MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  chrval Structured version   Visualization version   GIF version

Theorem chrval 19813
Description: Definition substitution of the ring characteristic. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
chrval.o 𝑂 = (od‘𝑅)
chrval.u 1 = (1r𝑅)
chrval.c 𝐶 = (chr‘𝑅)
Assertion
Ref Expression
chrval (𝑂1 ) = 𝐶

Proof of Theorem chrval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 chrval.c . 2 𝐶 = (chr‘𝑅)
2 fveq2 6158 . . . . . 6 (𝑟 = 𝑅 → (od‘𝑟) = (od‘𝑅))
3 chrval.o . . . . . 6 𝑂 = (od‘𝑅)
42, 3syl6eqr 2673 . . . . 5 (𝑟 = 𝑅 → (od‘𝑟) = 𝑂)
5 fveq2 6158 . . . . . 6 (𝑟 = 𝑅 → (1r𝑟) = (1r𝑅))
6 chrval.u . . . . . 6 1 = (1r𝑅)
75, 6syl6eqr 2673 . . . . 5 (𝑟 = 𝑅 → (1r𝑟) = 1 )
84, 7fveq12d 6164 . . . 4 (𝑟 = 𝑅 → ((od‘𝑟)‘(1r𝑟)) = (𝑂1 ))
9 df-chr 19794 . . . 4 chr = (𝑟 ∈ V ↦ ((od‘𝑟)‘(1r𝑟)))
10 fvex 6168 . . . 4 (𝑂1 ) ∈ V
118, 9, 10fvmpt 6249 . . 3 (𝑅 ∈ V → (chr‘𝑅) = (𝑂1 ))
12 fvprc 6152 . . . 4 𝑅 ∈ V → (chr‘𝑅) = ∅)
13 fvprc 6152 . . . . . . 7 𝑅 ∈ V → (od‘𝑅) = ∅)
143, 13syl5eq 2667 . . . . . 6 𝑅 ∈ V → 𝑂 = ∅)
1514fveq1d 6160 . . . . 5 𝑅 ∈ V → (𝑂1 ) = (∅‘ 1 ))
16 0fv 6194 . . . . 5 (∅‘ 1 ) = ∅
1715, 16syl6eq 2671 . . . 4 𝑅 ∈ V → (𝑂1 ) = ∅)
1812, 17eqtr4d 2658 . . 3 𝑅 ∈ V → (chr‘𝑅) = (𝑂1 ))
1911, 18pm2.61i 176 . 2 (chr‘𝑅) = (𝑂1 )
201, 19eqtr2i 2644 1 (𝑂1 ) = 𝐶
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  Vcvv 3190  c0 3897  cfv 5857  odcod 17884  1rcur 18441  chrcchr 19790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-iota 5820  df-fun 5859  df-fv 5865  df-chr 19794
This theorem is referenced by:  chrcl  19814  chrid  19815  chrdvds  19816  chrcong  19817  subrgchr  29621  ofldchr  29641  zrhchr  29844
  Copyright terms: Public domain W3C validator