Theorem rngcval 20537

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	⊢ (𝜑 → 𝑈 ∈ 𝑉)
rngcval.b	⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
rngcval.h	⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))

Assertion

Ref	Expression
rngcval	⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))

Proof of Theorem rngcval

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rngcval.c	. 2 ⊢ 𝐶 = (RngCat‘𝑈)
2		df-rngc 20536	. . 3 ⊢ RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))
3		fveq2 6830	. . . . 5 ⊢ (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
4	3	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
5		ineq1 4162	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑢 ∩ Rng) = (𝑈 ∩ Rng))
6	5	sqxpeqd 5653	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))
7		rngcval.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
8	7	sqxpeqd 5653	. . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))
9	8	eqcomd 2739	. . . . . . 7 ⊢ (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) = (𝐵 × 𝐵))
10	6, 9	sylan9eqr 2790	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = (𝐵 × 𝐵))
11	10	reseq2d 5934	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = ( RngHom ↾ (𝐵 × 𝐵)))
12		rngcval.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))
13	12	eqcomd 2739	. . . . . 6 ⊢ (𝜑 → ( RngHom ↾ (𝐵 × 𝐵)) = 𝐻)
14	13	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHom ↾ (𝐵 × 𝐵)) = 𝐻)
15	11, 14	eqtrd 2768	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = 𝐻)
16	4, 15	oveq12d 7372	. . 3 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((ExtStrCat‘𝑢) ↾cat ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
17		rngcval.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
18	17	elexd 3461	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
19		ovexd 7389	. . 3 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V)
20	2, 16, 18, 19	fvmptd2 6945	. 2 ⊢ (𝜑 → (RngCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
21	1, 20	eqtrid 2780	1 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3437   ∩ cin 3897   × cxp 5619   ↾ cres 5623  ‘cfv 6488  (class class class)co 7354   ↾_cat cresc 17719  ExtStrCatcestrc 18032  Rngcrng 20074   RngHom crnghm 20356  RngCatcrngc 20535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-res 5633  df-iota 6444  df-fun 6490  df-fv 6496  df-ov 7357  df-rngc 20536

This theorem is referenced by:  rngcbas  20540  rngchomfval  20541  rngccofval  20545  dfrngc2  20547  rngccat  20553  rngcid  20554  rngcifuestrc  20558  funcrngcsetc  20559