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

Theorem mthmval 31804
Description: A theorem is a pre-statement, whose reduct is also the reduct of a provable pre-statement. Unlike the difference between pre-statement and statement, this application of the reduct is not necessarily trivial: there are theorems that are not themselves provable but are provable once enough "dummy variables" are introduced. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mthmval.r 𝑅 = (mStRed‘𝑇)
mthmval.j 𝐽 = (mPPSt‘𝑇)
mthmval.u 𝑈 = (mThm‘𝑇)
Assertion
Ref Expression
mthmval 𝑈 = (𝑅 “ (𝑅𝐽))

Proof of Theorem mthmval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 mthmval.u . 2 𝑈 = (mThm‘𝑇)
2 fveq2 6332 . . . . . . 7 (𝑡 = 𝑇 → (mStRed‘𝑡) = (mStRed‘𝑇))
3 mthmval.r . . . . . . 7 𝑅 = (mStRed‘𝑇)
42, 3syl6eqr 2822 . . . . . 6 (𝑡 = 𝑇 → (mStRed‘𝑡) = 𝑅)
54cnveqd 5436 . . . . 5 (𝑡 = 𝑇(mStRed‘𝑡) = 𝑅)
6 fveq2 6332 . . . . . . 7 (𝑡 = 𝑇 → (mPPSt‘𝑡) = (mPPSt‘𝑇))
7 mthmval.j . . . . . . 7 𝐽 = (mPPSt‘𝑇)
86, 7syl6eqr 2822 . . . . . 6 (𝑡 = 𝑇 → (mPPSt‘𝑡) = 𝐽)
94, 8imaeq12d 5608 . . . . 5 (𝑡 = 𝑇 → ((mStRed‘𝑡) “ (mPPSt‘𝑡)) = (𝑅𝐽))
105, 9imaeq12d 5608 . . . 4 (𝑡 = 𝑇 → ((mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) = (𝑅 “ (𝑅𝐽)))
11 df-mthm 31728 . . . 4 mThm = (𝑡 ∈ V ↦ ((mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))))
12 fvex 6342 . . . . . 6 (mStRed‘𝑡) ∈ V
1312cnvex 7259 . . . . 5 (mStRed‘𝑡) ∈ V
14 imaexg 7249 . . . . 5 ((mStRed‘𝑡) ∈ V → ((mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) ∈ V)
1513, 14ax-mp 5 . . . 4 ((mStRed‘𝑡) “ ((mStRed‘𝑡) “ (mPPSt‘𝑡))) ∈ V
1610, 11, 15fvmpt3i 6429 . . 3 (𝑇 ∈ V → (mThm‘𝑇) = (𝑅 “ (𝑅𝐽)))
17 0ima 5623 . . . . 5 (∅ “ (𝑅𝐽)) = ∅
1817eqcomi 2779 . . . 4 ∅ = (∅ “ (𝑅𝐽))
19 fvprc 6326 . . . 4 𝑇 ∈ V → (mThm‘𝑇) = ∅)
20 fvprc 6326 . . . . . . . 8 𝑇 ∈ V → (mStRed‘𝑇) = ∅)
213, 20syl5eq 2816 . . . . . . 7 𝑇 ∈ V → 𝑅 = ∅)
2221cnveqd 5436 . . . . . 6 𝑇 ∈ V → 𝑅 = ∅)
23 cnv0 5676 . . . . . 6 ∅ = ∅
2422, 23syl6eq 2820 . . . . 5 𝑇 ∈ V → 𝑅 = ∅)
2524imaeq1d 5606 . . . 4 𝑇 ∈ V → (𝑅 “ (𝑅𝐽)) = (∅ “ (𝑅𝐽)))
2618, 19, 253eqtr4a 2830 . . 3 𝑇 ∈ V → (mThm‘𝑇) = (𝑅 “ (𝑅𝐽)))
2716, 26pm2.61i 176 . 2 (mThm‘𝑇) = (𝑅 “ (𝑅𝐽))
281, 27eqtri 2792 1 𝑈 = (𝑅 “ (𝑅𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1630  wcel 2144  Vcvv 3349  c0 4061  ccnv 5248  cima 5252  cfv 6031  mStRedcmsr 31703  mPPStcmpps 31707  mThmcmthm 31708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039  df-mthm 31728
This theorem is referenced by:  elmthm  31805  mthmsta  31807  mthmblem  31809
  Copyright terms: Public domain W3C validator