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Theorem ringcval 42536
Description: Value of the category of unital rings (in a universe). (Contributed by AV, 13-Feb-2020.) (Revised by AV, 8-Mar-2020.)
Hypotheses
Ref Expression
ringcval.c 𝐶 = (RingCat‘𝑈)
ringcval.u (𝜑𝑈𝑉)
ringcval.b (𝜑𝐵 = (𝑈 ∩ Ring))
ringcval.h (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
Assertion
Ref Expression
ringcval (𝜑𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))

Proof of Theorem ringcval
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 ringcval.c . 2 𝐶 = (RingCat‘𝑈)
2 df-ringc 42533 . . . 4 RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))))
32a1i 11 . . 3 (𝜑 → RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))))))
4 fveq2 6332 . . . . 5 (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
54adantl 467 . . . 4 ((𝜑𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
6 ineq1 3958 . . . . . . . 8 (𝑢 = 𝑈 → (𝑢 ∩ Ring) = (𝑈 ∩ Ring))
76sqxpeqd 5281 . . . . . . 7 (𝑢 = 𝑈 → ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)) = ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
8 ringcval.b . . . . . . . . 9 (𝜑𝐵 = (𝑈 ∩ Ring))
98sqxpeqd 5281 . . . . . . . 8 (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
109eqcomd 2777 . . . . . . 7 (𝜑 → ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) = (𝐵 × 𝐵))
117, 10sylan9eqr 2827 . . . . . 6 ((𝜑𝑢 = 𝑈) → ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)) = (𝐵 × 𝐵))
1211reseq2d 5534 . . . . 5 ((𝜑𝑢 = 𝑈) → ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))) = ( RingHom ↾ (𝐵 × 𝐵)))
13 ringcval.h . . . . . . 7 (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
1413eqcomd 2777 . . . . . 6 (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) = 𝐻)
1514adantr 466 . . . . 5 ((𝜑𝑢 = 𝑈) → ( RingHom ↾ (𝐵 × 𝐵)) = 𝐻)
1612, 15eqtrd 2805 . . . 4 ((𝜑𝑢 = 𝑈) → ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))) = 𝐻)
175, 16oveq12d 6811 . . 3 ((𝜑𝑢 = 𝑈) → ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
18 ringcval.u . . . 4 (𝜑𝑈𝑉)
19 elex 3364 . . . 4 (𝑈𝑉𝑈 ∈ V)
2018, 19syl 17 . . 3 (𝜑𝑈 ∈ V)
21 ovexd 6825 . . 3 (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V)
223, 17, 20, 21fvmptd 6430 . 2 (𝜑 → (RingCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
231, 22syl5eq 2817 1 (𝜑𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  cin 3722  cmpt 4863   × cxp 5247  cres 5251  cfv 6031  (class class class)co 6793  cat cresc 16675  ExtStrCatcestrc 16969  Ringcrg 18755   RingHom crh 18922  RingCatcringc 42531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-res 5261  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-ringc 42533
This theorem is referenced by:  ringcbas  42539  ringchomfval  42540  ringccofval  42544  dfringc2  42546  ringccat  42552  ringcid  42553  funcringcsetc  42563
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