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Theorem ringcval 42573
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 42570 . . . 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 6404 . . . . 5 (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
54adantl 469 . . . 4 ((𝜑𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
6 ineq1 4006 . . . . . . . 8 (𝑢 = 𝑈 → (𝑢 ∩ Ring) = (𝑈 ∩ Ring))
76sqxpeqd 5342 . . . . . . 7 (𝑢 = 𝑈 → ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)) = ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
8 ringcval.b . . . . . . . . 9 (𝜑𝐵 = (𝑈 ∩ Ring))
98sqxpeqd 5342 . . . . . . . 8 (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
109eqcomd 2812 . . . . . . 7 (𝜑 → ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) = (𝐵 × 𝐵))
117, 10sylan9eqr 2862 . . . . . 6 ((𝜑𝑢 = 𝑈) → ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)) = (𝐵 × 𝐵))
1211reseq2d 5597 . . . . 5 ((𝜑𝑢 = 𝑈) → ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))) = ( RingHom ↾ (𝐵 × 𝐵)))
13 ringcval.h . . . . . . 7 (𝜑𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
1413eqcomd 2812 . . . . . 6 (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) = 𝐻)
1514adantr 468 . . . . 5 ((𝜑𝑢 = 𝑈) → ( RingHom ↾ (𝐵 × 𝐵)) = 𝐻)
1612, 15eqtrd 2840 . . . 4 ((𝜑𝑢 = 𝑈) → ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))) = 𝐻)
175, 16oveq12d 6888 . . 3 ((𝜑𝑢 = 𝑈) → ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
18 ringcval.u . . . 4 (𝜑𝑈𝑉)
19 elex 3406 . . . 4 (𝑈𝑉𝑈 ∈ V)
2018, 19syl 17 . . 3 (𝜑𝑈 ∈ V)
21 ovexd 6904 . . 3 (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V)
223, 17, 20, 21fvmptd 6505 . 2 (𝜑 → (RingCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
231, 22syl5eq 2852 1 (𝜑𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2156  Vcvv 3391  cin 3768  cmpt 4923   × cxp 5309  cres 5313  cfv 6097  (class class class)co 6870  cat cresc 16668  ExtStrCatcestrc 16962  Ringcrg 18745   RingHom crh 18912  RingCatcringc 42568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-res 5323  df-iota 6060  df-fun 6099  df-fv 6105  df-ov 6873  df-ringc 42570
This theorem is referenced by:  ringcbas  42576  ringchomfval  42577  ringccofval  42581  dfringc2  42583  ringccat  42589  ringcid  42590  funcringcsetc  42600
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