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Theorem rngcval 45013
 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 45010 . . 3 RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))
3 fveq2 6664 . . . . 5 (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
43adantl 485 . . . 4 ((𝜑𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
5 ineq1 4112 . . . . . . . 8 (𝑢 = 𝑈 → (𝑢 ∩ Rng) = (𝑈 ∩ Rng))
65sqxpeqd 5561 . . . . . . 7 (𝑢 = 𝑈 → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))
7 rngcval.b . . . . . . . . 9 (𝜑𝐵 = (𝑈 ∩ Rng))
87sqxpeqd 5561 . . . . . . . 8 (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))
98eqcomd 2765 . . . . . . 7 (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) = (𝐵 × 𝐵))
106, 9sylan9eqr 2816 . . . . . 6 ((𝜑𝑢 = 𝑈) → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = (𝐵 × 𝐵))
1110reseq2d 5829 . . . . 5 ((𝜑𝑢 = 𝑈) → ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = ( RngHomo ↾ (𝐵 × 𝐵)))
12 rngcval.h . . . . . . 7 (𝜑𝐻 = ( RngHomo ↾ (𝐵 × 𝐵)))
1312eqcomd 2765 . . . . . 6 (𝜑 → ( RngHomo ↾ (𝐵 × 𝐵)) = 𝐻)
1413adantr 484 . . . . 5 ((𝜑𝑢 = 𝑈) → ( RngHomo ↾ (𝐵 × 𝐵)) = 𝐻)
1511, 14eqtrd 2794 . . . 4 ((𝜑𝑢 = 𝑈) → ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = 𝐻)
164, 15oveq12d 7175 . . 3 ((𝜑𝑢 = 𝑈) → ((ExtStrCat‘𝑢) ↾cat ( RngHomo ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
17 rngcval.u . . . 4 (𝜑𝑈𝑉)
1817elexd 3431 . . 3 (𝜑𝑈 ∈ V)
19 ovexd 7192 . . 3 (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V)
202, 16, 18, 19fvmptd2 6773 . 2 (𝜑 → (RngCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
211, 20syl5eq 2806 1 (𝜑𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1539   ∈ wcel 2112  Vcvv 3410   ∩ cin 3860   × cxp 5527   ↾ cres 5531  ‘cfv 6341  (class class class)co 7157   ↾cat cresc 17152  ExtStrCatcestrc 17453  Rngcrng 44925   RngHomo crngh 44936  RngCatcrngc 45008 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5174  ax-nul 5181  ax-pr 5303 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ral 3076  df-rex 3077  df-rab 3080  df-v 3412  df-sbc 3700  df-csb 3809  df-dif 3864  df-un 3866  df-in 3868  df-ss 3878  df-nul 4229  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4803  df-br 5038  df-opab 5100  df-mpt 5118  df-id 5435  df-xp 5535  df-rel 5536  df-cnv 5537  df-co 5538  df-dm 5539  df-res 5541  df-iota 6300  df-fun 6343  df-fv 6349  df-ov 7160  df-rngc 45010 This theorem is referenced by:  rngcbas  45016  rngchomfval  45017  rngccofval  45021  dfrngc2  45023  rngccat  45029  rngcid  45030  rngcifuestrc  45048  funcrngcsetc  45049
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