Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rngcval Structured version   Visualization version   GIF version

Theorem rngcval 43733
Description: Value of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020.) (Revised by AV, 8-Mar-2020.)
Hypotheses
Ref Expression
rngcval.c 𝐶 = (RngCat‘𝑈)
rngcval.u (𝜑𝑈𝑉)
rngcval.b (𝜑𝐵 = (𝑈 ∩ Rng))
rngcval.h (𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
Assertion
Ref Expression
rngcval (𝜑𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))

Proof of Theorem rngcval
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 rngcval.c . 2 𝐶 = (RngCat‘𝑈)
2 df-rngc 43730 . . 3 RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))
3 fveq2 6545 . . . . 5 (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
43adantl 482 . . . 4 ((𝜑𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
5 ineq1 4107 . . . . . . . 8 (𝑢 = 𝑈 → (𝑢 ∩ Rng) = (𝑈 ∩ Rng))
65sqxpeqd 5482 . . . . . . 7 (𝑢 = 𝑈 → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))
7 rngcval.b . . . . . . . . 9 (𝜑𝐵 = (𝑈 ∩ Rng))
87sqxpeqd 5482 . . . . . . . 8 (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))
98eqcomd 2803 . . . . . . 7 (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) = (𝐵 × 𝐵))
106, 9sylan9eqr 2855 . . . . . 6 ((𝜑𝑢 = 𝑈) → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = (𝐵 × 𝐵))
1110reseq2d 5741 . . . . 5 ((𝜑𝑢 = 𝑈) → ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = ( RngHomo ↾ (𝐵 × 𝐵)))
12 rngcval.h . . . . . . 7 (𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
1312eqcomd 2803 . . . . . 6 (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) = 𝐻)
1413adantr 481 . . . . 5 ((𝜑𝑢 = 𝑈) → ( RngHomo ↾ (𝐵 × 𝐵)) = 𝐻)
1511, 14eqtrd 2833 . . . 4 ((𝜑𝑢 = 𝑈) → ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = 𝐻)
164, 15oveq12d 7041 . . 3 ((𝜑𝑢 = 𝑈) → ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
17 rngcval.u . . . 4 (𝜑𝑈𝑉)
1817elexd 3460 . . 3 (𝜑𝑈 ∈ V)
19 ovexd 7057 . . 3 (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V)
202, 16, 18, 19fvmptd2 6649 . 2 (𝜑 → (RngCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
211, 20syl5eq 2845 1 (𝜑𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1525  wcel 2083  Vcvv 3440  cin 3864   × cxp 5448  cres 5452  cfv 6232  (class class class)co 7023  cat cresc 16911  ExtStrCatcestrc 17205  Rngcrng 43645   RngHomo crngh 43656  RngCatcrngc 43728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-res 5462  df-iota 6196  df-fun 6234  df-fv 6240  df-ov 7026  df-rngc 43730
This theorem is referenced by:  rngcbas  43736  rngchomfval  43737  rngccofval  43741  dfrngc2  43743  rngccat  43749  rngcid  43750  rngcifuestrc  43768  funcrngcsetc  43769
  Copyright terms: Public domain W3C validator