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Theorem rngcval 41754
 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 41751 . . . 4 RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))
32a1i 11 . . 3 (𝜑 → RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))))))
4 fveq2 6103 . . . . 5 (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
54adantl 481 . . . 4 ((𝜑𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
6 ineq1 3769 . . . . . . . 8 (𝑢 = 𝑈 → (𝑢 ∩ Rng) = (𝑈 ∩ Rng))
76sqxpeqd 5065 . . . . . . 7 (𝑢 = 𝑈 → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))
8 rngcval.b . . . . . . . . 9 (𝜑𝐵 = (𝑈 ∩ Rng))
98sqxpeqd 5065 . . . . . . . 8 (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))
109eqcomd 2616 . . . . . . 7 (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) = (𝐵 × 𝐵))
117, 10sylan9eqr 2666 . . . . . 6 ((𝜑𝑢 = 𝑈) → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = (𝐵 × 𝐵))
1211reseq2d 5317 . . . . 5 ((𝜑𝑢 = 𝑈) → ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = ( RngHomo ↾ (𝐵 × 𝐵)))
13 rngcval.h . . . . . . 7 (𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
1413eqcomd 2616 . . . . . 6 (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) = 𝐻)
1514adantr 480 . . . . 5 ((𝜑𝑢 = 𝑈) → ( RngHomo ↾ (𝐵 × 𝐵)) = 𝐻)
1612, 15eqtrd 2644 . . . 4 ((𝜑𝑢 = 𝑈) → ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = 𝐻)
175, 16oveq12d 6567 . . 3 ((𝜑𝑢 = 𝑈) → ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
18 rngcval.u . . . 4 (𝜑𝑈𝑉)
19 elex 3185 . . . 4 (𝑈𝑉𝑈 ∈ V)
2018, 19syl 17 . . 3 (𝜑𝑈 ∈ V)
21 ovex 6577 . . . 4 ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V
2221a1i 11 . . 3 (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V)
233, 17, 20, 22fvmptd 6197 . 2 (𝜑 → (RngCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
241, 23syl5eq 2656 1 (𝜑𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ↦ cmpt 4643   × cxp 5036   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549   ↾cat cresc 16291  ExtStrCatcestrc 16585  Rngcrng 41664   RngHomo crngh 41675  RngCatcrngc 41749 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-rngc 41751 This theorem is referenced by:  rngcbas  41757  rngchomfval  41758  rngccofval  41762  dfrngc2  41764  rngccat  41770  rngcid  41771  rngcifuestrc  41789  funcrngcsetc  41790
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