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Theorem rngcval 42287
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 = ( RngHomo ↾ (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 42284 . . . 4 RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))
32a1i 11 . . 3 (𝜑 → RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))))))
4 fveq2 6229 . . . . 5 (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
54adantl 481 . . . 4 ((𝜑𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
6 ineq1 3840 . . . . . . . 8 (𝑢 = 𝑈 → (𝑢 ∩ Rng) = (𝑈 ∩ Rng))
76sqxpeqd 5175 . . . . . . 7 (𝑢 = 𝑈 → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))
8 rngcval.b . . . . . . . . 9 (𝜑𝐵 = (𝑈 ∩ Rng))
98sqxpeqd 5175 . . . . . . . 8 (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))
109eqcomd 2657 . . . . . . 7 (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) = (𝐵 × 𝐵))
117, 10sylan9eqr 2707 . . . . . 6 ((𝜑𝑢 = 𝑈) → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = (𝐵 × 𝐵))
1211reseq2d 5428 . . . . 5 ((𝜑𝑢 = 𝑈) → ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = ( RngHomo ↾ (𝐵 × 𝐵)))
13 rngcval.h . . . . . . 7 (𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
1413eqcomd 2657 . . . . . 6 (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) = 𝐻)
1514adantr 480 . . . . 5 ((𝜑𝑢 = 𝑈) → ( RngHomo ↾ (𝐵 × 𝐵)) = 𝐻)
1612, 15eqtrd 2685 . . . 4 ((𝜑𝑢 = 𝑈) → ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = 𝐻)
175, 16oveq12d 6708 . . 3 ((𝜑𝑢 = 𝑈) → ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
18 rngcval.u . . . 4 (𝜑𝑈𝑉)
19 elex 3243 . . . 4 (𝑈𝑉𝑈 ∈ V)
2018, 19syl 17 . . 3 (𝜑𝑈 ∈ V)
21 ovexd 6720 . . 3 (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V)
223, 17, 20, 21fvmptd 6327 . 2 (𝜑 → (RngCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
231, 22syl5eq 2697 1 (𝜑𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  Vcvv 3231  cin 3606  cmpt 4762   × cxp 5141  cres 5145  cfv 5926  (class class class)co 6690  cat cresc 16515  ExtStrCatcestrc 16809  Rngcrng 42199   RngHomo crngh 42210  RngCatcrngc 42282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-rngc 42284
This theorem is referenced by:  rngcbas  42290  rngchomfval  42291  rngccofval  42295  dfrngc2  42297  rngccat  42303  rngcid  42304  rngcifuestrc  42322  funcrngcsetc  42323
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