Theorem ringcval 20550

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 20549	. . 3 ⊢ RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))))
3		fveq2 6826	. . . . 5 ⊢ (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
4	3	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
5		ineq1 4166	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑢 ∩ Ring) = (𝑈 ∩ Ring))
6	5	sqxpeqd 5655	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)) = ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
7		ringcval.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))
8	7	sqxpeqd 5655	. . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
9	8	eqcomd 2735	. . . . . . 7 ⊢ (𝜑 → ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) = (𝐵 × 𝐵))
10	6, 9	sylan9eqr 2786	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)) = (𝐵 × 𝐵))
11	10	reseq2d 5934	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))) = ( RingHom ↾ (𝐵 × 𝐵)))
12		ringcval.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
13	12	eqcomd 2735	. . . . . 6 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) = 𝐻)
14	13	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RingHom ↾ (𝐵 × 𝐵)) = 𝐻)
15	11, 14	eqtrd 2764	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))) = 𝐻)
16	4, 15	oveq12d 7371	. . 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 3462	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
19		ovexd 7388	. . 3 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V)
20	2, 16, 18, 19	fvmptd2 6942	. 2 ⊢ (𝜑 → (RingCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
21	1, 20	eqtrid 2776	1 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3438   ∩ cin 3904   × cxp 5621   ↾ cres 5625  ‘cfv 6486  (class class class)co 7353   ↾_cat cresc 17733  ExtStrCatcestrc 18046  Ringcrg 20136   RingHom crh 20372  RingCatcringc 20548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  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 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-res 5635  df-iota 6442  df-fun 6488  df-fv 6494  df-ov 7356  df-ringc 20549

This theorem is referenced by:  ringcbas  20553  ringchomfval  20554  ringccofval  20558  dfringc2  20560  ringccat  20566  ringcid  20567  funcringcsetc  20577