Theorem ringcval 44273

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 44270	. . 3 ⊢ RingCat = (𝑢 ∈ V ↦ ((ExtStrCat‘𝑢) ↾cat ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)))))
3		fveq2 6664	. . . . 5 ⊢ (𝑢 = 𝑈 → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
4	3	adantl 484	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → (ExtStrCat‘𝑢) = (ExtStrCat‘𝑈))
5		ineq1 4180	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → (𝑢 ∩ Ring) = (𝑈 ∩ Ring))
6	5	sqxpeqd 5581	. . . . . . 7 ⊢ (𝑢 = 𝑈 → ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)) = ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
7		ringcval.b	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = (𝑈 ∩ Ring))
8	7	sqxpeqd 5581	. . . . . . . 8 ⊢ (𝜑 → (𝐵 × 𝐵) = ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)))
9	8	eqcomd 2827	. . . . . . 7 ⊢ (𝜑 → ((𝑈 ∩ Ring) × (𝑈 ∩ Ring)) = (𝐵 × 𝐵))
10	6, 9	sylan9eqr 2878	. . . . . 6 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ((𝑢 ∩ Ring) × (𝑢 ∩ Ring)) = (𝐵 × 𝐵))
11	10	reseq2d 5847	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))) = ( RingHom ↾ (𝐵 × 𝐵)))
12		ringcval.h	. . . . . . 7 ⊢ (𝜑 → 𝐻 = ( RingHom ↾ (𝐵 × 𝐵)))
13	12	eqcomd 2827	. . . . . 6 ⊢ (𝜑 → ( RingHom ↾ (𝐵 × 𝐵)) = 𝐻)
14	13	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RingHom ↾ (𝐵 × 𝐵)) = 𝐻)
15	11, 14	eqtrd 2856	. . . 4 ⊢ ((𝜑 ∧ 𝑢 = 𝑈) → ( RingHom ↾ ((𝑢 ∩ Ring) × (𝑢 ∩ Ring))) = 𝐻)
16	4, 15	oveq12d 7168	. . 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 3514	. . 3 ⊢ (𝜑 → 𝑈 ∈ V)
19		ovexd 7185	. . 3 ⊢ (𝜑 → ((ExtStrCat‘𝑈) ↾cat 𝐻) ∈ V)
20	2, 16, 18, 19	fvmptd2 6770	. 2 ⊢ (𝜑 → (RingCat‘𝑈) = ((ExtStrCat‘𝑈) ↾cat 𝐻))
21	1, 20	syl5eq 2868	1 ⊢ (𝜑 → 𝐶 = ((ExtStrCat‘𝑈) ↾cat 𝐻))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Vcvv 3494   ∩ cin 3934   × cxp 5547   ↾ cres 5551  ‘cfv 6349  (class class class)co 7150   ↾_cat cresc 17072  ExtStrCatcestrc 17366  Ringcrg 19291   RingHom crh 19458  RingCatcringc 44268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-res 5561  df-iota 6308  df-fun 6351  df-fv 6357  df-ov 7153  df-ringc 44270

This theorem is referenced by:  ringcbas  44276  ringchomfval  44277  ringccofval  44281  dfringc2  44283  ringccat  44289  ringcid  44290  funcringcsetc  44300