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Theorem ringcval 44562
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 44559 . . 3 RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))))
3 fveq2 6661 . . . . 5 (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
43adantl 485 . . . 4 ((𝜑𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
5 ineq1 4166 . . . . . . . 8 (𝑢 = 𝑈 → (𝑢 ∩ Ring) = (𝑈 ∩ Ring))
65sqxpeqd 5574 . . . . . . 7 (𝑢 = 𝑈 → ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)) = ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
7 ringcval.b . . . . . . . . 9 (𝜑𝐵 = (𝑈 ∩ Ring))
87sqxpeqd 5574 . . . . . . . 8 (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
98eqcomd 2830 . . . . . . 7 (𝜑 → ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) = (𝐵 × 𝐵))
106, 9sylan9eqr 2881 . . . . . 6 ((𝜑𝑢 = 𝑈) → ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)) = (𝐵 × 𝐵))
1110reseq2d 5840 . . . . 5 ((𝜑𝑢 = 𝑈) → ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))) = ( RingHom ↾ (𝐵 × 𝐵)))
12 ringcval.h . . . . . . 7 (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
1312eqcomd 2830 . . . . . 6 (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) = 𝐻)
1413adantr 484 . . . . 5 ((𝜑𝑢 = 𝑈) → ( RingHom ↾ (𝐵 × 𝐵)) = 𝐻)
1511, 14eqtrd 2859 . . . 4 ((𝜑𝑢 = 𝑈) → ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))) = 𝐻)
164, 15oveq12d 7167 . . 3 ((𝜑𝑢 = 𝑈) → ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
17 ringcval.u . . . 4 (𝜑𝑈𝑉)
1817elexd 3500 . . 3 (𝜑𝑈 ∈ V)
19 ovexd 7184 . . 3 (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V)
202, 16, 18, 19fvmptd2 6767 . 2 (𝜑 → (RingCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
211, 20syl5eq 2871 1 (𝜑𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  cin 3918   × cxp 5540  cres 5544  cfv 6343  (class class class)co 7149  cat cresc 17078  ExtStrCatcestrc 17372  Ringcrg 19297   RingHom crh 19467  RingCatcringc 44557
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-res 5554  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7152  df-ringc 44559
This theorem is referenced by:  ringcbas  44565  ringchomfval  44566  ringccofval  44570  dfringc2  44572  ringccat  44578  ringcid  44579  funcringcsetc  44589
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