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Theorem rngcval 20556
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 = ( RngHom ↾ (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 20555 . . 3 RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))
3 fveq2 6900 . . . . 5 (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
43adantl 480 . . . 4 ((𝜑𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
5 ineq1 4205 . . . . . . . 8 (𝑢 = 𝑈 → (𝑢 ∩ Rng) = (𝑈 ∩ Rng))
65sqxpeqd 5712 . . . . . . 7 (𝑢 = 𝑈 → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))
7 rngcval.b . . . . . . . . 9 (𝜑𝐵 = (𝑈 ∩ Rng))
87sqxpeqd 5712 . . . . . . . 8 (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))
98eqcomd 2733 . . . . . . 7 (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) = (𝐵 × 𝐵))
106, 9sylan9eqr 2789 . . . . . 6 ((𝜑𝑢 = 𝑈) → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = (𝐵 × 𝐵))
1110reseq2d 5987 . . . . 5 ((𝜑𝑢 = 𝑈) → ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = ( RngHom ↾ (𝐵 × 𝐵)))
12 rngcval.h . . . . . . 7 (𝜑𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))
1312eqcomd 2733 . . . . . 6 (𝜑 → ( RngHom ↾ (𝐵 × 𝐵)) = 𝐻)
1413adantr 479 . . . . 5 ((𝜑𝑢 = 𝑈) → ( RngHom ↾ (𝐵 × 𝐵)) = 𝐻)
1511, 14eqtrd 2767 . . . 4 ((𝜑𝑢 = 𝑈) → ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = 𝐻)
164, 15oveq12d 7442 . . 3 ((𝜑𝑢 = 𝑈) → ((ExtStrCat‘𝑢) ↾cat ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
17 rngcval.u . . . 4 (𝜑𝑈𝑉)
1817elexd 3492 . . 3 (𝜑𝑈 ∈ V)
19 ovexd 7459 . . 3 (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V)
202, 16, 18, 19fvmptd2 7016 . 2 (𝜑 → (RngCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
211, 20eqtrid 2779 1 (𝜑𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3471  cin 3946   × cxp 5678  cres 5682  cfv 6551  (class class class)co 7424  cat cresc 17796  ExtStrCatcestrc 18117  Rngcrng 20097   RngHom crnghm 20378  RngCatcrngc 20554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-res 5692  df-iota 6503  df-fun 6553  df-fv 6559  df-ov 7427  df-rngc 20555
This theorem is referenced by:  rngcbas  20559  rngchomfval  20560  rngccofval  20564  dfrngc2  20566  rngccat  20572  rngcid  20573  rngcifuestrc  20577  funcrngcsetc  20578
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