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Theorem ringcval 20663
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 20662 . . 3 RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))))
3 fveq2 6906 . . . . 5 (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
43adantl 481 . . . 4 ((𝜑𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
5 ineq1 4220 . . . . . . . 8 (𝑢 = 𝑈 → (𝑢 ∩ Ring) = (𝑈 ∩ Ring))
65sqxpeqd 5720 . . . . . . 7 (𝑢 = 𝑈 → ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)) = ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
7 ringcval.b . . . . . . . . 9 (𝜑𝐵 = (𝑈 ∩ Ring))
87sqxpeqd 5720 . . . . . . . 8 (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
98eqcomd 2740 . . . . . . 7 (𝜑 → ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) = (𝐵 × 𝐵))
106, 9sylan9eqr 2796 . . . . . 6 ((𝜑𝑢 = 𝑈) → ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)) = (𝐵 × 𝐵))
1110reseq2d 5999 . . . . 5 ((𝜑𝑢 = 𝑈) → ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))) = ( RingHom ↾ (𝐵 × 𝐵)))
12 ringcval.h . . . . . . 7 (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
1312eqcomd 2740 . . . . . 6 (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) = 𝐻)
1413adantr 480 . . . . 5 ((𝜑𝑢 = 𝑈) → ( RingHom ↾ (𝐵 × 𝐵)) = 𝐻)
1511, 14eqtrd 2774 . . . 4 ((𝜑𝑢 = 𝑈) → ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))) = 𝐻)
164, 15oveq12d 7448 . . 3 ((𝜑𝑢 = 𝑈) → ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
17 ringcval.u . . . 4 (𝜑𝑈𝑉)
1817elexd 3501 . . 3 (𝜑𝑈 ∈ V)
19 ovexd 7465 . . 3 (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V)
202, 16, 18, 19fvmptd2 7023 . 2 (𝜑 → (RingCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
211, 20eqtrid 2786 1 (𝜑𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  Vcvv 3477  cin 3961   × cxp 5686  cres 5690  cfv 6562  (class class class)co 7430  cat cresc 17855  ExtStrCatcestrc 18176  Ringcrg 20250   RingHom crh 20485  RingCatcringc 20661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-res 5700  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-ringc 20662
This theorem is referenced by:  ringcbas  20666  ringchomfval  20667  ringccofval  20671  dfringc2  20673  ringccat  20679  ringcid  20680  funcringcsetc  20690
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