Theorem ringcval 20723

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	⊢ (𝜑 → 𝑈 ∈ 𝑉)
ringcval.b	⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))
ringcval.h	⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))

Assertion

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

Proof of Theorem ringcval

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ringcval.c	. 2 ⊢ 𝐶 = (RingCat‘𝑈)
2		df-ringc 20722	. . 3 ⊢ RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))))
3		fveq2 6871	. . . . 5 ⊢ (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
4	3	adantl 486	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
5		ineq1 4168	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑢 ∩ Ring) = (𝑈 ∩ Ring))
6	5	sqxpeqd 5684	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)) = ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
7		ringcval.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))
8	7	sqxpeqd 5684	. . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
9	8	eqcomd 2771	. . . . . . 7 ⊢ (𝜑 → ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) = (𝐵 × 𝐵))
10	6, 9	sylan9eqr 2822	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)) = (𝐵 × 𝐵))
11	10	reseq2d 5969	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))) = ( RingHom ↾ (𝐵 × 𝐵)))
12		ringcval.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
13	12	eqcomd 2771	. . . . . 6 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) = 𝐻)
14	13	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RingHom ↾ (𝐵 × 𝐵)) = 𝐻)
15	11, 14	eqtrd 2800	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))) = 𝐻)
16	4, 15	oveq12d 7418	. . 3 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
17		ringcval.u	. . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑉)
18	17	elexd 3480	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
19		ovexd 7435	. . 3 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V)
20	2, 16, 18, 19	fvmptd2 6988	. 2 ⊢ (𝜑 → (RingCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
21	1, 20	eqtrid 2812	1 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ∩ cin 3906   × cxp 5650   ↾ cres 5654  ‘cfv 6525  (class class class)co 7400   ↾_cat cresc 17855  ExtStrCatcestrc 18168  Ringcrg 20306   RingHom crh 20542  RingCatcringc 20721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-res 5664  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-ringc 20722

This theorem is referenced by:  ringcbas  20726  ringchomfval  20727  ringccofval  20731  dfringc2  20733  ringccat  20739  ringcid  20740  funcringcsetc  20750