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 46413
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 𝐢 = (RngCatβ€˜π‘ˆ)
rngcval.u (πœ‘ β†’ π‘ˆ ∈ 𝑉)
rngcval.b (πœ‘ β†’ 𝐡 = (π‘ˆ ∩ Rng))
rngcval.h (πœ‘ β†’ 𝐻 = ( RngHomo β†Ύ (𝐡 Γ— 𝐡)))
Assertion
Ref Expression
rngcval (πœ‘ β†’ 𝐢 = ((ExtStrCatβ€˜π‘ˆ) β†Ύcat 𝐻))
Proof of Theorem rngcval
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 rngcval.c . 2 𝐢 = (RngCatβ€˜π‘ˆ)
2 df-rngc 46410 . . 3 RngCat = (𝑒 ∈ V ↦ ((ExtStrCatβ€˜π‘’) β†Ύcat ( RngHomo β†Ύ ((𝑒 ∩ Rng) Γ— (𝑒 ∩ Rng)))))
3 fveq2 6862 . . . . 5 (𝑒 = π‘ˆ β†’ (ExtStrCatβ€˜π‘’) = (ExtStrCatβ€˜π‘ˆ))
43adantl 482 . . . 4 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ (ExtStrCatβ€˜π‘’) = (ExtStrCatβ€˜π‘ˆ))
5 ineq1 4185 . . . . . . . 8 (𝑒 = π‘ˆ β†’ (𝑒 ∩ Rng) = (π‘ˆ ∩ Rng))
65sqxpeqd 5685 . . . . . . 7 (𝑒 = π‘ˆ β†’ ((𝑒 ∩ Rng) Γ— (𝑒 ∩ Rng)) = ((π‘ˆ ∩ Rng) Γ— (π‘ˆ ∩ Rng)))
7 rngcval.b . . . . . . . . 9 (πœ‘ β†’ 𝐡 = (π‘ˆ ∩ Rng))
87sqxpeqd 5685 . . . . . . . 8 (πœ‘ β†’ (𝐡 Γ— 𝐡) = ((π‘ˆ ∩ Rng) Γ— (π‘ˆ ∩ Rng)))
98eqcomd 2737 . . . . . . 7 (πœ‘ β†’ ((π‘ˆ ∩ Rng) Γ— (π‘ˆ ∩ Rng)) = (𝐡 Γ— 𝐡))
106, 9sylan9eqr 2793 . . . . . 6 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ((𝑒 ∩ Rng) Γ— (𝑒 ∩ Rng)) = (𝐡 Γ— 𝐡))
1110reseq2d 5957 . . . . 5 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ( RngHomo β†Ύ ((𝑒 ∩ Rng) Γ— (𝑒 ∩ Rng))) = ( RngHomo β†Ύ (𝐡 Γ— 𝐡)))
12 rngcval.h . . . . . . 7 (πœ‘ β†’ 𝐻 = ( RngHomo β†Ύ (𝐡 Γ— 𝐡)))
1312eqcomd 2737 . . . . . 6 (πœ‘ β†’ ( RngHomo β†Ύ (𝐡 Γ— 𝐡)) = 𝐻)
1413adantr 481 . . . . 5 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ( RngHomo β†Ύ (𝐡 Γ— 𝐡)) = 𝐻)
1511, 14eqtrd 2771 . . . 4 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ( RngHomo β†Ύ ((𝑒 ∩ Rng) Γ— (𝑒 ∩ Rng))) = 𝐻)
164, 15oveq12d 7395 . . 3 ((πœ‘ ∧ 𝑒 = π‘ˆ) β†’ ((ExtStrCatβ€˜π‘’) β†Ύcat ( RngHomo β†Ύ ((𝑒 ∩ Rng) Γ— (𝑒 ∩ Rng)))) = ((ExtStrCatβ€˜π‘ˆ) β†Ύcat 𝐻))
17 rngcval.u . . . 4 (πœ‘ β†’ π‘ˆ ∈ 𝑉)
1817elexd 3479 . . 3 (πœ‘ β†’ π‘ˆ ∈ V)
19 ovexd 7412 . . 3 (πœ‘ β†’ ((ExtStrCatβ€˜π‘ˆ) β†Ύcat 𝐻) ∈ V)
202, 16, 18, 19fvmptd2 6976 . 2 (πœ‘ β†’ (RngCatβ€˜π‘ˆ) = ((ExtStrCatβ€˜π‘ˆ) β†Ύcat 𝐻))
211, 20eqtrid 2783 1 (πœ‘ β†’ 𝐢 = ((ExtStrCatβ€˜π‘ˆ) β†Ύcat 𝐻))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3459   ∩ cin 3927   Γ— cxp 5651   β†Ύ cres 5655  β€˜cfv 6516  (class class class)co 7377   β†Ύcat cresc 17720  ExtStrCatcestrc 18038  Rngcrng 46325   RngHomo crngh 46336  RngCatcrngc 46408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-pr 5404
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-res 5665  df-iota 6468  df-fun 6518  df-fv 6524  df-ov 7380  df-rngc 46410
This theorem is referenced by:  rngcbas  46416  rngchomfval  46417  rngccofval  46421  dfrngc2  46423  rngccat  46429  rngcid  46430  rngcifuestrc  46448  funcrngcsetc  46449
  Copyright terms: Public domain W3C validator