Theorem rngcval 20746

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 20745	. . 3 ⊢ RngCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)))))
3		fveq2 6885	. . . . 5 ⊢ (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
4	3	adantl 487	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
5		ineq1 4166	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑢 ∩ Rng) = (𝑈 ∩ Rng))
6	5	sqxpeqd 5695	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))
7		rngcval.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Rng))
8	7	sqxpeqd 5695	. . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)))
9	8	eqcomd 2771	. . . . . . 7 ⊢ (𝜑 → ((𝑈 ∩ Rng) × (𝑈 ∩ Rng)) = (𝐵 × 𝐵))
10	6, 9	sylan9eqr 2822	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((𝑢 ∩ Rng) × (𝑢 ∩ Rng)) = (𝐵 × 𝐵))
11	10	reseq2d 5980	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = ( RngHom ↾ (𝐵 × 𝐵)))
12		rngcval.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 = ( RngHom ↾ (𝐵 × 𝐵)))
13	12	eqcomd 2771	. . . . . 6 ⊢ (𝜑 → ( RngHom ↾ (𝐵 × 𝐵)) = 𝐻)
14	13	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHom ↾ (𝐵 × 𝐵)) = 𝐻)
15	11, 14	eqtrd 2800	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RngHom ↾ ((𝑢 ∩ Rng) × (𝑢 ∩ Rng))) = 𝐻)
16	4, 15	oveq12d 7434	. . 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 3480	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
19		ovexd 7451	. . 3 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V)
20	2, 16, 18, 19	fvmptd2 7002	. 2 ⊢ (𝜑 → (RngCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
21	1, 20	eqtrid 2812	1 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3457   ∩ cin 3905   × cxp 5661   ↾ cres 5665  ‘cfv 6540  (class class class)co 7416   ↾_cat cresc 17882  ExtStrCatcestrc 18195  Rngcrng 20253   RngHom crnghm 20541  RngCatcrngc 20744

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-res 5675  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7419  df-rngc 20745

This theorem is used by:  rngcbas  20749  rngchomfval  20750  rngccofval  20754  dfrngc2  20756  rngccat  20762  rngcid  20763  rngcifuestrc  20767  funcrngcsetc  20768