . . . . . . 10 (((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) ∧ ¬ 𝑣 ∈ dom 𝑓) → ⟨“𝑣”⟩ ∈ Word ((mCN‘𝑇) ∪ 𝑉))
2219, 21ifclda 4500 . . . . . . . . 9 ((((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) ∧ 𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉)) → if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
2322fmpttd 6878 . . . . . . . 8 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → (𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)):((mCN‘𝑇) ∪ 𝑉)⟶Word ((mCN‘𝑇) ∪ 𝑉))
24 wrdco 14192 . . . . . . . 8 ((𝑒 ∈ Word ((mCN‘𝑇) ∪ 𝑉) ∧ (𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)):((mCN‘𝑇) ∪ 𝑉)⟶Word ((mCN‘𝑇) ∪ 𝑉)) → ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒) ∈ Word Word ((mCN‘𝑇) ∪ 𝑉))
2513, 23, 24syl2anc 586 . . . . . . 7 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒) ∈ Word Word ((mCN‘𝑇) ∪ 𝑉))
26 eqid 2821 . . . . . . . . . . 11 (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉))) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉)))
275, 26frmdbas 18016 . . . . . . . . . 10 (((mCN‘𝑇) ∪ 𝑉) ∈ V → (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉))) = Word ((mCN‘𝑇) ∪ 𝑉))
284, 27ax-mp 5 . . . . . . . . 9 (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉))) = Word ((mCN‘𝑇) ∪ 𝑉)
2928eqcomi 2830 . . . . . . . 8 Word ((mCN‘𝑇) ∪ 𝑉) = (Base‘(freeMnd‘((mCN‘𝑇) ∪ 𝑉)))
3029gsumwcl 18002 . . . . . . 7 (((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) ∈ Mnd ∧ ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒) ∈ Word Word ((mCN‘𝑇) ∪ 𝑉)) → ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
317, 25, 30syl2anc 586 . . . . . 6 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)) ∈ Word ((mCN‘𝑇) ∪ 𝑉))
3231, 12eleqtrrd 2916 . . . . 5 (((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) ∧ 𝑒𝑅) → ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)) ∈ 𝑅)
3332fmpttd 6878 . . . 4 ((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) → (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))):𝑅𝑅)
3410fvexi 6683 . . . . 5 𝑅 ∈ V
3534, 34elmap 8434 . . . 4 ((𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) ∈ (𝑅m 𝑅) ↔ (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))):𝑅𝑅)
3633, 35sylibr 236 . . 3 ((𝑇𝑊𝑓 ∈ (𝑅pm 𝑉)) → (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) ∈ (𝑅m 𝑅))
3736fmpttd 6878 . 2 (𝑇𝑊 → (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))):(𝑅pm 𝑉)⟶(𝑅m 𝑅))
38 mrsubvr.s . . . 4 𝑆 = (mRSubst‘𝑇)
399, 2, 10, 38, 5mrsubffval 32754 . . 3 (𝑇𝑊𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
4039feq1d 6498 . 2 (𝑇𝑊 → (𝑆:(𝑅pm 𝑉)⟶(𝑅m 𝑅) ↔ (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝑅 ↦ ((freeMnd‘((mCN‘𝑇) ∪ 𝑉)) Σg ((𝑣 ∈ ((mCN‘𝑇) ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))):(𝑅pm 𝑉)⟶(𝑅m 𝑅)))
4137, 40mpbird 259 1 (𝑇𝑊𝑆:(𝑅pm 𝑉)⟶(𝑅m 𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398   = wceq 1533  wcel 2110  Vcvv 3494  cun 3933  wss 3935  ifcif 4466  cmpt 5145  dom cdm 5554  ccom 5558  wf 6350  cfv 6354  (class class class)co 7155  m cmap 8405  pm cpm 8406  Word cword 13860  ⟨“cs1 13948  Basecbs 16482   Σg cgsu 16713  Mndcmnd 17910  freeMndcfrmd 18011  mCNcmcn 32707  mVRcmvar 32708  mRExcmrex 32713  mRSubstcmrsub 32717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  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-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4838  df-int 4876  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-tr 5172  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6147  df-ord 6193  df-on 6194  df-lim 6195  df-suc 6196  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7580  df-1st 7688  df-2nd 7689  df-wrecs 7946  df-recs 8007  df-rdg 8045  df-1o 8101  df-oadd 8105  df-er 8288  df-map 8407  df-pm 8408  df-en 8509  df-dom 8510  df-sdom 8511  df-fin 8512  df-card 9367  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-nn 11638  df-2 11699  df-n0 11897  df-z 11981  df-uz 12243  df-fz 12892  df-fzo 13033  df-seq 13369  df-hash 13690  df-word 13861  df-concat 13922  df-s1 13949  df-struct 16484  df-ndx 16485  df-slot 16486  df-base 16488  df-sets 16489  df-ress 16490  df-plusg 16577  df-0g 16714  df-gsum 16715  df-mgm 17851  df-sgrp 17900  df-mnd 17911  df-submnd 17956  df-frmd 18013  df-mrex 32733  df-mrsub 32737
This theorem is referenced by:  mrsubrn  32760  mrsubff1  32761  mrsub0  32763  mrsubf  32764  mrsubccat  32765  mrsubcn  32766  elmrsubrn  32767  elmsubrn  32775  msubrn  32776  msubff  32777  msubff1  32803
  Copyright terms: Public domain W3C validator